1 quadratic functions - pbworksdustintench.pbworks.com/f/ch.1studenttext.pdf · introduction to...

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Chapter 1 ● Quadratic Functions 29

1

1.1 Lots and ProjectilesIntroduction to Quadratic

Functions ● p. 31

1.2 ParabolasProperties of the Graphs of

Quadratic Functions ● p. 37

1.3 ExtremesIncrease, Decrease, and

Rates of Change ● p. 45

1.4 Solving Quadratic EquationsReviewing Roots and Zeros ● p. 51

1.5 Finding the MiddleDetermining the Vertex of a

Quadratic Function ● p. 65

1.6 Other Forms of Quadratic FunctionsVertex Form of a Quadratic

Function ● p. 77

1.7 Graphing Quadratic FunctionsBasic Functions and

Transformations ● p. 85

Arches are used to support the weight of walls and ceilings in buildings. Arches were first used in

architecture by the Mesopotamians over 4000 years ago. Later, the Romans incorporated arches

into a wide range of structures, and they are commonly used in modern buildings. The arch in the

image above is in the shape of a parabola, which is a graphical representation of a quadratic

function. You will learn the properties and shapes of quadratic functions.

1C HA PT E R

Quadratic Functions

30 Chapter 1 ● Quadratic Functions

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Lesson 1.1 ● Introduction to Quadratic Functions 31

1

Problem 1 Plots and LotsIn a new housing development, every rectangular plot that is laid out must be six feet

longer than it is wide to accommodate a sidewalk and a tree lawn (the area between

the sidewalk and the road). Answer the following questions about this situation.

1. How long or wide would the plot be if the plot is

a. 50 feet wide?

b. 120 feet long?

c. 75 feet long?

2. What would be the area of the plot if the plot is

a. 60 feet wide?

b. 80 feet long?

c. 150 feet long?

3. Define a variable for the width of the plot.

ObjectivesIn this lesson, you will:

● Write quadratic functions.

● Use quadratic functions to model area.

● Use quadratic functions to model vertical

motion.

Key Terms● quadratic function

● vertical motion

1.1 Lots and ProjectilesIntroduction to Quadratic Functions


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4. Write an expression for the length of the plot.

5. Write an equation for the area of the plot.

6. Using this information, complete the following table. Then use the information in

the table to graph the area of the plot versus the width of the plot.

7. Is the graph linear? Explain.

8. If you haven’t done so already, use the distributive property to rewrite the equation

for the area without parentheses.

1 Quantity Name

Unit

Expression

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This equation is an example of a quadratic function. A quadratic function is defined

as any equation of the form

where a, b, and c are real-number constants and a � 0.

Problem 2 Galileo’s DiscoveryGalileo Galilei was a famous scientist who made many contributions in the areas

of astronomy and physics. One of his most important discoveries was verticalmotion––when an object is dropped or falls, the distance it travels is a quadratic

function of the time. Any object thrown, launched, or shot upward can be modeled

by the following equation:

where a is the acceleration from gravity, v0

is the initial upward velocity, s0

is the

initial distance off the ground, and s is the height after t seconds.

1. For instance, a cannon ball is launched directly upward from the ground with an

initial velocity of 320 feet per second. The acceleration due to gravity is 32 feet per

second squared. The following equation models this situation.

2. How high will the cannon ball be after

a. 2 seconds?

b. 10 seconds?

c. 3.1 seconds?

d. 20 seconds?

s � �16t2 � 320t

s � �1

2at2 � v0 t � s0

y � ax2 � bx � c

1


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3. At what time(s) will the cannon ball be

a. 304 feet above the ground?

b. 576 feet above the ground?

c. 2000 feet above the ground?

4. Using the information from Questions 2 and 3, complete the following table.

Then use the information in the table to graph the height of the cannon ball versus

the time.

1

Quantity Name

Unit

Expression

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5. Is this graph linear? Explain.

