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Key Words
What
You’ll Learn
To recognize quadratic
relations in tables, graphs,
and equations, and to use
these relationships to model
and solve problems
• parabola
• axis of symmetry
And Why
Quadratic relations are used
in many fields; for example, in
physics to model the path of a
projectile, in business to model
projected revenue, in environmental
science to model ecological systems,
and in engineering to model the cross
section of bridges and satellite dishes.
• vertex
• transformation
• translation
• reflection
• vertical stretch
• vertical compression
• standard form
• vertex form
• factored form
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CHAPTER
3
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Activate Prior Knowledge
Operations with Integers
Prior Knowledge for 3.1
Integers are any of the numbers . . . , 4, 3, 2, 1, 0, 1, 2, 3, 4, . . . .
They are ordered from left to right on a number line. So, 4 < 2.
−7 −6 −5 −4 −3 −2 −1
0
1
2
3
4
5
6
7
Positive integers
are often written
without the
symbol.
You can use a number line to add or subtract integers.
The product, or quotient of two integers with the same sign is positive,
and with opposite signs is negative.
Example
Evaluate.
a) 1 (3)
b)
4 3
c)
4 (5)
d)
15 (3)
Solution
−3
Start at 1. Draw an arrow to represent add 3.
−3 −2 −1 0 1 2
The arrow ends at 2. So, 1 (3) 2
b) Start at 4. Draw an arrow to represent
–3
subtract 3.
−8 −7 −6 –5 –4 –3
The arrow ends at 7. So, 4 3 7
c) Since 4 and –5 have opposite signs, their
product is negative. So, 4 (5) –20
d) Since 15 and 3 have the same sign, their quotient is positive. So, 15 (3) 5
a)
✓ Check
1. Evaluate.
6 (1)
e) 11 3
a)
5
f) (5) 8
b) 7
c) 7 5
g) 32 (–2)
d) 2 (3)
h) 27
2. Determine all pairs of integers with each product.
a)
8
b)
12
c) 8
3. Evaluate (4)(2)(5) and (9)(1)(7).
How can you determine the sign of each product without multiplying?
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d) 12
(9)
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Expanding and Factoring
Prior Knowledge for 3.1
To expand an expression, use the distributive property.
To factor an expression, determine the greatest common factor.
Example
a)
Expand 4x(3x 6).
b)
Factor 6x2 18.
Solution
a)
4x(3x 6)
4x(3x 6)
4x(3x) 4x(6)
12x2 24x
b) 6x2 18
6(x2) 6(3)
6(x2 3)
Use the distributive property. Multiply each term by 4x.
Multiply the numbers and variables.
Determine the signs.
The greatest common
Write each term as a product of 6
factor is 6.
and another factor.
Write the common factor outside the brackets.
Expand to verify the solution: 6(x2 3) 6(x2) 6(3) 6x2 18
✓ Check
1. Expand.
3(x 7)
d) –(2x2 8)
a)
7x(8x 9)
e) 2x(x 5)
b)
–3x(9 4x)
f) –5(6x2 1)
c)
2. Identify the greatest common factor in each expression.
a)
5x 15
b)
12x2 4
c)
12x2 16x 20
b)
12x2 4
c)
12x2 16x 20
56x 63
e) 9x 36
c)
3. Factor each expression.
a)
5x 15
4. Factor.
3x 21
d) 8x2 10
a)
b)
–27 12x
f) 2x2 4x 6
5. Compare your results in parts a to c of question 1 with results in parts a to c of question 4.
How can you relate expanding and factoring expressions?
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Graphing Relations on a Coordinate Grid
Prior Knowledge for 3.1
A point on a coordinate grid is described by its coordinates (x, y).
The x-coordinate represents the horizontal distance and direction from the origin.
The y-coordinate represents the vertical distance and direction from the origin.
To graph a relation, you can create a table of values for x and y, and then plot the points.
Example
Create a table of values for y 2x2 4 using values of x from 3 to 3.
b) Graph the relation on a coordinate grid.
a)
Solution
a)
Substitute each value of x.
Calculate y.
b)
Plot the points and join them
with a smooth curve.
x
y
(x, y)
16
3
2(3)2 4 14
(3, 14)
12
2
2(2)2 4 4
(2, 4)
1
2
(1, 2)
0
4
(0, 4)
1
2
(1, 2)
2
4
(2, 4)
3
14
(3, 14)
y
8
4
x
–4
–2 0
–4
2
4
✓ Check
1. Create a table of values, then graph each relation.
y 2x
d) y 2x2
a)
yx1
e) y (x 1)2
b)
y x 3
f) y x2 3
c)
2. Some graphs you drew in question 1 are linear and some are non-linear.
Explain how you can predict if a relation has a linear graph.
3. The cost of topsoil is $40 for delivery and $18 per cubic yard of soil.
Create a table of values to show the cost of up to 5 cubic yards of soil.
b) Graph the relation between the cost and volume of soil ordered.
a)
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Translations
Prior Knowledge for 3.3
When a shape is translated, it is moved along a straight line.
The orientation of a translation image is the same as the orientation of the original shape.
A translation image is congruent to the original shape.
A translation is also called a slide.
Example
Plot these points on a coordinate grid: A(5, 3), B(3, 1), C(6, 4), and D(7, 2)
Join the points to form quadrilateral ABCD.
Translate quadrilateral ABCD 7 units right and 4 units up. Describe the steps.
Label the coordinates of the vertices of the image.
y
Solution
Translate each point 7 squares right and
4 squares up.
Draw and label the image points
A(2, 7), B(4, 5), C(1, 0), and D(0, 6)
Join A, B, C, and D to form the image
quadrilateral.
8
A′(2, 7)
D′(0, 6)
6
B′(4, 5)
A(–5, 3)
D(–7, 2)
–8
4
2
–6
–4
A is read “A prime.”
C(–6, –4)
B(–3, 1)
–2 0
2
4
–2 C′(1, 0)
x
6
–4
–6
✓ Check
1. On a coordinate grid, draw the quadrilateral with vertices A(3, 3), B(4, 5), C(5, 1),
and D(2, 3).
a) Translate quadrilateral ABCD 5 units left and 3 units down.
b) Write the coordinates of the vertices of the image quadrilateral.
Compare the coordinates with those of the original quadrilateral.
What do you notice?
c) Determine the coordinates of the image of point E(3, 2) when it is translated 5 units
left and 3 units down. How did you determine the coordinates?
2. How do your diagrams show that the translation image and original shape are congruent?
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Reflections
Prior Knowledge for 3.4
When a shape is reflected in a line, its image is the same distance from the reflection line.
The orientation of the reflection image is opposite the orientation of the original shape.
A reflection image is congruent to the original shape.
A reflection is also called a flip.
Example
Plot these points on a coordinate grid: A(1, 5), B(2, 1), and C(2, 3)
Join the points to form ABC.
Reflect ABC in the x-axis. Describe the steps.
Label the coordinates of the vertices of the image.
y
Solution
The line of reflection is the x-axis.
Point A is 5 units above the x-axis.
Draw its reflection image 5 units below the
x-axis.
Label it A(1, 5).
Repeat the reflection for points B and C.
Label them B(2, 1) and C(2, 3).
Join A, B, and C to form the image triangle.
6 A(1, 5)
4
C(–2, 3)
2
B(2, 1)
–6
–4
–2 0
–2
C′(–2, –3)
–4
x
2
4
6
B′(2, –1)
–6 A′(1, –5)
✓ Check
1. On a coordinate grid, draw the quadrilateral with vertices P(3, 3), Q(4, 5), R(5, 1),
and S(2, 3).
a) Reflect quadrilateral PQRS in the x-axis.
b) Write the coordinates of the vertices of the image quadrilateral.
Compare the coordinates of the image with those of the original quadrilateral.
What do you notice?
c) Determine the coordinates of the image when point T(3, 2) is reflected in the x-axis.
How did you determine the coordinates?
2. How do your diagrams show that the reflection image and original shape are
congruent?
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Modelling Quadratic Relations
During a show at an aquarium, a dolphin jumps out of the water
to go through a ring held by the trainer. The dolphin’s height above
the water during the jump can be represented by a mathematical
model called a quadratic relation.
Investigate
Maximizing Area
Work with a partner.
You will need grid paper.
A rectangular swimming area is enclosed with
16 m of rope. One side is along the shore, so
the rope is only needed on the other three sides.
This is one possible swimming area.
➢ Sketch and label as many swimming areas
as you can. Label each swimming area
with its length and area.
➢ Organize your results in a table.
Swimming area (m2)
Chapter 03
Length of side perpendicular
to shore (m)
10 m
3m
3m
The 3-m sides are
perpendicular to the
shore.
Length of side perpendicular
to shore (m)
Length of
other side (m)
Swimming
area (m2)
3
10
3 10 30
➢ Plot Swimming area against Length of side perpendicular
to shore. Describe the graph.
➢ Which length perpendicular to the shore results in the greatest
swimming area? What is this area?
Reflect
➢ Should you join the points on the graph? Explain.
➢ Which point on the graph represents the greatest swimming
area? What do the coordinates of this point represent?
➢ How do the table and graph show that this is the greatest area?
