Secondary 2
Chapter 12
Secondary II
Chapter 12 – Introduction to Quadratic Functions
Date
Section
2014/2015
Assignment
Concept
- Worksheet 12.1 & 12.2
Exploring Quadratic Functions
Comparing Linear and Quadratic
Functions
- Worksheet 12.3
Domain, Range, Zeros and
Intercepts
A: 1/21
B: 1/22
12.1/12.2
A: 1/23
B: 1/26
12.3
A: 1/27
B: 1/28
12.4/12.5
- Worksheet 12.4 & 12.5
Factored Form and Vertex Form of
Quadratic Equation
A: 1/29
B: 1/30
12.6 /12.7
- Worksheet 12.6 & 12.7
Vertex Form of a Quadratic
Equation and Transformations of
Quadratic Functions
A: 2/2
B: 2/3
Review
Review Worksheet
Review
A: 2/4
B: 2/5
Chapter 12 TEST
Late and absent work will be due on the day of the review (absences must be excused). The review
assignment must be turned in on test day. All required work must be complete to get the curve on the test.
Remember, you are still required to take the test on the scheduled day even if you miss the review, so come
prepared. If you are absent on test day, you will be required to take the test in class the day you return. You
will not receive the curve on the test if you are absent on test day unless you take the test prior to your
absence.
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Secondary 2
Chapter 12
2
Secondary 2
Chapter 12
Chapter 12: Introduction to Quadratic Functions
12.1/12.2 – Exploring Quadratic Functions and Comparing Linear and Quadratic Functions
Example 1: A dog trainer is fencing in an enclosure, represented by the shaded region in the diagram.
The trainer will also have two square-shaped storage units on either side of the enclosure to store
equipment and other materials. She can make the enclosure and storage units as wide as she wants, but
she can’t exceed 100 feet in total length.
1. Let s represent a side length, in feet, of one of the square storage units.
a. Write an expression to represent the width of the enclosure. Label the width in the
diagram.
b. Write an expression to represent the length of the enclosure. Label the length in the
diagram.
c. Write an expression to represent the area of the enclosure.
2. Use the Distributive Property to rewrite the expression representing the area of the enclosure,
𝐴(𝑠), in standard form. Then, Identify a, b, and c for the function. Show your work.
The expression you wrote is a quadratic expression. A quadratic function written as
𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, where 𝑎 ≠ 0, is in Standard or General form.
(a and b are coefficients, and c represents a constant.)
3. Identify the independent and the dependent quantities and their units in this problem.
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Chapter 12
The progression of the diagrams on the right shows how the area of the enclosure, 𝐴(𝑠), changes as the
side length s of each square storage unit increases.
4. Describe how the area of the enclosure changes.
5. Use a graphing calculator to graph the function you
wrote in Question 3. Then sketch the graph and label the axes.
How can you tell the graph is a quadratic function?
To graph an equation on a graphing calculator:
1. Press Y= , then enter the function in y= form.
2. Press GRAPH.
3. You can use the WINDOW and ZOOM keys to adjust
window settings.
The shape that a quadratic function forms when graphed is called a Parabola.
6. Think about the possible areas of the enclosure. Is there a maximum area that the enclosure can
contain? Explain your reasoning in terms of the graph and in terms of the problem situation.
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Chapter 12
You can use a graphing calculator to find the maximum (or minimum) value on a graph.
1.
2.
3.
4.
Graph the function.
Press 2nd then CALC
Then 4: maximum press ENTER (3: minimum)
Move the cursor until it is on the left of the maximum, then press ENTER. It will then
ask you to do the same thing for the right side, then press ENTER.
5. Move the cursor until it is on the maximum, then press ENTER.
7. Use the instructions above to determine the absolute maximum of 𝐴(𝑠).
8. What do the x- and y- coordinate of the absolute maximum represent in this problem situation?
9. Determine the dimensions of the enclosure that will provide the maximum area. Show your work
and explain your reasoning.