6. From the graph, can you tell the maximum height that the cannon ball attains?

If so, what is this height and after how many seconds does the cannon ball

reach it?

7. Does this graph make sense based on your own understanding of the path of a

cannon ball? Explain.

Be prepared to share your work with another pair, group, or the entire class.

1


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Lesson 1.2 ● Properties of the Graphs of Quadratic Functions 37

1

1. Complete the table of values for the quadratic function y � x2. Then use the table

to construct a graph of the function.

2. Every quadratic function has a distinctive U-shape. Why?

The graph of a quadratic function is called a parabola. The vertex of a parabola is

the lowest or highest point on the curve. The axis of symmetry is the line that

passes through the vertex and divides the parabola into two mirror images. For the

parabolas we will be looking at in this chapter, the axis of symmetry is a vertical line.

3. Identify the vertex and the axis of symmetry for the graph of y � x2.


● Graph quadratic functions.

● Calculate the vertex, axis of symmetry,

zeros, and intercepts of quadratic functions.

Key Terms● parabola

● vertex

● axis of symmetry

● zeros

1.2 ParabolasProperties of the Graphs of Quadratic Functions

x y

0

1

2

3

�1

�2

�3

Problem 1 Exploring Quadratic Functions


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One way to determine a parabola’s vertex and axis of symmetry is graphically. On

the graph, locate the coordinates of the highest or lowest point. This point is the

vertex. For the parabolas we will be exploring, the axis of symmetry is the vertical

line that passes through the vertex.

The following table and graph are completed for the function y � x2 � 4x. The

coordinates of the vertex and intercepts are shown on the graph.

Vertex: (2, �4) x-intercepts: (0, 0) and (4, 0)

y-intercept: (0, 0) Axis of symmetry: x � 2

4. For each quadratic function, complete the table and sketch a graph.

Then, determine the coordinates of the vertex, x-intercept(s), y-intercept, and the

equation for the axis of symmetry. Label these key characteristics on the graph.

a. y � �x2

Vertex: y-intercept:

x-intercept(s): Axis of symmetry:

1

x y

0

�1

1

2

3

4

5

x1086

8

10

12

6

–2

–4

–2–4–6

y

(0, 0) (4, 0)

(2, –4)

x = 2 y = x

2 – 4xx y

0 0

1 �3

2 �4

�1 5

4 0

�2 12

5 5

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b. f(x) � x2 � 4x � 3

Vertex:

x-intercept(s):

y-intercept:

Axis of symmetry:

c. f(x) � �x2 � 4x

Vertex:

x-intercept(s):

y-intercept:

Axis of symmetry:

1

x y

x y


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d. y � x2 � 3x � 2

Vertex:

x-intercept(s):

y-intercept:

Axis of symmetry:

e. y � x2 � 4x � 3

Vertex:

x-intercept(s):

y-intercept:

Axis of symmetry:

1

x y

x y

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5. The x-intercepts of a quadratic function are also called zeros. Why?

6. The standard form of the quadratic function is y � ax2 � bx � c. Use your graphs

from Question 4 to answer the following questions.

a. How does the sign of a affect the graph of a quadratic function? Explain.

b. What does the value of c determine in the graph of a quadratic function?

Explain.

c. How is the x-value of the vertex related to the x-intercepts? Explain.

7. For each given axis of symmetry and point on a parabola, determine another point

on the parabola.

a. Axis of symmetry x � 2; given point (0, 5):

b. Axis of symmetry x � �5; given point (�7, 3):

c. Axis of symmetry x � ; given point (�2, 7):

d. Axis of symmetry x � 5; given point (10, 0):

1

2

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Problem 2 Key Characteristics ofQuadratic Functions

1. As we have seen, the graph of a quadratic function y � ax2 � bx � c is a parabola.

Some characteristics of parabolas include the vertex, intercepts, and the axis of

symmetry.