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Connect the Ideas
Ciara has 12 m of fencing to build a rectangular dog pen.
One side of the fence is beside a creek, but it still needs to be fenced.
Here are some possible dimensions and areas of the dog pen.
1m
2m
3m
5 m 5 m2
4m
4m
8 m2
3m
9 m2
5m
2m
8 m2
1m
5 m2
We can represent the relationship between the length of the side
beside the creek and the area of the dog pen in different ways.
Use an equation
In a quadratic
equation, the highest
power of x is x2 or
x-squared. The word
quadratic comes from
the Latin word
quadrare, which means
“to square.”
Use a table
The length of fencing along the four sides of the dog pen is 12 m.
So, 6 m of fencing must be used along any two adjacent sides,
as the above diagrams show.
Let x metres represent the length of the
side beside the creek. Then, the length
of the adjacent side is 6 x metres.
The area of the pen, A square metres,
is the product of these dimensions:
A x(6 x)
or
A 6x x2
We can list the possible lengths
in a table.
As the length of the side
beside the creek increases, the
area increases to a maximum
and then decreases.
The maximum area, 9 m2,
happens when the dog pen is
a square with side length 3 m.
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x
6–x
A = 6x – x 2
Length of
side beside
creek (m)
Length of
adjacent
side (m)
Area of
dog pen
(m2)
0
6
060
1
5
155
2
4
248
3
3
339
4
2
428
5
1
515
6
0
600
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Use a graph
Since it is possible to
have lengths that are
not whole numbers,
we join the points with
a smooth curve.
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The highest point on the graph
represents the dog pen with the
greatest area. The coordinates of
the point, (3, 9), show that the
maximum area, 9 m2, is when
the side length is 3 m.
Area of Dog Pen
The line x 3 is a line of symmetry.
Every point on the graph to the left
of the line has a matching point
to the right of the line.
Area of pen (m2)
Chapter 03
10
9
8
7
6
5
4
3
2
1
0
(3, 9)
x=3
2
1
3
4
5
6
Length of side beside creek (m)
The relationship between the length of the side beside the creek and the
area of the dog pen is an example of a quadratic relation.
Properties of
quadratic relations
Equation
A quadratic relation can be written in standard form as
y ax2 bx c, where a, b, and c are constants and a 0.
Table
The second differences of a quadratic relation are constant.
x
y
0
0
1
5
2
8
3
9
4
8
5
5
6
0
First differences
505
853
981
8 9 1
5 8 3
0 5 5
Second
differences
3 5 2
1 3 2
1 1 2
3 (1) 2
5 (3) 2
Graph
The graph of a quadratic relation is called
a parabola. Every parabola has a line of
symmetry called the axis of symmetry.
The point where the parabola intersects
the axis of symmetry is the vertex.
The graph of a quadratic relation in standard
form can open up or open down.
The direction of opening determines
whether the vertex is a maximum or a
minimum point.
axis of
symmetry
vertex is the
minimum point
vertex is the
maximum point
axis of
symmetry
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Practice
1. Inez has 16 m of edging to enclose a rectangular garden.
One side of the garden is beside a patio, but it needs edging.
a) Sketch and label some possible rectangles.
One possible rectangle is shown at the right.
b) Record your results in a table.
Length of side
beside the patio (m)
Length of
adjacent side (m)
Area of
garden (m2)
5
3
5 3 15
3m
3m
5m
Graph the data from the first and third columns of the table.
d) What are the dimensions and area of the largest garden?
How can you find the answer from the table? From the graph?
c)
2. A seagull drops a clam from the air. This diagram shows
the height of the clam above the ground every 0.5 s.
a) Copy and complete the table.
Time (s)
Height (m)
0
20
Use second differences to show that the relationship
between height and time is quadratic.
c) Plot Height against Time.
d) From what height did the seagull drop the clam?
Explain how to use the table and the graph
to determine the answer.
e) When does the clam hit the ground?
Explain how to use the table and the graph to determine the answer.
20.00 m
18.75 m
15.00 m
b)
3. A dolphin leaps out of the water.
The graph shows the path of the dolphin
during the leap.
a) What is the maximum height the dolphin
reaches?
b) How far did the dolphin jump?
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8.75 m
0m
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In business, the income from the sale of an item increases as the price of the item increases.
However, eventually the income may decrease because a higher cost can result in fewer sales.
Example
Jackie owns a clothing boutique. Last year, she sold 1200 T-shirts for $10 each.
Market research suggests that for every $5 increase in price, 200 fewer T-shirts
will be sold.
Price
Number of
Income
a) Copy and complete the table
T-shirts sold
until the price is $40.
$10
1200
$12 000
b) Graph the data. Plot Income against Price.
$15
1000
c) Which price results in the maximum income?
Solution
a) The income is the product of the price
b)
and the number of T-shirts sold.
Income from Sales of T-Shirts
Price
Number of
T-shirts sold
Income
$10
1200
$12 000
$15
1000
$15 000
$20
800
$16 000
$25
600
$15 000
$30
400
$12 000
$35
200
$7000
$40
0
$0
Income
(thousands of dollars)
20
18
16
14
12
10
8
6
4
2
0
10
15
20 25 30
Price ($)
35
c) A price of $20 results in the maximum revenue.
4. A craft store sold 800 ornaments for $2 each. A survey suggests
Price
Number of
ornaments sold
Income
$2
800
$1600
$3
700
that every $1 increase in price will reduce sales by 100.
a) Copy and complete the table until no ornaments are sold.
b) Graph the data.
c) Which price results in the maximum income?
5. A spark flew out of a star burst at a fireworks display.
Height of Fireworks Spark
This graph shows its path.
a) Estimate the height of the spark when it left the star burst.
b) What was the height of the spark after it travelled
a horizontal distance of 10 m?
c) What was the maximum height of the spark?
d) How far had the spark travelled horizontally
when it hit the ground?
Height (m)
Chapter 03
80
70
60
50
40
30
20
10
0
10 20 30 40 50
Horizontal distance (m)
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6. A diver jumps into the water from a spring board.
This graph shows her height above the surface of the water
during the dive. Four points are labelled on the graph.
Use the points to describe the motion of the diver.
7. Assessment Focus Each face of a cube is a square.
The surface area is the sum of the areas of the 6 faces.
Height of Diver
5
Height (m)
Chapter 03
4
a)
b)
c)
2 cm
(1.2, 3)
3 (0, 3)
2
1
0
1 cm
(0.6, 4.8)
(1.58, 0)
1
Time (s)
3 cm
Copy the table. Continue the table until the side length is 6 cm.
Side length
(cm)
Area of each
face (cm2)
Surface area
of cube (cm2)
1
12 1
616
2
22
4
6 4 24
Plot Surface Area against Side Length.
Is the relationship between the side length of a cube and its surface area quadratic?
Use the table and graph to justify your answer.
8. Take It Further A landscape architect wants to
make a rectangular garden with an area of 36 m2.
a) Use a table to list some possible dimensions
and perimeters of the garden.
b) Which dimensions give the minimum perimeter?
c) Show that the relationship between the width and
perimeter is not quadratic.
Do you find it helpful to model a quadratic relation with both a table and a graph?
Why or why not?
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3.2
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Graphing Quadratic Relations
Alison is planning a fireworks show
for Canada Day.
The show will be put to music,
so the fireworks must go off at specific times.
Alison uses a quadratic relation
to model the height of the fireworks
over time.
In general, the height, h metres, of an object t seconds
after it has been dropped or projected vertically upward
is given by the formula h = 4.9t2 + vt + s,
where v is the initial speed in metres per second
and s is the initial height.
Inquire
Analysing Quadratic Relations
You will need a TI-83 or TI-84 or similar graphing calculator.
The equation h 4.9t2 39.2t 2 represents the height, h metres,
of a fireworks rocket t seconds after it is launched.
We can graph the equation and use the graph to analyse the motion of the rocket.
1. Graphing the equation
➢ Enter and graph the equation.
Press o to access the
equation editor. Enter the
equation for the rocket’s
height.
Press · 4.9 „ ¡
‡39.2 „ ‡ 2.
The keystrokes are for a TI-83 or TI-84
graphing calculator. If you use a different
calculator, check the User’s Manual.
To clear any equations in the
list, use the down arrow key
to move to the equation.
Press M ¸. Then,
move the cursor back to
Y1.
3.2 Graphing Quadratic Relations
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Press s. Graph the
equation in a standard
window.
Press q 6.
Only part of the graph
appears in the window.
➢ Edit the viewing window.
Press p.
Change the window
settings as shown.
Since height and time are
only meaningful when they
are positive, we are only
interested in the part of the
graph that appears in the
first quadrant.
Press s.
Sketch the graph.
Label the axes and the
curve.
2. Determining the coordinates of the key points on the parabola
➢ Determine the y-intercept.
Use the TRACE feature.
At the y-intercept, x 0.
Press r 0 ¸Å.
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The y-intercept is the y-coordinate of the
point where the graph crosses the y-axis.
Record the y-intercept on
your sketch.
What does this value tell you
about the rocket?
How could you have found
the y-intercept from the
equation
y 4.9x2 39.2x 2?
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➢ Determine the coordinates of the vertex.
Use the MAXIMUM feature
in the CALC menu.
