3 Quadratic Functions...Aug 11, 2009 · Lesson 3.2 Quadratic Equations and Functions 143 3 2....

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3.1 Lots and Projectiles

Introduction to Quadratic

Functions ● p. 135

3.2 Intercepts, Vertices, and Roots

Quadratic Equations and

Functions ● p. 141

3.3 Quadratic Expressions

Multiplying and Factoring ● p. 147

3.4 More Factoring

Special Products and Completing

the Square ● p. 159

3.5 Quadratic Formula

Solving Quadratic Equations Using the

Quadratic Formula ● p. 169

3.6 Graphing Quadratic Functions

Properties of Parabolas ● p. 177

3.7 Graphing Quadratic Functions

Basic Functions and

Transformations ● p. 185

3.8 Three Points Determine a Parabola

Deriving Quadratic Functions ● p. 197

3.9 The Discriminant

The Discriminant and the Nature

of Roots/Vertex Form ● p. 203

The Chinese invented rockets over 700 years ago. Since then rockets have been used to propel

fireworks, weaponry, cargo, and even humans off the Earth’s surface. You will learn to use a

quadratic function to model the height of a launched object over time.

3C HA PT E R

Quadratic Functions

Chapter 3 ● Quadratic Functions 133

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134 Chapter 3 ● Quadratic Functions

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Lesson 3.1 ● Introduction to Quadratic Functions 135

3

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

3.1 Lots and ProjectilesIntroduction to Quadratic Functions

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.

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

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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.

Quantity Name

Unit

Expression

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.

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.

y � ax2 � bx � c

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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 vertical

motion––when an object is dropped or falls, the distance it travels is a function of

the time squared. 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. A more thorough

development and history of Galileo’s discovery of vertical motion can be found at

www.carnegielearning.com/Galileo’sDiscovery.

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?

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

a. 304 feet above the ground?

b. 576 feet above the ground?

s � �16t2 � 320t

s � �1

2at2 � v0 t � s0

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c. 2000 feet above the ground?

4. Using the information from Question 3, complete the following table. Then use the

information in the table to graph the height of the cannon ball versus the time.

Quantity Name

Unit

Expression

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?

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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.

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Lesson 3.2 ● Quadratic Equations and Functions 141

3


● Graph quadratic functions.

● Identify vertices of quadratic functions.

● Identify x- and y-intercepts of quadratic

functions.

Key Terms● vertex

● zero

● quadratic equation

● root

3.2 Intercepts, Vertices,and RootsQuadratic Equations and Functions

Problem 1 Intercepts and Vertices

x

0

1

�2

3

�4

y

1. For each of the following quadratic functions, complete the table and graph the

function.

a. y � x2

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Chapter 3 ● Quadratic Functions

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b. y � x2 � 2x

x

0

�1

1

�2

3

�4

y

c. y � �x2 � x

x

0

1

�2

3

�4

1

2

y

d. y � x2 � 4x � 3

x

0

�1

2

3

4

y

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2. Describe the shape of the graphs in Question 1.

3. For each graph in Question 1, solve for the y-intercept, the x-intercept(s), and the

lowest or highest point in the graph, called the vertex. The vertex represents the

maximum or minimum value of y. The plural of vertex is vertices.

a. y-intercept:

x-intercept(s):

Vertex:

b. y-intercept:

x-intercept(s):

Vertex:

c. y-intercept:

x-intercept(s):

Vertex:

d. y-intercept:

x-intercept(s):

Vertex:

4. Each x-intercept is called a zero of the function. Why?

5. How many zeros can a quadratic function have? Explain.

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Problem 2 Quadratic EquationsA quadratic equation is defined as any equation of the form

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

1. How does a quadratic equation relate to the quadratic function with the same

constants a, b, and c?

2. A solution to a quadratic equation is called a root. How are the roots of an

equation related to the zeros of a function? Explain.

3. Here is one method for determining roots. First, factor the quadratic equation.

Then, use the fact that if a product is zero, then one or both of the factors must be

zero. For instance, if

then

So, the roots are 0 and 2. For each of the following, factor the quadratic equation

and determine the roots or solutions.

a.

b. x2 � 4 � 0

x2 � 5x � 0

x � 0 or x � 2

x � 0 or x � 2 � 0

x(x � 2) � 0

x2 � 2x � 0

0 � ax2 � bx � c

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c.

d.

e.

f.