For each question, use the information provided to determine the remaining

characteristics, complete the table, and sketch a graph, if possible.

a. The vertex of a parabola is (�2, 4) and it passes through the point (0, 0).

Axis of symmetry: x-intercept(s):

y-intercept:

b. A parabola passes through the points (0, 4) and (8, 4).


y-intercept: Vertex:

1

x y

�2 4

0 0

x y

0 4

8 4

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c. The vertex of a parabola is (–6, –4) and it passes through the point (–8, 0).


y-intercept:


1

x y

�6 �4

�8 0


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Lesson 1.3 ● Increase, Decrease, and Rates of Change 45

1

Problem 1 Calculating Rates of ChangeConsider the table and graph for the quadratic function y � x2.

Earlier you learned that linear functions have a constant rate of change. This rate of

change, or slope, is calculated by dividing the vertical change by the horizontal

change: .m ��y�x


● Define extreme points.

● Determine intervals of increase and

decrease.

● Calculate rates of change.

Key Terms● second difference

● extreme points

● intervals

● open interval

● closed interval

● half-closed or half-open interval

1.3 ExtremesIncrease, Decrease, and Rates of Change

x y

0

1

2

3

�1

�2

�3


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2. What can you conclude about the rate of change of y � x2? Explain.

3. Consider the table and graph for the quadratic function y � �x2.

1

x y � x2 x� y� m ��y�x

0 0

1 1 1 1 1

2

3

x y � x2 x� y� m ��y�x

0 0

�1 1 �1 1 �1

�2

�3

x y

0

�1

1

2

3

4

5

1. Complete each table by calculating the unit rate of change, or slope, between each

pair of points on the graph of y � x2.

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4. Complete each table by calculating the unit rate of change, or slope, between each

pair of points on the graph of y � �x2.

5. What can you conclude about the rate of change of y � �x2? Explain.

6. The graphs of y � x2 and y � �x2 both have a vertex at the point (0, 0). Describe

the change in the rate of change from one side of the vertex to the other.

1

x y � �x2 x� y� m ��y�x

0 0

1 1 1 1 1

2

3

x y � �x2 x� y� m ��y�x

0 0

�1 1 �1 1 �1

�2

�3


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7. The unit rate of change in y for a function is called a first difference. The unit rate

of change in the first difference for a function is called a second difference.

Complete each table by calculating the first and second differences for y � x2.

8. Complete each table by calculating the first and second differences for y � �x2.

1

x y � x2 y� ( y)��

0 0

1 1 1

2 4 3 2

3 9 5

x y � �x2 y� ( y)��

0 0

1 �1 �1

2 �4 �3 �2

3 �9 �5

x y � x2 y� ( y)��

0 0

�1 1 1

�2 4 3 2

�3 9 5

x y � �x2 y� ( y)��

0 0

�1 �1 1

�2 �4 3 �2

�3 �9 5

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9. What conclusion can you make about the second difference of a quadratic

function? Explain.

Problem 2 Extreme Points1. Consider the graphs of quadratic functions of the form y � ax2 � bx � c

with positive a values.

a. Describe how the vertex relates to all the other points on the parabola.

b. Describe how the values of y change on each side of the vertex.

2. Consider the graphs of quadratic functions of the form y � ax2 � bx � c with

negative a values.

a. Describe how the vertex relates to all the other points on the parabola.

b. Describe how the values of y change on each side of the vertex.

A maximum or a minimum point on a graph, such as the vertex of a parabola, is

called an extreme point. To locate extreme points, it helps to use intervals.

An interval is defined as the set of real numbers between two given numbers.

The following notation is used for intervals:

• An open interval (a, b) is the set of all numbers between a and b, but not

including a or b.

• A closed interval [a, b] is the set of all numbers between a and b, including

a and b.

• A half-closed or half-open interval (a, b] is the set of all numbers between aand b, including b but not including a.

• A half-closed or half-open interval [a, b) is the set of all numbers between aand b, including a but not including b.

