Algebra 2 Unit 5 (Chapter 4)
0.
Spiral Review
Worksheet 0
1.
Graph quadratic equations in the form of y = a(x – h)2 + k
(Section 4.2)
Identify the axis (line) of symmetry, vertex, domain, and range. Writing
equations of parabolas from their graphs
Worksheet 1
1 – 18
2.
Converting quadratics form standard form into vertex form. (Section 4.1)
Identify line of symmetry, vertex, domain and range
Worksheet 2
1 – 12
3.
Translate parabolas in descriptive form. (Section 4.2)
State the domain and range.
Page 249
3 – 12, 33 – 35
Worksheet 3
1 – 17
4.
Compare widths, directions, maximums, minimums, domains and ranges of
various parabolas. Word problems
Worksheet 4
1 – 10
5.
Word problems using parabolas.
Worksheet 5
1–8
Review
Worksheet
1 – 18
1 – 30
Worksheet 0
Graph the following.
1.
2x + y = 4
2.
3x + y > –3
y ≥ − 2x
3.
y ≥1
Multiply in problems 4 – 6.
1
4.
7 3 3
(4x y )
5.
2
2x(x + 3)
6.
 6xy2 


10x 2 y 5 
Factor completely.
7. 5x2 + 6x + 1
8. 6x2 – 20x + 14
9. 9x4 – 9y4
Find the value of c that makes the expression a perfect square trinomial.
9.
x2 + 100x + c
10.
x2 – 9x + c
Solve using the quadratic formula. Express answers in simplest form.
11.
13.
x2 + 3x + 1 = 0
12.
2x2 + 5x = 7
The product of two consecutive positive integers is 26 more than twice their
sum. What are the integers?
14.
Find two consecutive positive even integers such that the square of the
smaller is 4 more than 4 times the larger.
Worksheet 1
For problems 1-15:
1.
y = (x – 5)2 – 2
2.
y=
4.
y=–
5.
17.
1
(x – 4)2 + 1
2
y = (x + 2)
2
f) State the range
6.
y = 3(x – 2)2 – 5
11.
y = – (x – 3)2
7.
y = –3x2 – 4
12.
y = 5(x + 3)2 + 2
8.
y = – (x – 6)2 – 7
13.
y = 2x2 + 4
9.
y=
2
(x + 1)2
3
14.
y = 3x2
10.
y=
1 2
x +1
5
15.
y=–
y = –2(x + 1)2 – 3
3.
16.
1
(x + 6)2 + 4
4
e) State the domain
d) State whether the
parabola has a maximum
or a minimum and give
that value
a) State the line of
symmetry
b) State the vertex
c) Sketch the parabola
Which multiple choice could be the equation of the graph shown?
A.
y = –2(x + 2)2 + 4
B.
y = – 2(x – 2)2 + 4
C.
y = –2(x + 2)2 – 4
D.
y = – 2(x – 2)2 – 4
Which multiple choice could be the equation of the graph shown?
A.
y = 2(x + 3)2 + 2
B.
y = 2(x – 3)2 + 2
C.
y = 2(x + 3)2 – 2
D.
y = 2(x – 3)2 – 2
(Continued on next page)
1 2
x –2
4
18.
Write the equation of the following graphs.
a)
b)
Worksheet 2
Convert the following
quadratics equations
a) Line of Symmetry
b) State the Vertex
from standard form into
vertex form. Then
sketch the following
c) State the Domain
d) State the Range
parabolas. Find and
label the following:
e) State whether the parabola
has a maximum or a minimum
and give that value
5.
g ( x ) = 3x 2 − 6x + 4
9.
f (x ) =
−x − 4x + 5
6.
−4x + 8x + 2
y =
10. h ( x =
)
y =
−x 2 − 2x − 1
7.
y = x 2 + 7x + 5
8.
f (x ) = x 2 − 9x − 2
1.
y = x 2 + 2x + 1
2.
y = x 2 − 6x + 3
3.
4.
2
2
y=
2 2
x − 3x + 6
3
1 2
x +x −3
2
8
− x 2 + 4x + 5
11. y =
5
( )
12. f x = 6x 2 − 4x − 5
Worksheet 3
1.
2.
3.
4.
5.