2

3x2 �

5

6x � 0

2x2 � 4x � 0

x2 � 5x � 6 � 0

4x � x2 � 0

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Lesson 3.3 ● Multiplying and Factoring 147

3


● Factor quadratic trinomials.

● Multiply binomials.

● Solve quadratic equations using factoring.

Key Terms● monomial

● binomial

● trinomial

● difference of two squares

3.3 Quadratic ExpressionsMultiplying and Factoring

Problem 1 Multiplying Quadratics: Monomials,Binomials, and Trinomials

From the definition of a quadratic expression,

where a, b, and c are real-number constants and a � 0, a quadratic expression can be

● One term, a monomial:

● Two terms, a binomial:

● Three terms, a trinomial: .

As we have seen, quadratic expressions can be used to model problem situations,

including area problems and vertical motion problems. Quadratic functions produce

graphs that are different from linear functions. In the last activity, you solved quadratic

equations by

● Setting the quadratic expression equal to zero.

● Factoring the expression into two linear expressions.

● Using the fact that if , then or (or both a and b equal 0).

● Setting each factor to zero and solving for the variable.

The following uses these steps to determine the roots of the quadratic expression

2x2 � 6x.

or

x � 0, �3

x � �3x � 0

x � 3 � 3 � 0 � 32x

2�

0

22x(x � 3) � 0

x � 3 � 02x � 02x2 � 6x � 0

b � 0a � 0ab � 0

ax2 � bx � c

ax2 � bx or ax2 � c

ax2

ax2 � bx � c

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Quadratic equations of the form can be solved by isolating and then

taking the square root of both sides:

However, not all quadratic expressions are as easily factored into two linear factors.

In order to understand when and how to factor quadratic expressions, we will first

examine in detail the multiplication of linear expressions.

1. For each of the following, use the distributive property to calculate the product of

the two linear expressions, a monomial times a binomial.

a.

b.

c.

d.

e.

f.

g.

When calculating the product of two binomials, there are a number of methods that

are commonly used:

● Applying the distributive property twice:

Distribute (x � 3) over x � 4

Distribute x over (x � 3) and �4 over (x � 3)

Combine like terms

x2 � x � 12

x2 � 3x � (�4x) � (�12)

(x � 3)x � (x � 3) (�4)

(x � 3) (x � 4)

2

3x ( 3

4x �

7

6 ) �

3.2x(3x � 5.3) �

4x(�2x � 7) �

�3x(x � 9) �

�x(x � 4) �

2x(x � 3) �

x(x � 7) �

x � �2

�x2 � ��4

x2 � 4

3x2

3�

12

3

3x2 � 12

3x2 � 12 � 12 � 0 � 12

3x2 � 12 � 0

x2ax2 � c � 0

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● Using FOIL (First, Outer, Inner, Last):

● Using a multiplication table:

(x � 3) (x � 4) � 12

)Last

(x � 3) (x � 4) 3x

)Inner

x2 � 4x � 3x � 12 � x2 � x � 12(x � 3) (x � 4) � 4x

)Outer

(x � 3) (x � 4) x2

)First

x 3

x 3x

�4 �4x �12

x2

Combine like terms:

Each of these methods has its advantages and disadvantages. The FOIL

method has the least applicability, as it only works when multiplying binomials.

2. For each of the following calculate the product.

a. (x � 4) (x � 4)

x2 � x � 12

b. (x � 5) (x � 6)

∂

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c. (2x � 3) (x � 6)

d. (�3x � 5) (x � 7)

e. (2x � 7)2

f. ( 2

3x �

2

5 ) ( 3

4x � 2 )

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g. (2.1x � 4.3) (5.1x � 3.2)

3. How is the answer to part (a) of Question 2 different from the other answers to

Question 2? Explain why this occurred.

4. This product is called the difference of two squares. Why?

Problem 2 Factoring QuadraticsUnderstanding the multiplication of two linear expressions to produce a quadratic

expression is necessary for understanding the use of factoring to solve a quadratic

equation. For instance, we know the following:

So, if we want to solve the equation

we factor the expression, set each factor equal to zero, and then solve for the

variable.

or

The whole procedure depends on being able to factor the quadratic expression.