Intervals that are unbounded can be written using the symbol for infinity, . For

instance, the interval [a, ) means all numbers greater than or equal to a.�

�

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3. In Problem 1 Question 1, the function y � x2 is decreasing over the interval ( , 0)

and increasing over the interval (0, ).

Complete the table and graph and determine the intervals over which each

quadratic function is increasing and over which it is decreasing.

a. f(x) � �x2 � 4x

b. y � x2 � 3x � 2

Be prepared to share your methods and solutions with the class.

�

�

1

x y

x y

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Lesson 1.4 ● Reviewing Roots and Zeros 51

1Problem 1 Solving Quadratic Equations by

FactoringThe solutions or roots of a quadratic equation are the same as the

x-intercepts or zeros of a quadratic function .

One method for calculating the roots of a quadratic equation or the zeros of a

quadratic function is by using factoring:

• If a function is given, set the function equal to zero.

• If an equation is given, perform transformations so that one side of the equation

is equal to zero.

• Factor the quadratic expression on the other side of the equation.

• Set each factor equal to zero.

• Solve each resulting equation for the roots or zeros.

For example, solve using factoring by performing the following steps:

or

or

or

Check:

x2 � 4x � (1)2 � 4(1) � 1 � 4 � �3

x2 � 4x � (3)2 � 4(3) � 9 � 12 � �3

x � 1x � 3

x � 1 � 1 � 0 � 1x � 3 � 3 � 0 � 3

(x � 1) � 0(x � 3) � 0

(x � 3) (x � 1) � 0

x2 � 4x � 3 � 0

x2 � 4x � 3 � �3 � 3

x2 � 4x � �3

x2 � 4x � �3

f(x) � ax2 � bx � cax2 � bx � c � 0


● Solve quadratic equations using factoring.

● Solve quadratic equations by extracting square roots.

1.4 Solving Quadratic EquationsReviewing Roots and Zeros


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1. Calculate the roots of .

2. Calculate the roots of .

3. Calculate the roots of .x2 � 5x � 13x � 81

x2 � 11x � 30

x2 � 8x � �7

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4. Calculate the zeros of .

5. Calculate the zeros of .

6. Calculate the roots of .x2 � 7x � 60

f(x) � 5x2 � 45x

y � x2 � 12x � 45

1

2x 3

5x

2


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Problem 2 Multiplication of BinomialsThe functions and equations in Problem 1 all had a coefficient of 1 on the x2 term.

Factoring when the coefficient of the x2 term is not 1 is more complex.

Before factoring quadratic expressions with a coefficient other than 1 on the x2 term,

let’s review multiplication of binomials.

For example, you can multiply using the distributive property.

Alternatively, you can multiply using a multiplication table.

You can also multiply using an area model.

1. Perform each multiplication using the distributive property.

a.

b. (8x � 3) (�5x � 11)

(�7x � 4) (�3x � 5)

10x2 � 19x � 6

10x2 � 4x � 15x � 6

2x(5x � 2) � 3(5x � 2)

(2x � 3) (5x � 2)

(2x � 3) (5x � 2)

1

• 2x 3

5x 10x2 15x

2 4x 6

10x2 15x

4x 6

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2. Perform each multiplication using a multiplication table.

a.

b.

3. Perform each multiplication using an area model.

a.

b.

4. Perform each multiplication using any method.

a.

b.

c.

d. (3x � 7) (3x � 7)

(9x � 4) (3x � 1)

(�x � 9) (3x � 1)

(�5x � 7) (4x � 11)

(�3x � 4) (3x � 7)

(2x � 5) (3x � 10)

(13x � 12) (�5x � 9)

(�x � 7) (�9x � 10)

1

•

•


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Problem 3 Factoring When a � 1In Problem 2, we saw that .

Remember that the general form of a quadratic is with factors

. Answer the following questions.