Given the equation y = x2 + 4. If the graph is shifted down 2 units, which equation
describes the new graph?
a.
y = x2 + 6
c.
y = (x – 2)2 + 2
e.
y = (x – 2)2 + 4
b.
y = x2 + 2
d.
y = (x + 2)2 + 4
2
Given y – 1 = 2(x +1) . If the equation is shifted left 5 units, which equation describes
the new graph?
a.
y – 6 = 2(x – 4)2
c.
y – 1 = 2(x + 6)2
e.
y – 1 = 2(x – 6)2
b.
y – 1 = 2(x – 4)2
d.
y – 1 = 2(x + 5)2
If the given function y = (x – 1)2 + 3 is shifted up 3 units and left 4 units, which equation
describes the new graph?
a.
y = (x – 4)2 + 3
c.
y – 7 = (x + 2)2
e.
y = (x + 3)2 + 6
b.
y – 3 = (x + 4)2
d.
y = (x – 5)2
If the given function y = (x + 3)2 + 4 is shifted down 5 units, which equation describes the
new function?
a.
y = (x + 3)2 + 9
c.
y = (x + 8)2 + 4
e.
y = – 5(x +3)2 + 4
b.
y = (x + 3)2 – 1
d.
y = (x – 2)2 + 4
If the graph of the equation y + 2=
equation describes the new graph?
2
2
( x + 4 ) is shifted to the right 3 units, which
3
2
2
( x + 7)
3
2
2
b. y + 2=
( x + 1)
3
2
2
c. y + 2=
( x − 1)
3
a.
y + 2=
d.
e.
2
2
( x − 3)
3
2
2
y + 2=
( x + 3)
3
y + 2=
6.
Given y = x2. If the function is shifted 4 units to the left, write an equation that
describes the new function.
7.
Given y − 2=
1
2
( x + 1) . If the function is shifted 4 units down and 2 units right, write an
2
equation that describes the new function.
Given y = – 2(x – 3)2 . If the function is shifted 8 units to the right and 3 units up, write
an equation that describes the new function.
9.
Given y – 7 = x2 . If the function is shifted left 1 unit and down 4 units, write an equation
that describes the new function.
10.
The function y = x2 – 2x + 2 may be formed by shifting the function y = x2 + 1 which way?
11.
The function y = x2 + 2x – 1 may be formed by shifting the function y = (x +2)2 – 3 which
way?
State the domain and range of the following:
8.
12.
y = 2(x + 4)2 – 7
13.
y=
14.
Y = x2 + 6x + 8
−1
2
( x − 3) − 6
2
15.
16.
17.
y = – 2x2 + 20x – 5
y = 4x2 + 7
y = – 3(x +1)2
Worksheet 4
In each problem below there are two parabolas f(x) and g(x). Graph each parabola
on a separate grid. Label the line of symmetry and vertex of each. Using both
your graphs determine all the statements that apply to the pair of parabolas from
this list:
a)
b)
c)
d)
e)
f)
g)
h)
Both parabolas open the same direction
The parabolas have the same line of symmetry
The parabolas have the same vertex
Both parabolas have maximums
Both parabolas have minimums
The parabolas are the same width
The f(x) parabola is wider than the g(x) parabola
The g(x) parabola is wider than the f(x) parabola
1
(x – 4)2 + 3
2
g(x) = 2(x – 4)2 + 3
1.
f(x) = –
2.
f(x) = 3(x + 1)2 – 2
g(x) = 3(x – 1)2 + 4
3.
f(x) = x2 + 6x + 8
g(x) = 3x2 + 6x – 9
4.
f(x) = –2x2 – 4
5.
f(x) = 4(x – 2)2
g(x) = –2x2 + 4
1
g(x) = –
(x – 2)2 – 6
4
6.
f(x) = 2x2 – 4x + 6
g(x) = –2x2 + 4x + 6
Solve the following word problems:
7.
The revenue, R, made selling phones at price, p, can be modeled by
−5 2
R=
p + 500p
2
a) What price will maximize the companies revenue?
b) What is the maximum revenue?
8.
The revenue, R, made selling widgets at price, p, can be modeled by
R = -5p2 + 100p
a) What price will maximize the companies revenue?
b) What is the maximum revenue?
(Continued on next page)
9.
The available power, P, is a function of the amount of current flowing in
amperes
P = 120A – 20A2
a) How many amperes will produce the maximum power?
b) What is the maximum power?
10.
The profit, P, a company makes depends on the ticket price, t, they charge.
P = –15t2 + 600 t + 50
a) What ticket price yields the maximum profit?
b) What is the maximum profit?
Worksheet 5
Solve the following word problems.
1.
Jason is standing on the ground and throws a ball vertically upward with an
initial speed of 80 ft/sec. Its height after t seconds is given by =
h 80t − 16t 2
a) How high does the ball go?
b) How many seconds does it take to reach maximum height?
c) When does the ball hit the ground?
2.
Kevin in standing in a field and shoots an arrow vertically upward with an
initial speed of 64 ft/sec. Its height after t seconds is given by
h = 64t – 16t2
a) How many seconds does it take to reach maximum height?
b) How high does the arrow go?
c) When does the arrow hit the ground?
3.