One method for factoring a trinomial is trial and error using the factors of the leading

term, , and the constant term, c:

Factor

Factors of the leading term:

�2x, �x

2x, x

2x2 � 3x � 5.

ax2

x � �6 x � �5

x � 6 � 6 � 0 � 6 x � 5 � 5 � 0 � 5

x � 6 � 0 x � 5 � 0

(x � 5) (x � 6) � 0

x2 � 11x � 30 � 0

x2 � 11x � 30 � 0

(x � 5) (x � 6) � x2 � 11x � 30

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correct2x2 � 3x � 5

Factors of the constant term:

Substitute combinations of each of these factor pairs into the binomials to

determine the combination that provides the correct middle term.

correct

You can also factor a trinomial using the multiplication table model presented earlier:

Start by writing the leading term and the constant term in the table.

(2x � 5) (x � 1) � 2x2 � 3x � 5

(2x � 5) (x � 1) � 2x2 � 3x � 5

(2x � 1) (x � 5) � 2x2 � 9x � 5

(2x � 1) (x � 5) � 2x2 � 9x � 5

5, �1

�5, 1

•

�5

2x2

• x

2x

�5

2x2

• x �1

2x �2x

5 5x �5

2x2

• x 1

2x 2x

�5 �5x �5

2x2

Factor the leading term.

Experiment with factors of the constant term until you find the pair that produces

the correct middle term.

2x2 � 3x � 5

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• x �5

2x �10x

1 x �5

2x2

2x2 � 9x � 5

• x 5

2x 10x

�1 �x �5

2x2

1. Factor each of the following quadratic expressions.

a. x2 � 9x � 10

2x2 � 9x � 5

b. x2 � 9x � 20

c. 3x2 � 8x � 3

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d. 4x2 � 25

2. Factor and solve each of the following quadratic equations.

a. x2 � 8x � 12 � 0

b. �x2 � 5x � 4 � 0

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c. x2 � 16 � 0

d. x2 � x � 30 � 0

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33. Determine the zeros of each of the following quadratic functions.

a. x2 � 8x � y

e. x2 � 5 � 0

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b. y � 2x2 � 7x � 3

c. y � x2 � 8x � 16

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d. x2 � 11x � 42 � y

e. 6x2 � 54 � y

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Lesson 3.4 ● Special Products and Completing the Square 159


● Factor perfect square trinomials.

● Factor the difference of two squares.

● Factor expressions using the largest

monomial.

● Solve quadratics by completing

the square.

Key Terms● perfect square trinomial

● completing the square

3.4 More FactoringSpecial Products and Completing the Square

Problem 1 Special Products1. Multiply each of the following pairs of binomials.

a.

b.

c.

d.

e.

2. Factor each of the following quadratic expressions.

a.

b.

c. x2 � 10x � 25

x2 � 25

x2 � 4x � 4

(x � 7)2

(x � 6) (x � 6)

(x � 3) (x � 3)

(x � 4) (x � 4)

(x � 4) (x � 4)

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d.

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In Questions 1 and 2, you should have observed a couple of special products. The

first, which you learned previously, is called the difference of two squares. The other

is called a perfect square trinomial.

3. Identify which expressions in Questions 1 and 2 are examples of the difference of

two squares. Write both the unfactored and factored forms of each expression.

4. Identify which expressions in Questions 1 and 2 are examples of a perfect square

trinomial. Write both the unfactored and factored forms of each expression.

The equation containing the difference of two squares can be solved by

● Factoring.

● Isolating the squared variable and taking the square root of each side.

Solve each of the following equations both ways.

a. x2 � 81 � 0

4x2 � 49

x2 � 4x � 4

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b.

The process for solving perfect square trinomials is similar. Perfect square trinomials

can be solved by

● Factoring and setting each factor equal to zero if the equation is equal

to zero.

● Factoring the perfect square on one side, taking the square roots of both sides,

and then solving for the variable.

7. For the following equations, solve by factoring and setting each factor equal

a. x2 � 12x � 36 � 0

9x2 � 25 � 0

3

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b. 4x2 � 12x � 9 � 0

Problem 2 A Method for Solving Any QuadraticEquation: Completing the Square

1. Factor and calculate the zeros of each of the following quadratic functions.

a.

b.

c.

d.