Press 2 r 4.
To answer the prompt “Left
bound?”, move the cursor
to the left of the maximum
and press ¸. The
prompt changes to “Right
bound?”. Move the cursor
to the right of the
maximum and press ¸.
At the prompt“Guess?”,
press ¸.
➢ Determine the x-intercept.
Use the ZERO feature
in the CALC menu.
Press 2 r 2.
Record the coordinates of
the maximum on your
sketch.
What do these coordinates
tell you about the rocket?
The x-intercept is the x-coordinate of a
point where the graph crosses the x-axis.
Another name for an x-intercept is a zero.
Record the x-intercept on
your sketch.
What does this value tell
you about the rocket?
To answer the prompt “Left
bound?”, move the cursor
to the left of the x-intercept
and press ¸. The
prompt changes to “Right
bound?”.
Move the cursor to the
right of the x-intercept.
Press ¸.
At the prompt “Guess?”,
press ¸.
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3. Using the graph to answer questions about the motion of the rocket
➢ What is the height of the rocket after 2 s?
Press r 2 ¸.
Record the coordinates
of the point on the sketch.
Write a statement about the
height of the rocket after 2 s.
➢ At what times is the rocket 50 m above the ground?
➢ Draw a horizontal line at y 50.
Press o to access the
equation editor.
Move the cursor to Y2 .
Press 50 s.
➢ Determine the coordinates of the points of intersection of the two equations.
Use the INTERSECT
feature in the CALC menu.
Press 2 r 5.
The prompt “First curve?”
and the equation in Y1 is
displayed. Move the cursor
close to the first point of
intersection.
Press ¸. The prompt
changes to “Second curve?”
and the equation in Y2 is
displayed.
Press ¸.
At the new prompt
“Guess?”, press ¸.
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Draw the line y 50 on your
sketch.
Record the coordinates of the
point of intersection.
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Move the cursor near
the second point of
intersection. Repeat the
previous step to determine
the coordinates of
this point.
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Record the coordinates of the
second point of intersection
on your sketch.
Write a statement about the
times at which the height of
the rocket is 50 m.
Practice
Use a graphing calculator.
Include a labelled sketch of a graphing calculator screen to support each answer.
1. Teresa heads the ball in soccer.
The path of the ball is described by the equation h 0.2d 2 0.8d 2,
where h metres is the height of the ball
and d metres is its distance measured horizontally from where Teresa hits it.
a) Graph the equation.
Why do we only use non-negative values of h and d?
b) What is the height of the ball when Teresa hits it?
c) What is the maximum height of the ball?
How far has the ball travelled horizontally when it reaches its maximum height?
d) How far has the ball travelled from its original position when it hits the ground?
What assumptions are you making?
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2. A company manufactures and sells computer games.
The daily profit, P dollars, is given by the equation
P 10x2 750x 9000,
where x dollars is the price of each game.
a) Graph the equation.
Are negative values of x meaningful in this situation? Explain.
b) For what range of prices does the company make a profit?
Explain.
c) What price gives the maximum profit? What is this profit?
The point where a
company neither
makes nor loses
money is called a
break-even point.
3. A quarterback throws a football.
The height, h metres, of the ball is given by the equation
h 5t2 20t 2,
where t is the time in seconds after the ball is thrown.
a) Graph the equation.
Why do we use only non-negative
values of h and t?
b) What is the height of the ball
1 s after it is thrown?
c) What is the maximum height of
the ball?
How long does it take for the ball
to reach the maximum height?
d) For how long is the ball more than
10 m above the ground?
Reflect
➢ The intercepts and vertex are key points on the graph of a
parabola. What do these points represent in real-world
situations? Explain using two examples.
➢ Which key points can be identified by looking at the equation in
standard form? Explain using an example.
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3.3
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The Role of h and k in
y ⴝ (x ⴚ h)2 ⴙ k
4
y
y = x2
3
2
This is the graph of the quadratic relation y x2.
The graphs of all other quadratic relations are transformations of
this parabola.
Investigate
1
–2
–1 0
–1
1
x
2
Graphing y ⴝ x 2 ⴙ k and y ⴝ (x ⴚ h)2
Work with a partner. You will need a TI-83 or TI-84 graphing calculator.
Part A: Comparing y ⴝ x2 and y ⴝ x2 ⴙ k
➢ Each partner should graph one of these sets of three equations
on the same screen.
y x2
y x2
2
yx 2
y x2 2
y x2 4
y x2 4
➢ Sketch each screen. Label each graph with its equation.
Record the coordinates of the vertex of each graph.
➢ Describe how each graph is related to the graph of y x2.
➢ Predict how the graphs of y x2 3 and y x2 5 compare
with the graph of y x2. Justify your predictions.
Check your predictions on a calculator.
Part B: Comparing y ⴝ x2 and y ⴝ (x ⴚ h)2
➢ Repeat the procedure in Part A with these sets of equations:
y x2
y x2
y (x 2)2
y (x 2)2
2
y (x 4)
y (x 4)2
➢ Predict how the graphs of y (x 3)2 and y (x 5)2 compare
with the graph of y x2. Justify your predictions and check them.
Reflect
➢ How are graphs of y x2 k related to the graph of y x2?
How do your sketches show this?
➢ How are graphs of y (x h)2 related to the graph of y x2?
How do your sketches show this?
3.3 The Role of h and k in y (x h)2 k
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A translation is an
example of a
transformation.
Here are the graphs of y x2,
y x2 3, and y x2 5.
12
10
Every point on the graph
of y x2 3 is 3 units above a point
on the graph of y x2.
So, the graph of y x2 3 is congruent
to the graph of y x2 and translated
3 units up.
8
y
6
5
The graph of
y x2 k
y=x2+
3
y=x2
Connect the Ideas
4
y=x2–
Chapter 03
2
x
–4
–2 0
–2
Similarly, every point on the graph of
y x2 5 is 5 units below
a point on the graph of y x2.
So, the graph of y x2 5
is congruent to the graph of y x2 and
translated 5 units down.
2
4
–4
–6
The role of k in the graph of y ⴝ x2 ⴙ k
The graph of y x2 k is a vertical translation of the graph y x2.
When k is positive, the graph is translated up.
When k is negative, the graph is translated down.
The coordinates of the vertex of the graph is (0, k).
The axis of symmetry is x 0.
The graph of
y (x h)2
Here are the graphs of y x2,
y (x 3)2, and y (x 5)2.
The graph of y (x 3)2 is
congruent to the graph of y x2
and translated 3 units right.
So, every point on the graph
of y (x 3)2 is 3 units to
the right of a point on the graph
of y x2.
10
y = (x + 5)2
8
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CHAPTER 3: Quadratic Relations
y = (x – 3)2
4
2
x
–8
–6
–4
–2 0
–2
The graph of y (x 5)2 is congruent to
the graph of y x2 and translated 5 units left.
So, every point on the graph of y (x 5)2
is 5 units to the left of a point on the graph of y x2.
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y = x2
y
2
4
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Notice that we write
y (x h)2 with
the sign . Since
(x [5])2 (x 5)2,
y (x 5)2 is a
translation to the left.
The step pattern
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The role of h in the graph of y ⴝ (x ⴚ h)2
The graph of y (x h)2 is a horizontal translation of
the graph y x2.
When h is positive, the graph is translated right.
When h is negative, the graph is translated left.
The coordinates of the vertex of the graph is (h, 0).
The axis of symmetry is x h.
There is a pattern in the first differences of the relation y x2.
On the graph of y x2, the pattern looks like steps.
So, it is called the step pattern.
From the vertex, go across 1 and up 1,
across 1 and up 3,
across 1 and up 5,
across 1 and up 7, and so on.
In the table, the first differences show the step pattern.
y
y ⴝ x2
x
y
4
16
3
9
2
4
1
1
0
0
1
1
2
4
3
9
4
16
First
differences
16
y = x2
14
7
7
3
10
1
8
1
7
12
5
5
5
6
3
4
5
3
7
–4
–3
–2
3
2
1
–1 0
x
1
1
2
3
4
Since the parabolas y (x h)2 and y x2 k are congruent
to the parabola y x2, they have the same step pattern.
3.3 The Role of h and k in y (x h)2 k
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Practice
1. Describe how to transform the graph of y x2 to produce the graph
of each relation.
a) y (x 4)2
b)
y x2 4
c)
y (x 4)2
d)
y x2 4
2. In each case the parabola y x2 is transformed as described.
Write the equation of the transformed parabola in the form y x2 k or
y (x h)2.
a) The parabola is translated 6 units up.
b) The parabola is translated 2 units right.
c) The parabola is translated 1 unit down.
d) The parabola is translated 3 units left.
3. a) Determine the coordinates of the vertex of each parabola in question 2.
b)
Choose one parabola in question 2.
Describe your strategy for writing the equation.
4. The equations of some quadratic relations are given below.
How are the graphs the same? How are the graphs different?
y (x 12)2
y x2 12
y (x 12)2
y x2 12
5. A parabola with equation y (x h)2 k has been translated
horizontally and vertically. Describe how to transform
the graph of y x2 to result in the graph of each relation.