An arrow is shot from the ground vertically upward with an initial speed of
96 ft/sec. Its height after t seconds is given by
h = 96t – 16t2
a) How high does the arrow go?
b) How long until the arrow reaches its maximum height?
c) When does the arrow hit the ground?
(Continued on next page)
4.
Cory is standing on a cliff that is 48 feet above the ground.
He throws a rock into the air. The height, h, of the rock
after t seconds is given by h = –2t2 + 4t + 48
a) How many seconds does it take to reach maximum height?
b) What is the rock’s maximum height?
c) How many seconds does it take for the rock to land on the
ground?
5.
Roger is standing at the top of a lighthouse.
He is 96 feet above the ocean. He throws a ball into
the air. The height, h, of the ball after t seconds
is given by h = –3t2 + 12t + 96
a) What is the ball’s maximum height?
b) How many seconds does it take to reach maximum
height?
c) How many seconds does it take for the ball to land in
the ocean?
6.
John throws a ball into the air from the top of a building that is 135 feet
tall.
The height, h, of the ball after t seconds is given by
h = –5t2 + 30t + 135
How many seconds does it take for the ball to land on the ground?
7.
Nicki tosses a frisbee into the air from the balcony of a building that is 64
feet tall. The height, h, of the frisbee after t seconds is given by
h = –8t2 + 16t + 64.
Find the maximum height of the Frisbee.
8.
Roger is standing on a hill that is 80 meters high. He throws a rock into the
air. The height, h, of the rock after t seconds is given by
h = – 4t2 + 32t + 80
Find the time needed for the rock to reach its maximum height
Review
Graph the following parabolas. Find and label the following:
a) Vertex b) Axis of Symmetry
c) Domain d) Range
1.
y = 3x 2
9.
y = 2x 2 + 4x + 5
2.
y = −2x 2
10.
y = 3x 2 − 18x + 26
3.
y =
−x 2 + 2
11.
f (x ) =
−2x 2 + 4x − 7
4.
=
y 5x 2 + 1
12.
y = 4x 2 − 16x + 17
13.
y = x2 + 8x + 2
14.
y = 4x2 – 8
5.
y =
− ( x + 3) + 5
2
( x − 3)
2
6.
=
y
7.
−4 ( x − 2 ) + 4
g (x ) =
15.
y=
8.
y = 2 (x + 1) − 3
16.
y = (x + 2)2
2
2
−1
(x+1)2 + 3
3
For problems 17 – 19 tell whether the function has a minimum value or a
maximum value. Then find the minimum or maximum value.
1
(x + 2)2 – 5
4
17.
y=
20.
Given:
18.
y = -5x2 + 6
19.
y = x2 – 4x + 7
y = –3(x + 4)2 – 8. If the function is shifted left 3 and up 4, write
an equation that describes the new function.
21.
Given: y – 6 =
1
(x – 2)2 .
4
If the function is shifted down 6 and left 3,
write an equation that describes the new function.
22.
If the two functions
f(x) = 2(x + 4)2 + 1
and
g(x) = –2(x – 4)2 + 3
were graphed on the same axis, which statement would be true.
a)
They have the same vertex
b)
They open the same direction.
c)
They have the same width
d)
They have the same line of sym
(Continued on next page)
23.
24.
Which statement is true about the graphs of y = 4x2 + 5
and
y = –2x2 + 5
a)
They have the same vertex
b)
They have the same width
c)
They both have maximums
d)
They both have minimums
The graph of y = 4x2 is translated so that the new vertex is (–5, –7). Write
the equation of the new parabola.
25
Tracy is standing on the ground and throws a ball vertically upward into the
air. Its height after t seconds is given by =
h 80t − 20t 2 .
a) How high does the ball go?
b) How many seconds does it take to reach maximum height?
c) When does the ball hit the ground?
26.
Susan is standing on a small hill 24 feet tall. She tosses an apple up into the
air the height of the apple is given by h = -3t2 + 6t + 24 , where h is the
height in feet and t is the time in seconds.
a) How many seconds for the apple to reach its maximum height?
b) What is the maximum height?
c) When does the apple hit the ground?
27.
The daily profit P of a camera company depends on the price x at which each
camera is sold. The profit is represented by the function
P = –x2 + 40x + 700
a) Find the price each camera should be sold at to maximize their profit.
b) Find the maximum profit.
(Continued on next page)
28.
Write the equation of the graph shown
if the ‘a’ value is 1.
29.
30.
Which multiple choice could be the equation of the graph shown?
A.
y = –2(x + 2)2 + 8
B.
y = – 2(x – 2)2 + 8
C.
y = –2(x + 2)2 – 8
D.
y = – 2(x – 2)2 – 8
Which multiple choice could be the graph of
A.
B.
C.
y = – x2 – 1 ?
D.