Example 2: Use the following function to answer the questions below.
𝐻(𝑛) =
𝑛(𝑛 − 1)
2
1. Rewrite the expression for the value n. as a quadratic function 𝐻(𝑛) in standard form.
2. Graph this function on the graphing calculator (use Y2), and
sketch the graph on the coordinate plane.
3.
Determine the absolute minimum o f 𝐻(𝑛).
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Chapter 12
Example 3: Looking at the same two functions in your calculator…
𝐴(𝑠) = −2𝑠 2 + 100𝑠
1
1
𝐻(𝑛) = 2 𝑛2 − 2 𝑛
1. How can you determine whether the graph of a quadratic function opens up or down based on
the equation?
2. How can you tell whether the graph of a quadratic function has an absolute minimum or an
absolute maximum?
Example 4: Two dog owners have 16 yards of fencing to build a dog run beside their house. The dog
owners want the run to be in the shape of a rectangle, and they want to use the side of the house as one
side of the dog run. A rough sketch of what they have in mind is shown below.
1. Complete the table to show the different widths, lengths, and area that can occur with sixteen
yards of fencing.
2. Describe what happens to the length as the width
of the dog run increases. Why do you think this
happens?
3. Describe what happens to the area as the width of
the dog run increases.
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Chapter 12
4. Write the equation of the line 𝐿(𝑤), using the two points
(0, 8) and (16, 0). Make sure your equation is in context.
(Using variables l and w). Then, sketch a graph of the
function. Label the axes.
5. Let 𝐴(𝑤) represent the area of the dog run as a function of the width.
a. How much fence do we have for the three sides?
b. So the three sides must add up to?
c. Using the total amount, write an equation for length l, in terms
of the width, w.
d. Since Area = (length)(width), use the equation found in (c) to write an equation for the area
𝐴(𝑤), in terms of the width.
6. Sketch a graph of the area equation using your
calculator. Label the axes.
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Chapter 12
7. Let’s compare and contrast the graphs of the two functions.
𝐿(𝑤): The length of the dog run as a function of the width
𝐴(𝑤): The area of the dog run as a function of the width.
a. Describe the type of function represented by each graph. Explain your reasoning.
b. State the domain in terms of each function and the problem situation.
c. Determine the y- intercepts of each graph and interpret the meaning of each in terms of the
problem situation.
8. Determine the dimensions that provide the greatest area.
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Chapter 12
Additional Notes
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Chapter 12
12.3 – Domain, Range, Zeros, and Intercepts
Use the graph below to draw an example of each of the following definitions.
Parabola: the “u” shape of a quadratic function.
Maximum: the highest y-value on a graph.
Minimum: the lowest y-value on a graph.
Domain: the set of x-values of a function. (The “input” values.)
Range: the set of y-values of a function. (The “output” values.)
Increasing: the interval where the graph going up from left to right.
Decreasing: the interval where the graph is going down from left to right.
x-intercepts: where the graph of the function crosses the x-axis.
Zeros: another name for x-intercepts. (Why do you think that they are also called zeros?)
Vertex: the lowest or highest point on the curve.
Axis of Symmetry: the vertical like that passes through the vertex and divides the parabola into two
mirror images.
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Chapter 12
Example 1: Suppose you launch a model rocket from the ground. You can model the motion of the
rocket using a vertical motion model. A vertical motion model is a quadratic equation that models height
of an object at a given time. The equation is of the form:
𝑔(𝑡) = −16𝑡 2 + 𝑣0 𝑡 + ℎ0
𝑔(𝑡) -represents the height of the object in feet
t -represents the time in seconds
𝑣0 -represents the initial velocity (speed) of the object in feet per second
ℎ0 -represents the initial height of the object in feet.
1. Why do you think that it makes sense that this situation is modeled be a quadratic equation?
Suppose the model rocket has an initial velocity of 160 feet per second.