Use a graphing calculator to check your answers.
a) y (x 3)2 4
b) y (x 6)2 4
c) y (x 3)2 4
d) y (x 1)2 3
We can plot the graph of a parabola efficiently using the step pattern.
Example
Sketch the graph of y (x 2)2 1
without making a table of values
or using a graphing calculator.
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Solution
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The graph of y (x 2)2 1 is
congruent to the graph of y x2
and translated 2 units left and 1 unit
down.
So, the coordinates of the vertex are at
(2, 1). Plot this point.
From (2, 1), move 1 right and 1 up,
1 right and 3 up, 1 right and 5 up.
Return to the vertex and repeat to
the left. From (2, 1), move 1 left
and 1 up, 1 left and 3 up, 1 left
and 5 up. Draw a smooth curve
through the points.
10
y = (x + 2)2 – 1
y
8
6
5
5
4
2
3
3
x
–4 1–3
–5
–2
–11 0
–2
1
2
6. Sketch a graph of each parabola without making a table of values
or using a graphing calculator.
a) y (x 5)2
b) y x2 4
c)
y (x 4)2 7
7. Explain what you learned from the Guided Example,
and how you used this to complete question 6.
8. Assessment Focus Describe what happens to the graph of each quadratic relation.
y (x h)2 2, as h varies
b) y (x 3)2 k, as k varies
a)
y
10
9. Take It Further A light is shining on a wall.
The shadow from the lampshade makes the light
form a parabola.
Determine an equation of the parabola.
Use the grid in the figure for coordinates.
5
2 1
0
1
2
3
4 x
Explain how to change the equation of y x2 to translate
its graph up, down, left, or right.
Include examples and graphs in your explanation.
3.3 The Role of h and k in y (x h)2 k
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Connecting Mathematical Ideas
It is easier to understand and remember new concepts in math
if you can connect them to concepts you already know.
Graphic organizers such as Frayer models and concept maps
are useful tools for connecting mathematical ideas.
Frayer model
Definition
A Frayer model can help you understand the
meaning of a term or a concept.
Talk with a partner about this Frayer model.
➢ What word should replace the question
mark?
➢ Is any of the information new to you?
Explain.
➢ Would you add anything to give a more
complete picture of the word? Explain.
A polygon with 4 sides.
Facts/
Characteristics
The sum of the interior
angles is 360° .
The sum of the exterior
angles is 360° .
?
Examples
Non-Examples
Trapezoid, parallelogram,
rectangle, square
Concept map
A concept map can be used to provide a
visual summary of the connections between
different terms or concepts.
Talk with a partner about this concept map.
➢ What would you use as a title for this
concept map?
➢ What characteristics do all rectangles,
rhombi, and squares have in common?
➢ Are all squares rectangles?
Why or why not?
Quadrilateral
with 1 pair of
parallel sides
with 2 pairs of
parallel sides
Trapezoid
Parallelogram
with 4
right angles
with 4
equal sides
Rectangle
with equal
sides
Rhombus
Square
with 4
right angles
• Create a Frayer model or concept map for the term quadratic
relation. Start with what you have learned.
• Add to your model or map as you progress through this chapter.
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3.4
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The Role of a in y ⴝ ax2
The equation y 5x2 can be used to estimate the distance, y metres,
that an object falls x seconds after it is dropped.
The graph of y 5x2 is a transformation of the graph of y x2.
Investigate
Graphing y ⴝ ax 2
Work with a partner. You will need a TI-83 or TI-84 graphing calculator.
Part A: The graph of y ⴝ ax2 for positive values of a
➢ Each partner should graph one of these sets of three equations on
the same screen.
y x2
y x2
y 2x2
y 0.3x2
y 3x2
y 0.5x2
➢ Sketch each screen. Label each graph with its equation.
Describe how each graph is related to the graph of y x2.
➢ Predict how the graphs of y 6x2 and y 0.6x2
compare with the graph of y x2. Justify your predictions.
Check your predictions on a calculator.
Part B: The graph of y ⴝ ax2 for negative values of a
➢ Repeat the procedure in Part A with these sets of equations.
y x2
y x2
y 2x2
y 0.3x2
2
y 3x
y 0.5x2
➢ Predict how the graphs of y 6x2 and y 0.6x2 compare
with the graph of y x2.
Justify your predictions and check them.
Reflect
➢ How does the graph of y ax2 change as the value of a changes?
How do your sketches show this?
3.4 The Role of a in y ax2
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Connect the Ideas
The graph of
y ⴝ ax2, a > 0
Here are the graphs of y x2, y 2x2,
and y 0.5x2.
Each graph opens up and has its vertex
at the origin.
y
12
y = 2x 2
10
8
x
6
2
y=
y=
2
x
0.5
The graph of y 2x2 is narrower than the
graph of y x2. For any x-coordinate,
the point on the graph of y 2x2 has
a y-coordinate that is 2 times the
y-coordinate of the point on the graph of
y x2. We say that the graph of y 2x2 is
transformed from the graph of y x2 by a
vertical stretch of factor 2.
14
4
2
x
–3
–2
–1 0
1
2
3
2
x
3
The graph of y 0.5x2 is wider than the graph of y x2. For any
x-coordinate, the point on the graph of y 0.5x2 has a y-coordinate
that is one-half the y-coordinate of the point on the graph of y x2.
We say that the graph of y 0.5x2 is transformed from the graph of
y x2 by a vertical compression of factor 0.5.
This relationship is also true for the graphs
of y 2x2 and y 2x2, and the graphs of
y 0.5x2 and y 0.5x2.
A reflection is a
transformation.
–2
y=
–0
.5
x
2
–3
–1 0
–2
–6
–8
–10
–12
–14
The role of a in the graph of y ⴝ ax2
The graph of y ax2 is a vertical stretch or compression of the
graph of y x2. If a < 0, the graph is also reflected in the x-axis.
The graph is vertically stretched when a > 1 or a < –1.
The graph is vertically compressed when 1 < a < 1, a 0.
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1
–4
x2
The graph of y x2 is congruent to the
graph of y x2 and is reflected in the
x-axis. For any x-coordinate, the point on
the graph of y x2 has a y-coordinate
that is the negative of the y-coordinate
of the point on the graph of y x2.
y
y=–
Here are the graphs of y x2, y 2x2,
and y 0.5x2. Each graph opens down
and has its vertex at the origin.
x2
The graph of
y ⴝ ax2, a < 0
y = –2
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Practice
1. Describe how to transform the graph of y x2 to produce the graph
of each relation.
a) y 8x2
b)
y 8x2
c)
y 5x2
d)
y 5x2
2. In each case, the parabola y x2 is transformed as described.
Write the equation of the transformed parabola in the form y ax2.
a) The parabola is stretched vertically by a factor of 5.
b) The parabola is reflected in the x-axis.
c) The parabola is compressed vertically by a factor of 0.25.
d) The parabola is stretched vertically by a factor of 2,
then reflected in the x-axis.
3. For each relation:
Describe the transformations needed to change the graph of y x2
to the graph of the relation.
b) Sketch the graph of y x2 and the relation on the same grid.
c) Use a graphing calculator to check your graphs in part b.
i) y 4x2
ii) y 6x2
iii) y 0.25x2
a)
We can use a step pattern to graph y ax2 without first graphing y x2.
Example
Use a step pattern to graph y 2x2.
Solution
The step in the
x-direction is still 1.
Only the y-step
changes.
For the graph of y x2, the step pattern
for the next y-values is 1, 3, 5, and so on.
For any x-coordinate, the y-coordinate
of the point on the graph of y 2x2 is
2 times the y-coordinate of the point
on the graph of y x2.
So, in the graph of y 2x2, the step
pattern for the next y-values is 1 2 2,
3 2 6, 5 2 10, and so on.
Plot the vertex (0, 0).
Go 1 right and 2 up, 1 right and 6 up,
1 right and 10 up, and so on.
Then go 1 left and 2 up, 1 left and 6 up,
1 left and 10 up, and so on.
Draw a smooth curve through the points.
30
y
y = 2x 2
20
10
10
10
6
–4
–3
2
–2 –1 0
6
2
1
3.4 The Role of a in y ax2
2
3
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x
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4. Use a step pattern to graph each relation.
a)
y 4x2
b)
y 0.5x2
c)
y 3x2
d)
1.5x2
5. Assessment Focus Without graphing, tell whether the graph for each equation
opens up or down and whether it is narrower or wider than the graph of y x2.
Justify your answers.
Then use a graphing calculator to check. Sketch each screen.
a) y 3x2
b) y 2x2
c) y 0.5x2
6. Sabine says that the parabola whose equation is of the form y ax2
always has its vertex at the origin. Do you agree? Explain.
7. Take It Further The rainbow in the picture has the shape of a parabola.
Use the axes and choose a suitable scale to determine its equation. Explain your answer.
Use a graphing calculator to check.
y
0
The graph of y ax2 is a parabola.
What happens to the parabola as a increases from 0 to 1?
What happens to the parabola as a decreases from 0 to 1?
Include graphs in your explanation.
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3.5
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The Vertex Form of a Quadratic Relation
When a player shoots a free-throw in basketball,
the path of the ball is a parabola.
The path can be modelled by an equation of the form
y a(x h)2 k.