2. Were you able to determine the zeros of each function in Question 1 by factoring?

Explain.

x2 � 5x � 2 � y

x2 � 4x � 5 � y

x2 � 5x � 4 � y

x2 � 4x � 2 � y

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3. Graph the following quadratic function.

x2 � 4x � 2 � y

4. Is it possible to calculate the zeros of this function by factoring? Does this function

have zeros? Explain.

This quadratic function has zeros but cannot be factored, so we must find another

method for calculating the zeros of a quadratic function or solving a quadratic

equation. We can construct a procedure for solving any quadratic equation by using

our understanding of the relationship among the coefficients of a perfect square

trinomial.

5. For each of the following perfect square trinomials, describe how the coefficient

of the middle term, b, is related to the constant term, c.

a.

b.

c.

d. x2 � 10x � 25 � 0

x2 � 6x � 9 � 0

x2 � 12x � 36 � 0

x2 � 20x � 100 � 0

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6. For each of the following, write the value of b or c that would make the

expression a perfect square trinomial.

a.

b.

c.

d.

We can use our understanding of the relationship among the coefficients of a perfect

square trinomial to construct a procedure to solve any quadratic equation. This

procedure is called completing the square.

x2 � x � 144 � 0

x2 � x � 100 � 0

x2 � 10x � � 0

x2 � 8x � � 0

Let’s solve the following equation:

x2 � 6x � 5 � 0

First, use an algebraic transformation to

remove the constant term from the quadratic

expression. Write the resulting equation.

Next, examine the first two terms of the

quadratic expression. Determine what the

constant term would be if this expression

were a perfect square trinomial. Add that

constant term to each side of the equation.

Write the resulting equation.

Factor the left side of the equation, which

should be a perfect square trinomial. Write

the resulting equation.

Take the square root of each side of the

equation. Make sure to write both the positive

and negative roots of the constant. Write

the resulting equation.

Finally, set the factor of the perfect square

trinomial equal to each of the square roots

of the constant. Solve each of these equations

for x, and write the results.

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7. Solve the same equation by factoring. Do you get the same answers?

8. Solve each of the following equations by completing the square.

a. x2 � 2x � 1 � 0

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b. x2 � 12x � 6 � 0

c. x2 � 9x � 52 � 0

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d.


x2 � 10x � 299 � 0

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Lesson 3.5 ● Solving Quadratic Equations Using the Quadratic Formula 169

ObjectiveIn this lesson, you will:

● Use the Quadratic Formula to solve

quadratic equations.

Key Term● Quadratic Formula

3.5 Quadratic FormulaSolving Quadratic Equations Using theQuadratic Formula

Problem 1By completing the square, we can solve any quadratic equation. Some, however, are

more difficult to solve than others.

1. Solve each of the following equations by completing the square.

a. x2 � 8x � 4 � 0

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b. (Hint: Begin by dividing each side of the equation by 3.)

When a is any value other than zero and b is not evenly divisible by a, completing

the square becomes more difficult and time consuming. To solve these challenging

quadratics, we can use a formula based on the general quadratic,

By solving the general quadratic by completing the square, we can determine the

following solutions:

These solutions can be combined and written as the Quadratic Formula:

The derivation, a short history, and a discussion of the quadratic formula can be

found at www.carnegielearning.com/QuadraticFormula.

To use this formula, first identify the values of a, b, and c. Then, just substitute the

values into the formula:

Solve with a � 5, b � 11, and c � �7

x ��11 � �121 � 140

10

x ��11 � �112 � 4(5) (�7)

2(5)

x ��b � �b2 � 4ac

2a

5x2 � 11x � 7 � 0

x ��b ��b2 � 4ac

2a

x ��b ��b2 � 4ac

2a,

�b ��b2 � 4ac

2a

ax2 � bx � c � 0.

3x2 � 12x � 6 � 0

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Make sure to check your work by substituting the answers into the original equation.