2. Write the function, 𝑔(𝑡), to describe the height of the model rocket in terms of t.
3. Describe the dependent and independent quantities.
4. Use the graphing calculator to graph the
function. Sketch the graph and label the axes.
Include the window settings.
You can use a graphing calculator to find any y-value on a graph.
1. Press 2nd the CALC
2. Then 1: value press ENTER
3. Then type the x-value you would like to fine the y-value for. x=? then ENTER
5. Use the graphing calculator to answer each question.
a. What is the height of the model rocket at 6 seconds?
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Secondary 2
Chapter 12
You can use a graphing calculator to find any x-value on a graph.
1. Graph the function.
2. Press Y= , then in Y2= , enter in the y-value that you would like to find the x-value for.
Press 2nd then CALC
3. Then 5: intersect press ENTER
4. Move the cursor until it is on the intersection of the two graphs, then press ENTER.
It will then ask you to do the same thing for the second graph, then press ENTER.
5. Press ENTER one more time.
b. After approximately how many seconds is the model rocket at a height of 200 feet?
c. What is the maximum height of the model rocket?
d. When is the rocket at its maximum height?
6. You can use a graphing calculator and intersection points to determine the x-intercepts of a
function.
a. What linear function would you use to determine the x-intercepts of the quadratic
function? Explain your reasoning.
b. Determine the x-intercepts of 𝑔(𝑡). Then, explain what this means in terms of this
problem situation.
You can use a graphing calculator to determine the zeros of a function.
1. Press 2nd then CALC
2. Then 2: zero press ENTER
3. Move the cursor until it is on the left of the zero, then press ENTER. Move the cursor
until it is on the right side of the zero, then press ENTER.
4. Press ENTER one more time.
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Chapter 12
7. Identify and describe the domain of the function in terms of the contextual situation.
8. Identify and describe the range of the function in terms of the contextual situation.
An interval is defined as the set of real numbers between two given numbers. To describe an interval,
this notation is used:
-
An open interval (𝑎, 𝑏) describes the set of all numbers between a and b, but NOT
including a and b.
A closed interval [𝑎, 𝑏] describes the set of all numbers between a and b, including a and b.
A half closed or half open interval (𝑎, 𝑏] describes the set of all numbers between a and b,
including b, but not including a. Or, [𝑎, 𝑏) describes the set of all numbers between a and
b, including a, but not including b.
Intervals that are unbounded are written using (), and the symbol for infinity, ∞.
9. Use interval notation to describe the interval which all numbers that are:
a. less than a.
b. less than or equal to a.
c. a is any real number.
10. Use interval notation to describe the interval of the domain in which the model rocket is:
a. increasing
b. decreasing.
11. How does the absolute minimum or absolute maximum help you determine each interval?
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Chapter 12
For each function shown, identify the domain, range, zeros, max or min, and the intervals of increase and
decrease.
Example 2:
𝑓(𝑥) = −2𝑥 2 + 4𝑥
Domain:
Range:
y-intercept:
Zeros:
Max/min:
Increasing:
Decreasing:
Example 3:
𝑓(𝑥) = 𝑥 2 + 5𝑥 + 6
Domain:
Range:
y-intercept:
Zeros:
Max/min:
Increasing:
Decreasing:
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Secondary 2
Chapter 12
Additional Notes
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Secondary 2
Chapter 12
12.4/12.5 – Factored Form and Vertex Form of a Quadratic Function
To factor an expression means to use the distributive property in reverse.
When factoring an expression, you are taking out the greatest common factor.
If the leading coefficient is negative, factor out the negative sign as well. Pay attention to the signs!!!
Example 1: Factor out the GCF.
a. 5𝑥 + 10
b. −5𝑥 + 10
c. 2𝑥 2 + 4𝑥 − 6
Example 2: A group of students are working together on the problem shown.