Graphing y ⴝ a(x ⴚ h)2 ⴙ k
Investigate
You can use the
TABLE feature to
determine the step
pattern.
Relation
y x2
y 2(x 1)2
y (x y 3(x Graph each equation on the graphing calculator.
Copy and complete the table.
Value
of a
Value
of h
Value
of k
Vertex
Axis of
symmetry
Direction of
opening
Step pattern
1
0
0
(0, 0)
x0
up
1, 3, 5, 7 . . .
3
2)2
2)2
Work with a partner.
You will need a TI-83 or TI-84 graphing calculator.
1
3
y 1.5(x 3)2 1
Reflect
In your table, which of a, h, or k determines:
➢ the direction of opening of the parabola y a(x h)2 k?
➢ the x-coordinate of the vertex of the parabola y a(x h)2 k?
➢ the y-coordinate of the vertex of the parabola y a(x h)2 k?
➢ the step pattern?
Explain your reasoning for each answer.
3.5 The Vertex Form of a Quadratic Relation
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Connect the Ideas
The vertex form
of a quadratic
relation
The numerical value
of a is the value of a
when you ignore the
sign.
Graph an
equation in
vertex form
The general equation of a quadratic relation in vertex form
is y a(x h)2 k. The graph has these properties.
➢
➢
➢
➢
➢
➢
➢
It is a parabola.
The vertex is (h, k).
The axis of symmetry is x h.
The stretch or compression factor is the numerical value of a.
If a is positive, the parabola opens up and has a minimum.
If a is negative, the parabola opens down and has a maximum.
The step pattern for the parabola is a, 3a, 5a, . . . .
We can use these properties to graph the parabola y 2(x 1)2 3.
Compare y 2(x 1)2 3 with the equation y a(x h)2 k
to identify the values of a, h, and k.
a = 2 so the
So, a 2, h 1 and k 3.
numerical value of
a is 2.
Graph the equation.
The parabola opens down.
Its vertex is (1, 3).
The stretch factor is 2, so the steps from the vertex are:
1 (2) 2, 3 (2) 6, 5 (2) 10, and so on.
Start at (1, 3).
Plot points to the right,
and then to the left,
using these steps:
across 1 and down 2,
across 1 and down 6,
across 1 and down 10.
y
2
–2
–1 0
–2
1
2
3
–4
–6
–8
–10
–12
–14
–16
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y = –2(x – 1)2 + 3
x
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Determine the
equation from
a graph
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We can also determine the equation of
a parabola when given its graph.
10
The equation of a parabola in vertex
form is y a(x h)2 k.
8
6
In the graph, the vertex is (2, 1).
So, h 2 and k 1.
Since the parabola opens up, the value of
a is positive.
On the graph of y a(x h)2 k, from
the vertex, move 1 right and a up.
y
4
2
–6
–4
–2 0
A(–1, 1.5)
x
2
4
On the graph, point A is at (1, 1.5).
From the vertex, this represents moving 1 right and 0.5 up.
So, a 0.5.
Substitute the values of a, h, and k into y a(x h)2 k.
The equation of the parabola is y 0.5(x 2)2 1.
Practice
1. For each parabola, determine:
the values of a, h, and k
ii) the coordinates of the vertex
iii) the direction of opening
iv) the axis of symmetry
v) the first 3 terms of the step pattern
a) y 15(x 12)2 11
b) y 6(x 8)2 9
c) y 7(x 6)2 4
d) y x2 1
e) y 0.35(x 1.5)2 2.5
f) y 0.45(x 4.2)2
i)
2. Graph each relation without creating a table of values.
y 12 (x 6)2 4
b) y 3(x 7)2 6
c) y 2(x 5)2 3
d) y 0.5(x 8)2 2
a)
3.5 The Vertex Form of a Quadratic Relation
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3. Determine the vertex form equation of the parabola with
vertex at (11, 12) and a stretch factor of 3.
1
b) vertex at (5, 0) and a compression factor of 3 .
c) vertex at the origin and a stretch factor of 4.
d) a minimum value of 3, axis of symmetry x 2, and congruent to y x2.
a)
4. The suspension cable of the bridge in the picture forms a parabola.
Estimate the values of a, h, and k and write the equation of the parabola.
y
Use a scale of
1 unit = 10 m
x
0
We can use what we know about the relation between a vertex form of a quadratic equation
and its graph to solve problems.
Example
On a rainy day, the profit,
P dollars, of an umbrella seller can be modelled
by the equation P 30(x 10)2 300,
where x dollars is the price of an umbrella.
a) What should the price be to maximize the day’s profit?
b) What is the maximum profit?
Solution
124
Since a is negative, the parabola opens down and has a maximum.
So, the profit is maximized at the vertex.
Since h 10 and k 300,
the vertex is at (10, 300).
a) Since the x-coordinate of the vertex is 10, the price should be $10.
b) Since the y-coordinate of the vertex is 300, the maximum profit is $300.
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5. On a sunny day, the daily profit, P dollars, of a popcorn vendor
is modelled by the equation P 60(x 4)2 120,
where x is the price of a bag of popcorn.
a) What should the price be to maximize the daily profit?
b) What is the maximum daily profit?
6. Assessment Focus For each relation, determine each of these.
Then explain how your strategy in part a
is different from your strategy in part b.
i) the values of a, h, and k
ii) the coordinates of the vertex
iii) the axis of symmetry
iv) the direction of opening
v) the y-intercept
vi) the first 3 terms of the step pattern
a) y 3(x 4)2 5
b)
6
y
4
2
x
–2 0
–2
2
4
6
–4
7. Take It Further Use a graphing calculator to graph
the quadratic relations y1 14 x2 2x 2
and y2 14 (x 4)2 2 on the same screen.
What do you notice? Give a reason for this.
Explain how to use an equation in the form y a(x h)2 k to identify each of these:
the coordinates of the vertex,
the direction of opening,
and the stretch or compression factor of a parabola.
Include examples and graphs in your explanation.
3.5 The Vertex Form of a Quadratic Relation
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Mid-Chapter Review
3.1 1. A rectangular swimming area is to be
Swimming area (m2)
enclosed by 60 m of rope. One side of
the swimming area is along the shore
so the rope will only be used on three
sides. The graph shows how the
swimming area is related to the length
of the side perpendicular to the beach.
400
3.4
of y x2 to produce the graph of each
relation.
i) y (x 2)2
ii) y x2 2
iii) y 2x2
iv) y x2 2
b) Graph each relation in part a.
c) How are the graphs the same?
d) How are they different?
3.5 4. In each case, the parabola y x2 is
300
200 A(5, 250)
B(25, 250)
100
0
3.3 3. a) Describe how to transform the graph
5
10 15 20 25 30
Length of side
perpendicular to beach (m)
What are the dimensions of the
largest swimming area? Sketch the
swimming area and label the sketch.
b) Draw the swimming areas that
correspond to points A and B, and
label their dimensions. What is the
same about these swimming areas?
Which would you prefer, and why?
a)
transformed as described. Write the
equation of the new parabola in the form
y a(x h)2 k.
a) The parabola is translated 2 units up
and 5 units right.
b) The parabola is stretched vertically by
a factor of 4.
c) The parabola is translated 2 units
right, then reflected in the x-axis.
d) The parabola is compressed vertically
by a factor of 0.5.
5. Match each parabola with its equation.
Explain your answers.
a) y 2(x 3)2 7 b) y 3x2 7
i)
ii)
3.2 2. The stopper in a bathtub is released and
the water begins to drain. The volume of
water, V litres, in the tub t minutes after
the stopper is pulled is given by the
equation V 5t2 8t 120.
a) Graph the equation using a graphing
calculator. Why do we only need to
use the first quadrant?
b) How many litres of water are in the
tub when it begins to drain?
c) How much time does it take for all the
water to drain?
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4
y
4
y
2
2
x
x
–2
–1 0
–2
1
2
0
–2
–4
–4
–6
–6
2
4
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3.6
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Multiplying Polynomials
The graph of the parabola y x2 6x 5 is shown.
We can use the graph to write the equation
of the parabola in vertex form as y (x 3)2 4.
Since both equations represent the same parabola,
they are the same equation written in different forms.
So, (x 3)2 4 is equivalent to x2 6x 5.
6
4
1
x
–2 0
–2
3
x
x
1
4
6
8
(3, –4)
Work with a partner. You will need algebra tiles.
x
2x
2
Multiplying with Algebra Tiles
x
x
1
1
1-tile x-tile x 2-tile
x
y = x 2 – 6x + 5
2
–4
Investigate
y
111
Part A: Using algebra tiles
➢ The guiding tiles for the product (2x 3)(x 1) are shown.
Build a rectangle whose length and width are represented
by these tiles. Sketch your rectangle.
What is the sum of the areas of the tiles in the rectangle?
Use the result to expand and simplify (2x 3)(x 1).
➢ Build a second rectangle with length 2x 3 and width x 1.
➢ Explain how the rectangles represent the product 2(2x 3)(x 1).
What is the sum of the areas of the tiles in the two rectangles?
Expand and simplify 2(2x 3)(x 1).
➢ Simplify 3(2x 3)(x 1) and 4(2x 3)(x 1).
Part B: Explaining the steps
➢ Explain how you can use algebra tiles to simplify each product.