1. Complete the multiplication table for the product .

2. Use the general form of a quadratic equation, , to write an equivalent

expression for each value.

a. a

b. c

c. b

Knowing the relationship between the coefficients of the quadratic equation and the

coefficients of the binomial factors can help to factor.

To factor we know the following:

• The coefficient of the x2 term, 4, is the product of d and f, the coefficients of the

x terms of the binomials.

• The constant term, 15, is the product of e and g, the constant terms of the

binomials.

• The product ac is equal to edfg, the product of all the coefficients in the factors.

3. The coefficient of the x term, b, is the sum of a pair of factors of ac. Why?

4x2 � 23x � 15

ax2 � bx � c

(dx � e) (fx � g)

(cx � d) (ex � f )ax2 � bx � c

(2x � 3) (5x � 2) � 10x2 � 19x � 6

1

• 2x 3

5x 10x2 15x

2 4x 6

•

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4. Calculate ac for this expression and list all the possible factor pairs for this product.

Remember to include negative factor pairs.

ac �

Factor pairs:

5. Which of these factor pairs has a sum that is equal to b?

6. List all the factors of 4 and 15.

7. Identify the factors of 4 and the factors of 15 that, when multiplied, result in a

product of 3 and 20.

8. Complete the multiplication table for .

9. Use the process in Questions 4 through 8 to factor each quadratic expression.

a.

ac �

Factor pairs of ac:

Pair with sum of �41:

Factors of 5:

Factors of 8:

5x2 � 41x � 8

4x2 � 23x � 15

Factors of 15:

Factors of 4:

1

•

4x2

15


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Factors of 5 and 8 that produce the desired products of �1 and �40:

b.

ac �

Factor pairs of ac:


Factors of 6:

Factors of �10:

Factors of 6 and �10 that produce the desired products of 4 and �15:

6x2 � 11x � 10 �

6x2 � 11x � 10

5x2 � 41x � 8 �

1

•

5x2

8

•

6x2

�10

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10. Use the process in Questions 4 through 8 to solve each quadratic equation.

a.

ac �

Factor pairs of ac:


Factors of 3:

Factors of 7:

Factors of 3 and 7 that produce the desired products of �1 and �21:

3x2 � 22x � 7 � 0

1

•


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b.

ac �

Factor pairs of ac:

Pair with sum of 2:

Factors of 8:

Factors of �21:

Factors of 8 and �21 that produce the desired products of �12 and 14:

8x2 � 2x � 21 � 0

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Problem 4 Solving for a Perfect SquareA second method for calculating the roots of a quadratic equation or the zeros of a

quadratic function is by solving for a perfect square and then extracting the square

roots.

For example, solve using perfect squares by performing the following

steps:

Check:

1. Solve each equation by solving for a perfect square and then extracting the square

roots.

a. �3x2 � �147

5(�3)2 � 45 � 45 � 45 � 0

x � �3

�x2 � ��9

x2 � 9

5x2

5�

45

5

5x2 � 45 � 45 � 0 � 45

5x2 � 45 � 0

5x2 � 45 � 0

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b.

c. 3(x � 3)2 � 219 � �18x

x(x � 5) � 144 � 5x

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d.

Be prepared to share your methods and solutions.

5x2 � 35

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Lesson 1.5 ● Determining the Vertex of a Quadratic Function 65

1

Problem 1 Exploring the VertexIn previous activities, we found the coordinates of the vertex from the graph of a

quadratic function. We also explored the importance of the vertex as a maximum or

minimum point and in determining intervals of increase and decrease.

1. Graph each of the following quadratic functions using a graphing calculator and

sketch each on the grid.

a. y � x2

b. y � 2x2 � 2

c. y � 3x2 � 4


● Determine the vertex of a parabola given the equation of a quadratic function.

● Determine the vertex for the standard form of the quadratic function.