1. Write a quadratic function in factored form to represent a parabola that opens downward and
has zeros at (4, 0) and (-1, 0).
a. What is wrong with Judy’s function?
2. Use your graphing calculator to graph Maureen’s and Michael’s functions.
a. What are the similarities?
b. What are the differences?
c. Is it possible to have more than one correct function?
d. How many possible functions can you write with the given characteristics? Explain.
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Chapter 12
3. For a quadratic function written in factored form 𝑓(𝑥) = 𝑎(𝑥 − 𝑟1 )(𝑥 − 𝑟2 ):
a. What does the sign of a tell you about the graph?
b.
4.
What do the values 𝑟1 and 𝑟2 tell you about the graph?
Use the given information to write a quadratic equation in factored form
𝑓(𝑥) = 𝑎(𝑥 − 𝑟1 )(𝑥 − 𝑟2 ).
a. The parabola opens upward and the zeros are (2, 0) and (4, 0).
b. The parabola opens downward and the zeros are (-3, 0) and (1, 0).
c. The parabola opens downward and the zeros are (0, 0) and (5, 0).
Example 3: Use a graphing calculator to determine the zeros of each function. Sketch each graph using
the zeros and the minimum/maximum. Remember to include the window setting.
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Chapter 12
You can model the motion of a pumpkin release from a catapult using a vertical motion model.
Remember, a vertical motion model is a quadratic equation that models the height of an object at a given
time. The equation is in the form:
𝑦 = −16𝑡 2 + 𝑣0 𝑡 + ℎ0
𝑦
t
𝑣0
ℎ0
-represents the height of the object in feet
-represents the time in seconds
-represents the initial velocity (speed) of the object in feet per second
-represents the initial height of the object in feet.
1. Why do you think it makes sense that this situation is modeled by a quadratic function?
Example 4: Suppose that a catapult hurls a pumpkin from a height of 68 feet at an initial velocity of 128
feet per second.
1. Write a function for the height of the pumpkin ℎ(𝑡) in terms of t.
2. Does the function you wrote have an absolute minimum or an absolute maximum? How can you
tell from the function?
3. Use a graphing calculator to determine the zeros of the function. Then explain what each means
in terms of the problem situation.
4. Determine the y-intercept and explain its meaning in terms of this problem situation.
5. Use a graphing calculator to determine the absolute minimum or maximum. Then explain what it
means in terms of this problem situation.
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Chapter 12
Reminder:
- The vertex of the parabola is the lowest or highest point on the curve.
- The axis of symmetry is the vertical line that passes through the vertex and divides the
parabola into two mirror images. A parabola is said to be symmetric.
a. Where is there symmetry in the parabola?
b. What kind of line would be the line of symmetry?
c. What do equations of those lines look like?
6. Write the equation for the axis of symmetry.
7. Identify the coordinates of the vertex of the graph.
8.
Using a graphing calculator, when does the pumpkin reach a height of 180 feet?
9. Use the information from above to construct a graph.
a. Plot and label the vertex.
b. Draw and label the axis of symmetry.
c. Plot and label the point from #13.
d. Plot the point symmetric to the point from
#13.
e. Plot and label the zeros.
10. Analyze the symmetric points.
a. What do you notice about the y- coordinates?
b. What do you notice about each point’s horizontal distance from the axis of symmetry?
c. How does the x-coordinate of each symmetric point compare to the x-coordinate of the
vertex?
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Chapter 12
Example 5:
1. Determine the axis of symmetry of the parabola.
a. The x-intercepts are (1, 0) and (5, 0).
b. Two symmetric points on the parabola are (-7, 2) and (0, 2).
2. Determine the location of the vertex of each parabola.
a. The function 𝑓(𝑥) = 𝑥 2 + 4𝑥 + 3 has the axis of symmetry at 𝑥 = −2.
b. The function 𝑓(𝑥) = 𝑥 2 + 6𝑥 − 5 has two symmetric points (-1, -12) and (7, -12).