(2x 3)2
2(2x 3)2
3(2x 3)2
4(2x 3)2
➢ Simplify each product.
Reflect
➢ What patterns do you notice in each set of products?
➢ Use these patterns to simplify 5(2x 3)(x 1) and 5(2x 3)2.
Explain your thinking.
3.6 Multiplying Polynomials
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Connect the Ideas
Expressions like 3, 5x 3, and 2x2 5x 3 are called polynomials.
The expression 3 has only one term, so it is a monomial.
The expression 5x 3 has two terms, so it is a binomial.
The expression 2x2 5x 3 has three terms, so it is a trinomial.
The expression (3x 1)(x 2) is the product of two binomials.
Here are three ways to determine this product.
Use algebra tiles
Use rectangles
Expand
The product of a
number and two
binomials
Make a rectangle with length 3x 1
and width x 2.
x
The area of the rectangle is (3x 1)(x 2).
1
Fill in the rectangle with tiles.
1
2
The sum of the areas of the tiles is 3x 7x 2.
So, (3x 1)(x 2) 3x2 7x 2.
Sketch a rectangle with length 3x 1
and width x 2.
The area of the rectangle is (3x 1)(x 2).
Divide the rectangle into 4 smaller rectangles.
The sum of their areas is 3x2 6x x 2, or,
collecting like terms, 3x2 7x 2.
So, (3x 1)(x 2) 3x2 7x 2.
Multiply each term in the first binomial
by each term in the second binomial.
Draw arrows to show which terms
are multiplied.
Recall that this process is called
expanding.
x
x
1
x2
x2
x2
x
x
x
x
x
x
x
3x
1
x
3x 2
x
2
6x
2
(3x 1)(x 2)
3x(x) 3x(2) 1(x) 1(2)
3x2 6x x 2
3x2 7x 2
The expression 3(3x 1)(x 2) is the product of 3 and two binomials.
To determine the product of the binomials, visualize two rectangles
with length 3x 1 and width x 2.
Each rectangle has area 3x2 7x 2,
so the sum of the areas is 3(3x2 7x 2), or 9x2 21x 6.
We write:
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x
CHAPTER 3: Quadratic Relations
3(3x 1)(x 2)
3(3x2 7x 2)
9x2 21x 6
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Practice
1. Simplify.
(x 7)(x 2)
1
d) ( x 10)(x 9)
2
a)
b)
(x 3)(x 3)
e) (3x 1)(x 5)
c)
(x 5)2
f) (2x 3)2
2(x 7)2
1
e) (x 2)( x 5)
2
c)
What tools can you
select?
2. Expand and simplify.
4(x 1)(x 3)
1
d)
(x 4)(x 12)
2
a)
6(x 2)(x 2)
f) 3(x 1)2
b)
3. Expand each set of binomials. What patterns do you notice?
Write and expand the next two binomials in the pattern.
a) (x 1)(x 2)
b) (x 1)2
c)
2
(x 1)(x 3)
(x 2)
(x 1)(x 4)
(x 3)2
(x 1)(x 5)
(x 4)2
2(x 1)(x 1)
4(x 2)(x 2)
6(x 3)(x 3)
8(x 4)(x 4)
We cannot use
an area model
to multiply
binomials with
a negative
coefficient. We
use a table
instead.
Knowing how to multiply binomials allows you to convert the equation of a quadratic relation
from vertex form to standard form.
Example
Write the equation y 2(x 3)2 5 in standard form.
Use technology to verify your answer.
Solution
y 2(x 3)2 5
2(x 3)(x 3) 5
2(x2 3x 3x 9) 5
2(x2 6x 9) 5
2x2 12x 18 5
2x2 12x 13
Press o. Enter
x
x
3
x2
3x 2
3 3x
9
A table can help you
keep track of the four
products in the
expansion of (x 3)2.
2(x 3)2 5 in Y1 and
2x2 12x 13 in Y2.
Press q 6 to graph
the equations in
a standard window.
Two equations are entered, but it seems the graphs for the equations coincide.
This shows that the same graph represents y 2(x 3)2 5 and
y 2x2 12x 13, so the equations are equivalent.
3.6 Multiplying Polynomials
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4. Write each equation in standard form. Check your answer by graphing
the original equation and your answer on a graphing calculator.
a) y (x 2)2 3
b) y 2(x 1)2
c) y (x 4)2 1
5. Determine whether the equations in each set are equivalent.
Justify each answer.
a) y (x 2)2
y x2 4
b)
y (x 1)2 3
y x2 2x 4
c)
y 3(x 2)2 4
y 3x2 12x 8
6. The equation h 5(t 1)2 7.5 models the height, h metres,
of a baseball t seconds after it is thrown.
a) What is the maximum height of the ball?
b) How long does it take for the ball to reach its maximum height?
c) Write an equivalent equation in standard form.
d) From what height is the ball thrown? How do you know?
e) Which form of the equation do you find easier to use? Why?
7. Assessment Focus
Which relation has the same graph as y 2x2 12x 19?
i) y 2(x 2)2 17
ii) y 2(x 2)2 11
iii) y 2(x 3)2 1
Justify your choice.
b) Determine the coordinates of the vertex for the parabola y 2x2 12x 19.
Explain how you determined these coordinates.
a)
8. Take It Further The graph of the parabola y x2 2x 3
is shown.
a) What are the coordinates of the vertex?
b) Write the equation in vertex form.
c) Check that the equation in part b is correct by converting it
to standard form.
y
4 y = x 2 – 2x – 3
2
x
–4
–2 0
–2
–4
You can use different tools and strategies to expand the product of two binomials.
Do you use the same method for the product of all binomials, or do you use
different methods depending on the binomials in the product? Explain.
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Parabola Patterns
Materials
• TI-83 or TI-84 graphing calculators
➢ The design on each graphing calculator screen
is made with parabolas.
Create the same design on your graphing calculator.
Explain your thinking.
Design 1
Design 2
➢ Create a design of your own.
Trade your design with a classmate.
Describe the transformations of the parabola y x2
used in your classmate’s design.
Determine the equation of each parabola
in your classmate’s design.
PUZZLE: Parabola Patterns
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Factoring Trinomials Using Models
and Technology
Factoring is the reverse of expanding.
We expand (x 2)(x 3) to get x2 5x 6.
We factor x2 5x 6 to get (x 2)(x 3).
Inquire
Factoring Using Models and Technology
Work with a partner.
You will need algebra tiles and a TI-89 or similar calculator.
Part A: Factoring trinomials of the form ax2 ⴙ bx ⴙ c , where a ⴝ 1
1. We can use algebra tiles to factor x2 2x 1.
➢ Use one x2-tile, two x-tiles, and one 1-tile.
➢ Arrange the tiles to form a rectangle.
Place the x2-tile and the 1-tile first.
Then place the x-tiles to complete a rectangle.
➢ Write the length and width of the rectangle as binomials.
➢ Write the area of the rectangle as the product of the length and width.
2. We can also use a CAS to factor x2 2x 1.
➢ Clear the home screen.
Press " ƒ8 M.
➢ Factor x2 2x 1.
Press „2 Ù Z2 «
2 Ù«1d ¸.
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CHAPTER 3: Quadratic Relations
If these keystrokes and screens do not match
those on your calculator, check the User’s Manual.
Does this answer agree with
the answer obtained with
algebra tiles? Explain.
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3. Copy the table. Then use algebra tiles and CAS to complete the table.
Trinomial
x2 2x 1
Factored form from algebra tiles
Factored form from CAS
(x 1)(x 1)
(x 1)2
x2 3x 2
x2 4x 3
x2 5x 4
x2 6x 5
4. Describe the patterns in the factored form of the trinomials.
5. The next trinomial in the pattern is x2 7x 6.
Predict the factors of this trinomial.
b) Use algebra tiles or a graphing calculator to verify your prediction.
c) Which tool did you choose? Why?
a)
6. a) What patterns do you see in the trinomials and the numbers in each factor?
Use these patterns to factor the next two trinomials in the pattern.
c) How do you know that your answers are correct?
b)
3.7 Factoring Trinomials Using Models and Technology
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Part B: Factoring trinomials of the form ax2 ⴙ bx ⴙ c , where a is a common factor
7. How are the trinomials in the first column related to the trinomials
in the third column?
Trinomial
Factored form
Trinomial
x2 2x 1
2x2 4x 2
x2 3x 2
2x2 6x 4
x2 4x 3
2x2 8x 6
5x 4
2x2 10x 8
x2
Factored form
8. Use your table from Part A to complete the second column in this table.
Use CAS to complete the fourth column in the table.
9. Compare the factors in the second and fourth columns. What do you notice?
10. Following the pattern, the next trinomial in the third column is 2x2 12x 10.
Predict the factors of this trinomial. Justify your prediction.
b) Verify your prediction using CAS or algebra tiles. Explain your choice of tool.
a)
11. Copy and complete the table without using technology.
Trinomial
Factored form
3x2 3x 6
3x2 6x 9
2x2 8x 6
3x2 15x 18
2x2 6x 4
3x2 6x 9
12. Verify your results using CAS.
Explain any mistakes you made.
How could you avoid making the same mistakes in the future?