1.5 Finding the MiddleDetermining the Vertex of a Quadratic Function


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2. Graph each of the following quadratic functions using a graphing calculator and

sketch each on the grid.

a. y � �x2

b. y � �3x2 � 2

c. y � �2x2 � 4

3. All of the functions in Questions 1 and 2 are in the form y � ax2 � c. What do you

notice about the relationship between the vertex and axis of symmetry of the

parabola and the equation of the function?

4. What are the coordinates of the vertex for a quadratic function in the form

y � ax2 � c with a 0? Is the vertex a maximum or a minimum? Explain.

5. What are the coordinates of the vertex for a quadratic function in the form

y � ax2 � c with a 0? Is the vertex a maximum or a minimum? Explain.

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Problem 2 The Vertex and Other KeyCharacteristics

For a quadratic function in the form y � ax2 � bx� c with b � 0, determining the

coordinates of the vertex and the equation of the line of symmetry is more difficult.

1. Consider the quadratic function y � x2 � 4x � 3. Complete the table for the

function. Then, graph the quadratic function using a graphing calculator and

sketch the graph on the grid.

a. What is the vertex of the function y � x2 � 4x � 3? Explain how you determined

the coordinates.

1x y

1

0

�1

�2

�3

�4

�6


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b. What is the y-intercept of this function?

c. What are the x-intercepts of this function?

d. What is the equation of the axis of symmetry of this function?

The axis of symmetry divides the parabola into two halves that are mirror images of

each other. Every point of the parabola on one side of the axis of symmetry has a

symmetric point on the other side of the axis of symmetry. These symmetric points are

equidistant from the axis of symmetry.

e. The points (1, 8) and (�6, 15) are points on the parabola. Determine the point on

the parabola that is symmetric to each.

f. Explain how you determined these symmetric points.

g. Verify that the points you identified in Question 1(e) are on the parabola by

substituting the coordinates of each into the function.

h. What do you notice about the y-coordinates of each pair of symmetric points?

i. If you draw a line segment connecting each pair of symmetric

points, the midpoint of these segments lies on the axis of

symmetry. Why?

1

Take NoteThe midpoint of a segment

with endpoints (x1, y1

) and

(x2, y2

) is .(x1 � x2

2, y1 � y2

2)

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j. Use the midpoint formula to calculate the midpoint of the line segments

connecting each pair of symmetric points.

k. What is the equation for the axis of symmetry? How do you know?

2. Explain how to calculate the equation for the axis of symmetry if you know the

coordinates of two symmetric points on the parabola.

3. Calculate the equation for the axis of symmetry using each pair of symmetric

points.

a. (4, 8) and (10, 8)

b. (–5, �6) and (�2, �6)

c. (d, e) and (f, e)

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4. Consider the quadratic function . Complete the table for the

function. Then, graph the quadratic function using a graphing calculator and sketch

the graph on the grid.

a. Use the graph to estimate the coordinates of the vertex.

b. Based on the estimate for the vertex, what is the equation for the axis of

symmetry?

c. What are the x-intercepts of this function?

d. Calculate the equation for the axis of symmetry of ?y � 2x2 � 7x � 3

y � 2x2 � 7x � 3

1

x y

1

0

�1

�2

�3

�4

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e. How does the calculated value compare to the estimated value using the graph?

f. Explain why it is difficult to use a graph to determine exact values for the vertex

and axis of symmetry.

Problem 3 Calculating the VertexCoordinates

1. Consider the function .

a. What is the y-intercept?

b. The y-intercept has a symmetric point on the parabola that has the same

y-coordinate. Substitute the y-coordinate of the y-intercept into the equation and

determine the coordinates of the point symmetric to the y-intercept.

y � 2x2 � 7x � 3

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c. Use these symmetric points to calculate the equation of the axis of symmetry for

this function.

d. The vertex lies on the axis of symmetry, so the x-coordinate of the vertex is the

x-coordinate of the axis of symmetry. You can calculate the y-coordinate of the

vertex by evaluating the function for this x-coordinate. Calculate the vertex

coordinates for .