3. Determine another point on each parabola.
a. The vertex is (0.5, 9).
An x-intercept is (-2.5, 0).
b. The vertex is (-2, -8).
A point on the parabola is (-1, -7).
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Chapter 12
Additional Notes
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Secondary 2
Chapter 12
12.6/12.7 – Vertex Form of a Quadratic Function and Transformations of Quadratic Functions
In examples 1 & 2, complete table below without using a calculator. Then, use a graphing calculator to
fill in the rest of the missing information, and graph the function.
Example 1: 𝑓(𝑥) = −2𝑥 2 + 6𝑥 + 20 (standard form)
x
f(x)
-2
-1
0
1
2
Parabola opens:___________________
Zero(s):
___________________
Vertex:
y-intercept:
___________________
Parabola opens:___________________
Zero(s):
___________________
Vertex:
y-intercept:
___________________
___________________
Example 2: 𝑓(𝑥) = −2(𝑥 + 2)(𝑥 − 5)
x
f(x)
-2
-1
0
1
2
___________________
a. Compare your answers in Example 1 and Example 2. What do you notice?
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Chapter 12
Example 3: 𝑓(𝑥) = −2(𝑥 − 1.5)2 + 24.5
x
f(x)
-2
-1
0
1
2
Parabola opens:___________________
Zero(s):
___________________
Vertex:
y-intercept:
___________________
___________________
a. Compare your answers in Example 3 with Examples 1 & 2. What do you notice?
The quadratic function in Example 3 is written in vertex form.
A quadratic function written in vertex form is in the form:
𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘, where 𝑎 ≠ 0.
a. What does the variable h represent in the vertex for of a quadratic function?
b. What does the variable k represent in the vertex for of a quadratic function?
Example 4: Use a graphing calculator to rewrite each quadratic function. First, determine the vertex of
each and write the function in vertex form. Then, determine the zero(s) of each and write the function in
factored form.
a. 𝑓(𝑥) = 𝑥 2 − 8𝑥 + 12
Vertex:__________________________
Vertex form: __________________________
Zero(s):__________________________
Factored form: _________________________
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Chapter 12
b. 𝑓(𝑥) = −𝑥 2 − 4𝑥
Vertex:__________________________
Vertex form: __________________________
Zero(s):__________________________
Factored form: __________________________
Example 5:
1. Consider the three quadratic functions shown, where 𝑔(𝑥) is the basic function.
- 𝑔(𝑥) = 𝑥 2
- 𝑐(𝑥) = 𝑥 2 + 3
- 𝑑(𝑥) = 𝑥 2 − 3
a. Graph each of the functions on your calculator using the
window x: [-5, 5] and y: [-5, 5].
b. Compare the graphs of 𝑐(𝑥) and 𝑑(𝑥) to the graph of the basic function. What do you notice?
2. Consider the three quadratic functions shown, where 𝑔(𝑥) is the basic function.
-
𝑔(𝑥) = 𝑥 2
𝑗(𝑥) = (𝑥 + 3)2
𝑘(𝑥) = (𝑥 − 3)2
a. Graph each of the functions on your calculator using the
window x: [-5, 5] and y: [-5, 5].
b. Compare the graphs of 𝑗(𝑥) and 𝑘(𝑥) to the graph of the basic function. What do you notice?
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Chapter 12
3. Consider the three quadratic functions shown, where 𝑔(𝑥) is the basic function.
-
𝑔(𝑥) = 𝑥 2
𝑚(𝑥) = −𝑥 2
𝑛(𝑥) = (−𝑥)2
a. Graph each of the functions on your calculator using the
window x: [-5, 5] and y: [-5, 5].
b. Compare the graphs of 𝑚(𝑥) and 𝑛(𝑥) to the graph of the basic function. What do you notice?
c. Describe each of the transformations listed below, in relation to the parent function, 𝑔(𝑥) = 𝑥 2 .
i.