Reflect
➢ How can you factor a trinomial without using algebra tiles
or CAS? How do you know your answer is correct?
➢ How can you factor a trinomial when the terms have
a common factor?
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Factoring Polynomials
Alex is a manager at a computer firm. She uses quadratic relations
to model income from the sales of different computer models.
The factored form of each relation can be used to determine
the break-even points.
Investigate
Using Strategies for Factoring
Work with a partner.
You may need algebra tiles or a TI-89 calculator.
Each of these expressions can be rewritten
as the product of three factors.
Use a tool or strategy of your choice
to determine the correct factors.
Factor.
➢ 30
➢ 7x2 14x
➢ 2x2 6x 4
➢ 3x2 12x 12
➢ 5x2 20
Reflect
➢ Compare your answers and your strategies with another pair.
If you have different answers, find out why.
➢ What tools or strategies did you use?
Why did you choose the tools or strategies?
➢ How is factoring an algebraic expression similar to factoring
a whole number? How is it different?
3.8 Factoring Polynomials
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Connect the Ideas
When we multiply two binomials, the result is often a trinomial.
12 is the product of 6 and 2
(x 6)(x 2) x2 4x 12
4 is the sum of 6 and 2.
These relationships suggest a way to factor trinomials.
To factor a trinomial in the form x2 bx c,
determine two integers that have a product of c and a sum of b.
Factor x2 bx c
To factor x2 7x 12, determine two integers
with product 12 and sum 7.
The product is positive and the sum is negative,
so both integers are negative.
List factors of 12 that have a negative sum.
The integers 3 and 4 have product 12
and sum –7.
So, x2 7x 12 (x 3)(x 4).
Factors
of 12
Sum
of factors
1, 12
13 2, 6
8 3, 4
7 ✓
Factors
of ⴚ6
Sum
of factors
To factor x2 5x 6, determine two integers
with product 6 and sum 5.
The product is negative, so one of the integers is negative.
The sum is positive, so the integer with the greater numerical value is
positive.
List factors of 6 that have a positive sum.
The integers 1 and 6 have product 6
and sum 5.
So, x2 5x 6 (x 1)(x 6).
Factor
ax2 ⴙ bx ⴙ c,
where a is a
common factor
136
1, 6
5✓
2, 3
1
If the terms of a trinomial have a common factor,
determine the common factor before determining the binomial factors.
To factor 4x2 24x 108,
first determine the common factor, 4.
Divide by the common factor,
then factor the remaining trinomial.
4x2 24x 108 4(x2 6x 27)
4(x 3)(x 9)
CHAPTER 3: Quadratic Relations
Factors
of ⴚ27
Sum
of factors
1, 27
26 3, 9
6 ✓
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You can check factoring by expanding.
4(x 3)(x 9) 4(x2 9x 3x 27)
4(x2 6x 27)
4x2 24x 108
Since this is the trinomial we started with, the factors are correct.
Practice
1. Factor each trinomial. Use algebra tiles or a rectangle diagram to represent the factored product.
a)
x2 3x 2
b)
x2 6x 8
c)
x2 8x 15
d)
x2 6x 9
2. a) Factor each trinomial.
x2 7x 6
ii) x2 8x 7
iii) x2 9x 8
iv) x2 10x 9
b) Describe a pattern in the trinomials in part a.
c) Extend the pattern by writing three more trinomials, then factor them.
i)
3. a) Factor.
x2 6x 9
ii) x2 12x 36
iii) x2 10x 25
b) Why is each trinomial in part a called a binomial square?
i)
4. a) Factor. What is the coefficient of x in each polynomial?
x2 16
ii) x2 64
iii) x2 1
b) Why is each binomial in part a called a difference of squares?
i)
5. Factor, using any strategies or tools you choose.
5x2 15x
d) 2x2 4x 6
a)
2x2 18x 40
e) 6x2 2x
b)
4x2 12x 40
f) 7x2 42x 63
c)
You can use a graphing calculator to check factoring.
Example
Use a graphing calculator to check that 4x2 24x 108 factors to 4(x 3)(x 9).
Solution
Graph the original and factored expressions on the same set of axes.
If the original and factored expressions are equivalent, their graphs will coincide.
Only one parabola is displayed, so it seems 4x2 24x 108 factors
to 4(x 3)(x 9).
3.8 Factoring Polynomials
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6. Use a graphing calculator to check the factoring.
Sketch the screens and explain how they justify your answer.
a) Does 3x2 3 factor to 3(x2 1)?
b) Does 2x2 4x 4 factor to 2(x 2)2 ?
c) Does 5x2 10x 15 factor to 5(x 1)(x 3)?
d) Does x2 3x 4 factor to (x 4)(x 1)?
7. a) Factor these trinomials.
2x2 4x 2
3x2 12x 12
4x2 24x 36
5x2 40x 80
b) Describe the pattern in the factors.
c) The trinomial 10x2 180x 810 continues the above pattern.
Factor it, then check your answer by expanding.
8. Assessment Focus
Factor completely each set of polynomials.
x2 x 2
5x2 10x 5
5x2 5
x2 2x 3
5x2 20x 20
5x2 20
x2 3x 4
5x2 30x 45
5x2 45
b) Describe the patterns in each set. Include patterns in the given polynomials
and in the factored form.
a)
9. Take It Further A student council wants to raise money by selling tickets
to a dinner. The income, T dollars, for the evening depends on the number n
of students who buy tickets, according to the equation
1 2
T 10
n 10n 2000
a) Find the income if 100 students buy tickets.
b) What is the income if no tickets are sold? What might this represent?
c) Factor the right side of the equation.
d) How many tickets must the student council sell to make a profit?
What do you think is most difficult about factoring?
Which tools and strategies could someone use to help with this difficulty?
Use examples in your explanation.
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The Factored Form of a Quadratic Relation
Companies use equations to model financial situations.
By interpreting the equations and graphs representing the equations,
they can make business decisions.
Investigate
Relating Factors and Intercepts
Work with a partner.
You will need a TI-83 or TI-84 graphing calculator.
➢ Copy the table.
x-intercepts
Equation
y (x 4)(x 2)
y (x 1)(x 3)
y 3(x 3)(x 5)
Each equation in the
table is written in
factored form.
The x-intercepts are
the x-coordinates of
the points where the
graph intersects the
x-axis.
y 0.5(x 4)(x 4)
y 2x(x 6)
y (15 x)(10 x)
➢ Graph each equation in a standard window, and
complete the table.
➢ Compare the factors of each equation with the x-intercepts
of the graph it represents. What do you notice?
➢ For each graph, each partner should choose
a different x-intercept.
Substitute the x-intercept into the equation, and
evaluate the equation.
Compare answers. What do you notice?
Reflect
➢ What is the relationship between the factored form of the
equation of a quadratic relation and the x-intercepts of its
graph? Justify your answer.
3.9 The Factored Form of a Quadratic Relation
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Connect the Ideas
The graph shows the height of a football,
y metres, x seconds after it is kicked.
The equation of the graph can be written
in 3 different forms.
Vertex form
y 5(x 2)2 20
Standard form
y 5x2 20x
Factored form
y 5x(x 4)
Height of football
20
Height (m)
Chapter 03
15
10
5
0
1
3
2
Time (s)
Each form gives different information
about the graph.
Vertex form
Standard form
Factored form
Some quadratic
relations cannot be
factored using
integers, so they
cannot be written in
factored form.
140
The vertex form tells us that the graph of y 5(x 2)2 20
results from reflecting the graph of y x2 in the x-axis,
vertically stretching it by a factor of 5,
and translating it 2 units right and 20 units up.
So, the graph opens down and has vertex (2, 20).
The standard form y 5x2 20x can be rewritten
as y 5x2 20x 0.
Recall that the constant term is the y-intercept.
This shows that the graph has a y-intercept of 0.
The factored form of the equation y 5x(x 4)
shows the x-intercepts of the graph.
At the x-intercepts, y 0.
So, at the x-intercepts, 0 5x(x 4).
The product of two factors is 0 if one or both of the factors are zero.
One x-intercept is 0 because 5x 0 when x 0.
The other x-intercept is 4 because x 4 0 when x 4.
The values shown by each form of the equation
can be checked by looking at the graph.
The equation of a quadratic relation in factored form is
y a(x r)(x s). The x-intercepts are r and s.
CHAPTER 3: Quadratic Relations
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Practice
1. Determine the x-intercepts of the graph of each equation.
y (x 2)(x 5)
c) y (x 4)(x 4)
y (x 7)(x 1)
d) y x(x 10)
a)
b)
2. Write each equation in factored form.
Then determine the x-intercepts of its graph. Justify your answers.
a) y x2 10x 9
b) y x2 x 20
c) y x2 4
d) y x2 4x
3. Write the equation of each quadratic relation in factored form.
Each graph is congruent to the graph of y x2.
a)
b)
y
4
y
4
2
2
–4
x
–2 0
–2
2
4
x
6
0
–2
2
4
6
8
–4
–4
c)
2
–2 0
–2
d)
y
2
4
6
x
8
10
y
x
–8
–6
–4
–2 0
–10
–4
–20
–6
–30
–8
–40
2
4
4. a) Sketch the graph of y x2 4 for x from 3 to 3.
b)
Use the graph to explain why y x2 4 cannot be written in
factored form.