Check:

2. Use the Quadratic Formula to solve the following quadratic equations.

a. x2 � 11x � 10 � 0

� 0

� 37 � 36.92

� 37 � 29.92 � 7

5(�2.72)2 � 11(�2.72) � 7 � 5(7.40) � 29.92 � 7

� 0

� 7.07 � 7

� 1.35 � 5.72 � 7

5(0.52)2 � 11(0.52) � 7 � 5(0.27) � 5.72 � 7

5x2 � 11x � 7 � 0

x � �11 � 16.16

10� �1.1 � 1.62 � �2.72, 0.52

x ��11 � �261

10� �

11

10�

�261

10, �

11

10�

�261

10

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b.

c. �2x2 � 7x � 3 � 0

x2 � x � 7 � 0

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d.

e. 4x2 � 7x � 8 � 0

5x2 � 11 � 0

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f.

g. 2x2 � 5x � 7 � 0

3x2 � 11x � 4 � 0

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h.


2.4x2 � 3x � 5.2 � 0

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Lesson 3.6 ● Properties of Parabolas 177

3


● Graph parabolas.

● Determine properties of parabolas.

● Determine the vertex of a parabola.

● Determine the axis of symmetry of a

parabola.

Key Terms● parabola

● axis of symmetry

● second difference

3.6 Graphing QuadraticFunctionsProperties of Parabolas

Problem 1 Parabolas1. Complete the table of values for the quadratic function Then use the table to

construct a graph of the function.

y � x2.

x y

0

1

2

3

�1

�2

�3

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

This U-shaped graph is called a parabola. A more thorough discussion of the

parabola’s properties, including how it received its name, is in Chapter 11. Recall

that the lowest or highest point on the curve is called the vertex. In any parabola,

a line drawn vertically through the vertex divides the parabola into two mirror

images. The line is called the axis of symmetry.

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Earlier we discovered that the distinctive property of linear functions was a

constant rate of change or constant slope, which was calculated by dividing

the vertical change by the horizontal change: .m ��y

�x

3. Complete the tables to calculate the slope between each pair of points in the

function y � x2.

x �x �y

0 0

�1 1 �1 1 �1

�2

�3

m ��y

�xy � x2

x �y �(�y)

0 0

1 1 1

2 4 3 2

3 9 5

y � x2

x �x �y

0 0

1 1 1 1 1

2

3

m ��y

�xy � x2

4. What conclusion can you make about a quadratic function and its rate of

change? Explain.

5. Complete the tables to calculate the differences in the rate of change in y for

Each of these differences is called a second difference.y � x2.

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6. What conclusion can you make about a quadratic function and the rate of

change of the slope? Explain.

7. For each of the following quadratic functions, complete the tables to calculate

the second differences in y.

a. y � 2x2

x �y �(�y)

0 0

�1 1 1

�2 4 3 2

�3 9 5

y � x2

x �y �(�y)

0 0

1 2

2

3

y � 2x2

x �y �(�y)

1 0

2

3

4

y � 3x2 � 5x � 2

b. y � 3x2 � 5x � 2

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c. y � �2x2 � 3x

x �y �(�y)

1 �5

2

3

4

y � �2x2 � 3x

8. Was your conclusion in Question 6 correct? Explain.

Problem 2 Properties of ParabolasOne way to determine a parabola’s vertex and axis of symmetry is to graph the

parabola, locate the maximum (highest) or minimum (lowest) point (the vertex), and

determine the equation of the vertical line that passes through the vertex (the axis of

symmetry).

The following table and graph are completed for the function y � x 2 � 4x. As

shown, we can use the graph to determine the vertex, x- and y-intercepts, and axis

of symmetry.

x y

0 0

1 �3

2 �4

�1 5

4 0

�2 12

5 5

Vertex: (2, �4)

x-intercepts: (0, 0) and (4, 0)

y-intercept: (0, 0)

Axis of symmetry: x � 2

x

–2

–4

y

10

8

6

12

6 8 10–6 –4 –2

x = 2 y = x 2

– 4x

(2, –4)

(4, 0)(0, 0)

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1. For each of the following quadratic functions:

● Graph the quadratic function.

● Determine the vertex.

● Determine the x-intercept(s).

● Determine the y-intercept.

● Determine the axis of symmetry.

a. y � �x2

Vertex:

x-intercept(s):

y-intercept:

Axis of symmetry: x �

x y

0

�1

1

2

3

4

5

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b. y � x2 � 4x � 3

Vertex:

x-intercept(s):

y-intercept:


x y

x y

c. y � �x2 � 4x

Vertex:

x-intercept(s):

y-intercept:


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d. y � x2 � 3x � 2

x y

Vertex:

x-intercept(s):

y-intercept:


e. y � x2 � 4x � 3

x y

Vertex:

x-intercept(s):

y-intercept:


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Based on the graphs in Question 1 and your knowledge of the standard form of the

the quadratic function, , answer the following questions.