𝑓(𝑥) = 𝑥 2 + 𝑏, 𝑤ℎ𝑒𝑟𝑒 𝑏 > 0
ii. 𝑓(𝑥) = 𝑥 2 + 𝑏, 𝑤ℎ𝑒𝑟𝑒 𝑏 < 0
iii. 𝑓(𝑥) = (𝑥 + 𝑏)2 ,
v. 𝑓(𝑥) = −𝑥 2
𝑤ℎ𝑒𝑟𝑒 𝑏 > 0
iv. 𝑓(𝑥) = (𝑥 + 𝑏)2 ,
𝑤ℎ𝑒𝑟𝑒 𝑏 < 0
vi. 𝑓(𝑥) = (−𝑥)2
4. Consider the three quadratic functions shown, where 𝑔(𝑥) is the basic function.
- 𝑔(𝑥) = 𝑥 2
- 𝑝(𝑥) = 3𝑥 2
-
1
3
𝑞(𝑥) = 𝑥 2
a. Graph each of the functions on your calculator using the
window x: [-5, 5] and y: [-5, 5].
b. Compare the graphs of 𝑝(𝑥) and 𝑞(𝑥) to the graph of the basic function. What do you notice?
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Chapter 12
c. Complete a table of values for 𝑔(𝑥) and sketch the parabola below.
x
f(x)
-2
-1
0
1
2
d. A parabola with no change in the “stretch” has the first movement of:
𝑦 = 𝑥 2 has a horizontal move of ______
then
vertical move of ______
𝑦 = 3𝑥 2 has a horizontal move of ______
then
vertical move of ______ (from the vertex)
then
vertical move of ______ (from the vertex)
1
𝑦 = 3 𝑥 2 has a horizontal move of ______
(from the vertex)
A vertical dilation of a function is a transformation in which the y-coordinate of every point on
the graph of the function is multiplied by a common factor called the dilation factor. A vertical
dilation stretches or shrinks the graph of a function.
e. Describe each of the transformations listed below, in relation to the parent function, 𝑔(𝑥) = 𝑥 2 .
a. 𝑓(𝑥) = 𝑎𝑥 2 , 𝑤ℎ𝑒𝑟𝑒 𝑎 > 1
b. 𝑓(𝑥) = 𝑎𝑥 2 , 𝑤ℎ𝑒𝑟𝑒 0 < 𝑎 < 1
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Chapter 12
Example 6: Use the given characteristics to write a function and sketch the graph of 𝑓(𝑥).
1. Write a function in vertex form and sketch a graph that has these characteristics:
 The function is quadratic.
 The function is continuous.
 The parabola opens upward.
 The function is translated 5 units to the right of
𝑓(𝑥) = 𝑥 2 .
Equation: 𝑓(𝑥) = _____________________
2. Write a function in vertex form and sketch a graph that has these characteristics:
 The function is quadratic.
 The function is continuous.
 The parabola opens downward.
 The function is translated 1 unit down from
𝑓(𝑥) = −𝑥 2 and is vertically dilated with a
dilation factor of 2.
Equation: 𝑓(𝑥) = _____________________
3. Write a function in vertex form and sketch a graph that has these characteristics:
 The function is quadratic.
 The function is continuous.
 The parabola opens downward.
 The function is translated 8 units up and 2 units
to the right of 𝑓(𝑥) = 𝑥 2 .
Equation: 𝑓(𝑥) = _____________________
4. Based on the equation of each function, describe how the graph of each function compares to
the graph of 𝑔(𝑥) = 𝑥 2 .
a. 𝑤(𝑥) = (𝑥 + 2)2
b. 𝑡(𝑥) = 3𝑥 2 + 4
c. 𝑟(𝑥) = −(𝑥 − 1)2 − 10
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Chapter 12
Additional Notes
28