5. Assessment Focus The equation of a quadratic relation in standard
form is y 2x2 12x 18.
a) Use the factored form of the equation to explain why the graph
of this relation has only one x-intercept.
b) What is the x-intercept? How would the graph show this?
3.9 The Factored Form of a Quadratic Relation
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6. Write each equation in factored form.
y 3x2 6x 24
c) y 2(x 1)2 8
a)
y x2 4
d) y (x 3)2 9
b)
7. The daily profit made by a game manufacturer is modelled by the equation
P 10(x 60)(x 15), where P dollars is the profit
and x dollars is the price of each game.
a) Determine the x-intercepts of this equation.
b) What do the x-intercepts represent in this situation? Explain.
Interpreting quadratic
relations provides a
strategy business can
use to make decision.
We can use the factored form of a quadratic relation to solve problems.
Example
The height, h metres, of a football t seconds after it is kicked is given
by the equation h 5t 2 30t. How long is the ball in the air?
Solution
Determine the x-intercepts. Write h 5t 2 30t in factored form: h 5t(t 6)
The factor 5t equals 0 when t 0.
The factor t 6 equals 0 when t 6.
So, the ball is at ground level at t 0 s and at t 6 s.
So, the ball is in the air for 6 s.
8. The equation h 5t 2 20t 25 gives the height, h metres,
8
of a flare t seconds after it is fired.
For how long is the flare in the air?
y
6
4
9. Take It Further The graph of a quadratic relation
is shown. Determine the equation of the relation in factored form.
Explain your strategy.
Suppose a classmate is confused about the three forms of the equation
for a quadratic relation.
For each form, create a Frayer model or concept map
that would help your classmate figure out the form of an equation,
and that describes the information you can see in each form.
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2
–6
–4
–2 0
2
x
4
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Chapter Review
What Do I Need To Know?
Tables, Graphs, and Equations for Quadratic Relations
A quadratic relation can be represented using a table, graph, or equation.
Table
x
y
2
7
1
0
0
5
1
8
2
9
3
8
4
5
5
0
6
7
Graph
y (2, 9)
Second
differences
First differences
0 (7) 7
505
853
8
5 7 2
6
3 5 2
4
1 3 2
981
8 9 1
5 8 3
0 5 5
7 0 7
2
1 1 2
x
3 (1) 2
–4
5 (3) 2
7 (5) 2
–2 0
–2
Vertex form
Standard form
Factored form
y x2 4x 5
y (x 1)(x 5)
Transforming the graph of y ⴝ x2
x-intercepts: 1 and 5
y-intercept
y x2 2
Move the graph up by 2 units
y
Move the graph down by 2 units
2
Some quadratic
relations cannot
be written in
factored form.
Draw the graph of y ⴝ x2 and:
To graph:
x2
6
x-intercepts: 1 and 5; y-intercept: 5
y (x 2)2 9
opens down
4
Vertex: (2, 9)
Equation of axis of symmetry: x 2
Maximum value is 9.
Equation The equation can be written in several forms.
vertex (2, 9)
2
y (x 2)2
Move the graph left by 2 units
y (x 2)2
Move the graph right by 2 units
y 2x2
Stretch the graph vertically by a factor of 2
y
0.5x2
Compress the graph vertically by a factor of 0.5
y
x2
Reflect the graph in the x-axis
Expanding and Factoring
Multiplying two binomials
3(2x 3)(4x 1)
3(8x2 2x 12x 3)
3(8x2 10x 3)
24x2 30x 9
Factoring a trinomial
2x2 6x 20
2(x2 3x 10)
2(x 5)(x 2)
(5) (2) 3
(5)(2) 10
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What Should I Be Able to Do?
3.1
models her height, h metres, above
the surface of the water after t seconds.
a) Graph the equation using a graphing
calculator.
b) Estimate the maximum height and the
time required to reach it.
c) How long is the diver in the air?
d) Estimate the time period when the
diver was higher than the board.
e) Are negative values of t and h
meaningful? Explain.
1. Last year, 500 people paid $6 a ticket to
attend the school play.
They estimate that they will sell
50 fewer tickets for every $1 increase
in price.
a) Copy and complete the table.
Price
($)
Number of
tickets sold
6
500
Income
($)
Plot Income against Price.
c) What ticket price should the drama
club charge to obtain the maximum
income?
Justify your answer.
b)
3.3
4. Match each parabola with its equation.
Explain.
y
4
2 iv
2. The graph shows the height of the ball
ii
–4
Height (m)
as it rolls down a ramp.
a) Estimate the height of the ball
after 1.5 s.
b) Estimate the time when the ball is at a
height of 3 m.
0
c)
3.2
4
iii
y x2 1
y (x 2)2
c) y x2 1
d) y (x 1)2 5
b)
2
4
6
8 10
Time (s)
12
Estimate when the ball reaches the
ground.
3. A diver jumps into the water from a spring
board. The equation h 5t2 8.8t 5
144
x
2
–4
a)
5
4
3
2
1
–2 0
–2
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CHAPTER 3: Quadratic Relations
5. Write the equation of the parabola
congruent to the graph of y x2
that has:
a) its vertex at (0, 3)
b) its vertex at (2, 4)
c) its vertex translated 5 units left and
3 units down from (0, 0)
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3.4
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6. Match each parabola with its equation.
a)
y
4
2
b)
c)
i
ii
x
–4
d)
–2 0
2
4
iv
–2
iii
e)
–4
y 2x2
c) y x2
a)
y 12 x2
d) y x2
3.6
b)
2(x 5)(x 14)
1
b) 2(x 6)( 2 x 12)
1
c) ( x 5)2
3
d) 4(x 8)(x 5)
y 3x2.
Follow your description.
9. The parabola y x2 is vertically
stretched by a factor of 4.
Then, it is translated 2 units down
and 3 units to the right. Write the
equation of the new parabola.
3.2 10. A stock on the TSX-Venture Index
3.5
12. Write in standard form:
y (x 7)2 20
b) y 2(x 6)2 1
c) y (x 5)(x 5)
d) y 3(x 1)(x 9)
a)
8. Identify a, h, and k in each relation, and
describe the transformation of the graph
of y x2 that each relation represents.
a) y 3(x 2)2 7
b) y (x 5)2 1
c) y 4x2 2
d) y (x 10)2
changes its price rapidly during the first
25 days after its initial listing. The price,
P cents, on day D is modelled by
P 14 (D 12)2 80.
Graph the equation on a TI-83 or TI-84
graphing calculator.
11. Expand and simplify.
a)
7. Describe how to sketch the graph of
3.5
Describe the price trends of the stock
over the first 25 days.
What is the initial listing price?
What is the highest value of the stock
in the 25 days?
What is the lowest value of the stock
in the 25 days?
On what day does the stock return to
its initial price?
3.7
3.8
13. Factor each expression.
a)
b)
c)
d)
e)
f)
x2 5x 36
3x2 12x
16 x2
x2 14x 49
2x2 18x 40
5x2 45
14. Explain how to factor 4x2 4x 8.
3.9 15. Write each equation in factored form.
y x2 11x 28
b) y x2 5x 24
c) y (x 2)2 9
d) y 2(x 1)2 8
a)
16. A quadratic relation has equation
y (x 2)(x 8). Determine its:
a) x-intercepts
b) y-intercept
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Practice Test
Multiple Choice: Choose the correct answers for questions 1 and 2. Justify each choice.
1. What is the factored form of 3x2 6x 9?
A.
3(x 3)(x 1)
B.
3(x 3)(x 1)
C.
3(x 3)(x 1)
2. Which equation represents the parabola?
4
y (x 4
2
B. y (x 1) 4
C. y (x 3)(x 1)
D. y (x 3)(x 1)
1)2
A.
D.
3(x 3)(x 1)
y
2
x
–4
–2 0
–2
2
4
6
–4
Show your work for questions 3 to 6.
3. Knowledge and Understanding
Expand.
b) Factor.
a)
(7x 3)(x 4)
i) x2 13x 40
i)
2(x 3)2
ii) 5x2 5x 30
ii)
4. Communication The winning design for a school crest contains 3 parabolas.
Each parabola is a transformation of y x2.
Parabola A: Translation 2 units down
Parabola B: Vertical stretch by a factor of 2
Parabola C: Vertical stretch by a factor of 2, reflection in the x-axis, and translation 4 up
a) Draw the crest on 1-cm grid paper.
b) Describe how you drew each parabola.
5. Application A group of engineering students constructs a potato cannon.
The height, h metres, of a potato t seconds after it is shot from the
cannon that is on the roof of a building is modelled by the equation
h 5(t 2)2 45.
a) How high is the potato after 1 s?
b) When does the potato reach its maximum height?
What is this height?
c) How long is the potato in the air?
6. Thinking The track of a roller coaster passes over several
hills and then goes into a tunnel. Near the tunnel, it follows
1
(d 12) (d 12), where h metres is its height
the path h 72
above the ground and d metres is a measure of horizontal distance.
a) Find the horizontal distance between the entrance and the exit of the tunnel.
b) How deep is the tunnel? Justify your answer.
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