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

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

4. List four ways you can determine the zeros, or x-intercepts, without graphing. Explain.

5. How is the x-value of the vertex related to the x-intercepts, or zeros? Explain.


ax2 � bx � c � y

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Lesson 3.7 ● Basic Functions and Transformations 185


● 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

3.7 Graphing QuadraticFunctionsBasic Functions and Transformations

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


a.

b.




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

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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.


h(�1) �f(2) �

f(�3) �h(2) �

h(4) �f(0) �

h(x) � (x � 2)2.f(x) � x2


a.

b.




y �1

2x2

y � 2x2

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a.

b.




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

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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,

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

y � ax2 � bx � c,

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2. Vertex: y � x2 � 5x � 4

3. Vertex: y � x2 � 5x � 4

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3

4. Vertex: y � 2x2 � 12x � 7

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

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b. Vertex: y � x2 � 5x � 4

c. Vertex: y � x2 � 5x � 4

d. Vertex: y � 2x2 � 12x � 7

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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: y � x2 � 10x � 24

3

Graphical transformations:

b. Vertex: y � x2 � 5x � 4


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c. Vertex: y � x2 � 5x � 4


d. Vertex: y � 2x2 � 12x � 7


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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

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Lesson 3.8 ● Deriving Quadratic Functions 197

3

ObjectiveIn this lesson, you will:

● Derive a quadratic equation given three

points.

3.8 Three Points Determine a ParabolaDeriving Quadratic Functions

Problem 1 Rhonda’s ClaimOne day in math class, Rhonda, after studying quadratic functions and parabolas,

made the following claim: “Because it takes two points to determine a line, and we

are able to derive the equation of the line from those two points, it seems to me that

three points must determine a parabola, and we ought to be able to derive its

equation from those three points.” To prove her claim, she picked three points,

P(1, 2), Q(�3, 2), and R(�1, 0), and showed the following work.

P(1, 2), Q (–3, 2), R(–1, 0)y = ax2 + bx + cBy substituting in each pointSubstituting P

(I) 2 = a(1)2 + b(1) + c = a + b + cSubstituting Q

(II) 2 = a(–3)2 + b(–3) + c = 9a – 3b + cSubstituting R

(III) 0 = a(–1)2 + b(–1) + c = a – b + cSubtract (I) from (II)

2 = 9a – 3b + c (II)2 = a + b + c (I)

(IV) 0 = 8a – 4bSubtract (III) from (I)

2 = a + b + c (I)0 = a – b + c (III)2 = 2bb = 1

Substitute in (IV)0 = 8a – 4(1)8a = 4

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3 1. Examine Rhonda’s method. Did she make any mistakes?

2. Do you think her conjecture works for all sets of three distinct points? Why or

why not?

a =

2 = + 1 + c

Substitute in (I)

c =

a = b = 1 c =

y = x2 + x +

Check:(1, 2) 2 = (1)2 + 1 + = + 1 + = 2

(–3, 2) 2 = (–3)2 – 3 + = – 3 + = 2

(–1, 0) 0 = (–1)2 – 1 + = – 1 + = 012

12

12

12

12

92

12

12

12

12

12

12

12

12

12

12

12

12

12

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Lesson 3.8 ● Deriving Quadratic Functions 199

3. Use Rhonda’s method to determine the equation of the parabola that passes

through each of the following sets of points.

a. (1, 0), (�2, 15), (3, 10)

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b. (2, 1), (�1, �2), (3, �10)

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Lesson 3.8 ● Deriving Quadratic Functions

c. (2, 12), (�2, 32), (1, 8)

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d. (2, �2), (�3, 4), (2, 8)

e. Do you think Rhonda’s claim is correct for any three ordered pairs? If not,

under what conditions would it be correct?


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Lesson 3.9 ● The Discriminant and the Nature of Roots/Vertex Form

3

ObjectivesIn this lesson, you will

● Classify the number and nature of the

roots/zeros of quadratic equations/

functions.

● Use the discriminant to classify

the roots/zeros of quadratic

equations/functions.

● Solve for the vertex form of

quadratic functions.