2. Calculate the vertex for each of the following quadratic functions.

a.

y-intercept:

Coordinates of the point symmetric to the y-intercept:

y � x2 � 4x � 3

y � 2x2 � 7x � 3

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b.

y-intercept:


y � x2 � 7x � 6

1


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c. y � x2 � 8x � 15

y-intercept:


1

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d.

y-intercept:


y � �3x2 � 2x � 5

1


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e.

y-intercept:


Be prepared to share your solutions and methods.

y � ax2 � bx � c

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Lesson 1.6 ● Vertex Form of a Quadratic Function 77

1

Problem 1 Vertex and Axis of SymmetryFor each quadratic function:

a. Calculate the equation for the axis of symmetry.

b. Calculate the coordinates of the vertex.

c. Complete the table by including the x-coordinate of the vertex and the

x-coordinates one and two units to the left and right of the vertex.

d. Sketch a graph of the function.

1. f(x) � x2 � 8x + 9

a. Axis of symmetry:

b. Vertex:

c. Table:

ObjectiveIn this lesson, you will:

● Write quadratic functions in vertex form.

Key Term● vertex form of a quadratic

function

1.6 Other Forms of QuadraticFunctionsVertex Form of a Quadratic Function

x y

Vertex:


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d. Graph:

2. f(x) � �2x2 � 12x + 7

a. Axis of symmetry:

b. Vertex:

c. Table:

d. Graph:

1

Vertex:

x y

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Problem 2 Different Forms of QuadraticFunctions

For each quadratic function, complete the table and sketch a graph.

1. f(x) � (x � 4)2 � 7

2. f(x) � �2(x � 3)2 � 25

3. Compare the tables and graphs from Problem 1 Question 1 and Problem 2

Question 1. What do you notice?

4. Simplify the expression .(x � 4)2 � 7

1

x y

2

3

4

5

6

x y

2

0

�2

�4

�6

�8


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5. What can you conclude about the functions from Problem 1 Question 1 and

Problem 2 Question 1?

6. Compare the tables and graphs from Problem 1 Question 2 and Problem 2

Question 2. What do you notice?

7. Simplify the expression �2(x � 3)2 + 25.

8. What can you conclude about the functions from Problem 1 Question 2 and

Problem 2 Question 2?

Problem 3 Working with Standard FormIn Problem 1, the quadratic functions are written in standard form, f(x) � ax2 � bx + c.

1. Consider the function f(x) � x2 � 8x � 9.

a. Identify the values of the constants a, b, and c.

b. What information does the value of a provide about the graph of the function?

c. What information does the value of b provide about the graph of the function?

d. What information does the value of c provide about the graph of the function?

In Problem 2 the quadratic functions are written in the form f(x) � a(x � h)2 + k.

2. Consider the function f(x) � (x � 4)2 � 7.

a. Identify the values of the constants a, h, and k.

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b. What information does the value of a provide about the graph of the function?

c. What information does the value of h provide about the graph of the function?

d. What information does the value of k provide about the graph of the function?

The form f(x) � a(x � h)2 + k is called the vertex form of a quadratic function.

Problem 4 Converting between Vertex andStandard Forms

To convert from vertex form to standard form, simplify the quadratic expression in the

function definition.

For example, convert to standard form:

1. Convert each quadratic function from vertex form to standard form.

a. f(x) � 2(x � 8)2 + 1

b. f(x) � �4(x � 3)2 + 7

� �5x2 � 40x � 65

� �5x2 � 40x � 80 � 15

� �5(x2 � 8x � 16) � 15

f(x) � �5(x � 4)2 � 15

f(x) � �5(x � 4)2 � 15

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c. f(x) � �4 � 7

To convert from standard form to vertex form, calculate the coordinates of the vertex.

Use the vertex to rewrite the function.