Key Terms● discriminant

● vertex form of a

quadratic function

3.9 The DiscriminantThe Discriminant and the Natureof Roots/Vertex Form

Problem 1Use the Quadratic Formula to calculate the zero(s) of each quadratic function.

Next, solve for the average of the zeros to identify each function’s vertex and axis

of symmetry. Then, graph all three functions in the grid shown.

1.

a. Zero(s):

b. Axis of symmetry:

c. Vertex:

y � 2x2

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2.

a. Zero(s):


c. Vertex:

3.

a. Zero(s):


c. Vertex:

y � 2x2 � 4

y � 2x2 � 2

4. What are the similarities among the graphs? The axes of symmetry? The vertices?

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Lesson 3.9 ● The Discriminant and the Nature of Roots/Vertex Form 205

3

5. How many zeros does each of the three functions have?

a. Examine the Quadratic Formula. What portion of the formula determines the

number of zeros? How do you know?

b. Because this portion of the formula “discriminates” the number and nature

of the roots, it is called the discriminant. Using the discriminant, write three

inequalities to describe when a quadratic function has

i. no real roots/zeros.

ii. one real root/zero.

iii. two real roots/zeros.

6. Use the discriminant to determine the number of roots/zeros that each equation/

function has. Then solve for the roots/zeros.

a.

b.

c. y � x2 � 12x � 36

0 � 2x2 � 12x � 20

y � 2x2 � 12x � 2

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d.

e.

f.

7. Examine each of the roots/zeros of the equations/functions in Question 6.

Describe the “nature” of the roots. In other words, what characteristic of the

discriminant determines if the roots will be rational or irrational?

8. Based on the number and nature of each of the following roots/zeros, decide if

(i) the discriminant is positive, negative, or zero and (ii) the discriminant is a or is

not a perfect square.

a. no real roots/zeros

i.

ii.

b. one rational root/zero

i.

ii.

0 � 9x2 � 12x � 4

y � 4x2 � 9

y � 3x2 � 7x � 20

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c. two rational roots/zeros

i.

ii.

d. two irrational roots/zeros

i.

ii.

Problem 2From previous activities, we know that all quadratic functions with the same leading

coefficient, a, have the same shape. When graphing any quadratic function with a

leading coefficient of a, we can solve for the vertex using the formula and

then graph the function, using the vertex as the initial point.

1. For each of the following quadratic functions, sketch a graph using the given

vertex.

a. Vertex: ( 5

2, 0 )y � 4x2 � 20x � 25

y � ax2,

x � �b

2a

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b. Vertex: (4, �4)y � x2 � 8x � 12

c. Vertex: (0, 4)y � 4x2 � 4

2. Factor each of the corresponding quadratic equations.

a.

b.

c. 4x2 � 4 � 0

x2 � 8x � 12 � 0

4x2 � 20x � 25 � 0

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3. For each of the following, complete the square without changing the function.

Then solve for the vertex. An example is shown:

Vertex: (1, 16)

a. 4x2 � 20x � 25 � y

9(x � 1)2 � 16 � y

9(x2 � 2x � 1) � 25 � 9 � y

9(1)2 � 18(1) � 25 � 9 � 18 � 25 � 16 9(x2 � 2x) � 25 � y

�b

2a�

�(�18)

18� 1 9x2 � 18x � 25 � y

b.

c.

d. �x2 � 6x � 8 � y

4x2 � 4 � y

x2 � 8x � 12 � y

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e.

4. What do you notice about the final form of the quadratic function and the

coordinates of the vertex?

5. The form is called the vertex form of a quadratic function.

In terms of this form, what are the coordinates of the vertex?

a. What does a tell you about the graph of this function?

b. What do h and k tell you about the graph of the quadratic function?

6. For each of the following quadratic functions in vertex form, determine its vertex

and sketch its graph.

a. y � 2(x � 3)2 � 4

y � a(x � h)2 � k

3x2 � 12x � 8 � y

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b. y � �(x � 2)2 � 3

c. y � 3(x � 1)2 � 5

7. Rewrite each of the following quadratic functions in vertex form.

a. y � 3x2 � 6x � 4

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b.

c.

8. Describe how the variables of the vertex form of a quadratic function tell you how

the graph of the function will be transformed.


y � �4x2 � 16x � 1

y � 5x2 � 15x � 2

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