For example, to convert f(x) � 3x2 � 12x � 5 to vertex form:

vertex: (2, �17)

2. Convert each quadratic function from standard form to vertex form.

a. f(x) � x2 � 8x � 6

b. f(x) � 2x2 � 12x � 13

f(x) � 3(x � 2)2 � 17

f(2) � 3(2)2 � 12(2) � 5 � 12 � 24 � 5 � �17

x � �b2a

� ��12

2(3)� 2

( x �3

2 )2

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c. f(x) � �5x2 � 6x � 3

Be prepared to share your methods and solutions.

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Lesson 1.7 ● Basic Functions and Transformations 85

1


● Graph the basic quadratic function.

● Transform the graph of the quadratic

basic function.

● Dilate the graph of the basic quadratic

function.

Key Term● basic quadratic function

1.7 Graphing QuadraticFunctionsBasic Functions and Transformations

Problem 1 Basic Function1. The basic quadratic function is Graph this function. y � x2.

2. On the same grid, graph the following functions.

a.

b.

3. What do you notice about these three graphs? Explain.

y � x2 � 3

y � x2 � 4


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4. For each part of Question 2, describe how the graph and the equation have been

transformed from the basic function.

5. The basic quadratic function is Graph this function.


a.

b.


8. For each part of Question 6, describe how the graph and the equation have been

transformed from the basic function.

9. Factor:

a. Calculate the following values of and

i. iv.

ii. v.

iii. vi. g(�5) �f(2) �

f(�3) �g(�2) �

g(0) �f(0) �

g(x) � (x � 2)2.f(x) � x2

y � x2 � 4x � 4 �

y � x2 � 4x � 4 �

y � x2 � 4x � 4

y � x2 � 4x � 4

y � x2.

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b. Calculate the following values of and

i. iv.

ii. v.

iii. vi.

c. Describe how the factors (x � 2) and (x � 2) are related to the transformations

you described in Question 8.



a.

b.


13. For each part of Question 11, describe how the graph and the equation have

been transformed from the basic function.

y �1

2x2

y � 2x2

y � x2.

h(�1) �f(2) �

f(�3) �h(2) �

h(4) �f(0) �

h(x) � (x � 2)2.f(x) � x2

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a.

b.


17. For each part of Question 15, describe how the graph and the equation have

been transformed from the basic function.

In each of the previous graphs of quadratic functions, one or more of the four

following transformations were performed on the quadratic basic function:

A. Vertical shift

B. Horizontal shift

C. Reflection

D. Dilation

18. Which one of these transformations changed the “shape” of the parabola? Explain.

y � �2x2 � 8x � 8

y � �x2 � 2x � 1

y � x2.

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19. Given that only one transformation changed the shape of the parabola, which

of the coefficients of the standard quadratic function, y � ax2 � bx � c,

determines the “shape” of the parabola? Why?

Remember that the zeros of a quadratic function can be determined by using the

quadratic formula and that the x-value of the vertex of the function is the average of

the zeros.

Problem 2For each of the following, determine the vertex. First determine the x-value by

calculating the average of the zeros. Then use substitution to determine the y-value.

1. Vertex: y � x2 � 10x � 24

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4. Vertex:

5. Use the vertices you calculated in Questions 1 to 4 and your knowledge of the

shape determined by the value of a to graph each of the following functions.

a. Vertex: y � x2 � 10x � 24

y � 2x2 � 12x � 7

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b. Vertex:

c. Vertex:

d. Vertex: y � 2x2 � 12x � 7

y � x2 � 5x � 4

y � x2 � 5x � 4

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6. Using the vertices you have already calculated in Questions 1–4, graph the

following functions. Then describe the graphical transformations that can be used

to transform the basic function to each function.

a. Vertex:

Graphical transformations:

b. Vertex:


y � x2 � 5x � 4

y � x2 � 10x � 24

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7. Using the Quadratic Formula, derive a general formula for the average of the zeros

in terms of a, b, and c.

8. Using your result from Question 7, what is the x-coordinate of the vertex of

?


y � ax2 � bx � c

1

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