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© Houghton Mifflin Harcourt Publishing Company © Houghton Mifflin Harcourt Publishing Company Name Class Date 8-5 Solving Quadratic Equations by Graphing Going Deeper Essential question: How can you solve a quadratic equation by graphing? Finding Intersections of Lines and Parabolas The graphs of three quadratic functions are shown. Parabola A is the graph of f (x) = x 2 . Parabola B is the graph of f (x) = x 2 + 4. Parabola C is the graph of f (x) = x 2 + 8. A On the same coordinate grid, graph the function g(x) = 4. What type of function is this? Describe its graph. B At how many points does the graph of g(x) intersect each parabola? Intersections with parabola A: Intersections with parabola B: Intersections with parabola C: C Use the graph to find the x-coordinate of each point of intersection of the graph of g(x) and parabola A. Show that each x-coordinate satisfies the equation x 2 = 4. D Use the graph to find the x-coordinate of each point of intersection of the graph of g(x) and parabola B. Show that each x-coordinate satisfies the equation x 2 + 4 = 4. REFLECT 1a. Describe how you could solve an equation like x 2 + 5 = 7 graphically. EXPLORE 1 PREP FOR A-REI.4.11 0 6 2 -2 -4 -2 4 2 10 12 x y A B C Chapter 8 437 Lesson 5

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Page 1: Solving Quadratic Equations by Graphing · PDF fileSolving Quadratic Equations by Graphing Going Deeper Essential question: How can you solve a quadratic equation by graphing? Finding

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Name Class Date 8-5Solving Quadratic Equations by GraphingGoing DeeperEssential question: How can you solve a quadratic equation by graphing?

Finding Intersections of Lines and Parabolas

The graphs of three quadratic functions are shown.

Parabola A is the graph of f (x) = x 2 .

Parabola B is the graph of f (x) = x 2 + 4.

Parabola C is the graph of f (x) = x 2 + 8.

A On the same coordinate grid, graph the function 

g(x) = 4. What type of function is this? Describe

its graph.

B At how many points does the graph of g(x) intersect

each parabola?

Intersections with parabola A:

Intersections with parabola B:

Intersections with parabola C:

C Use the graph to find the x-coordinate of each point of intersection of the graph of g(x)

and parabola A. Show that each x-coordinate satisfies the equation x 2 = 4.

D Use the graph to find the x-coordinate of each point of intersection of the graph of g(x)

and parabola B. Show that each x-coordinate satisfies the equation x 2 + 4 = 4.

REFLECT

1a. Describe how you could solve an equation like x 2 + 5 = 7 graphically.

E X P L O R E1PREP FOR A-REI.4.11

0

6

2

-2-4 -2 42

10

12

x

y

A

B

C

Chapter 8 437 Lesson 5

Page 2: Solving Quadratic Equations by Graphing · PDF fileSolving Quadratic Equations by Graphing Going Deeper Essential question: How can you solve a quadratic equation by graphing? Finding

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You can solve an equation of the form a(x - h ) 2 + k = c, which is called a quadratic equation, by graphing the functions f (x) = a(x - h ) 2 + k and g(x) = c and finding the

x-coordinate of each point of intersection.

Solving Quadratic Equations Graphically

Solve 2(x - 4 ) 2 + 1 = 7.

A Graph f (x) = 2(x - 4 ) 2 + 1.

What is the vertex?

If you move 1 unit right or left from the vertex, how

must you move vertically to be on the graph of f (x)?

What points are you at?

B Graph g(x) = 7.

C At how many points do the graphs of f (x) and g(x) intersect? If possible, find the

x-coordinate of each point of intersection exactly. Otherwise, give an approximation

of the x-coordinate of each point of intersection.

D For each x-value from part C, find the value of f (x). How does this show that you have

found actual or approximate solutions of 2(x - 4 ) 2 + 1 = 7?

REFLECT

2a. If you solved the equation 4(x - 3 ) 2 + 1 = 5 graphically, would you be able to

obtain exact or approximate solutions? Explain.

2b. For what value of c would the equation 4(x - 3 ) 2 + 1 = c have exactly one solution?

How is that solution related to the graph of f (x)?

E X AM P L E2A-REI.4.11

20

2

-2-2

4

4

6 8

6

8

x

y

Chapter 8 438 Lesson 5

Page 3: Solving Quadratic Equations by Graphing · PDF fileSolving Quadratic Equations by Graphing Going Deeper Essential question: How can you solve a quadratic equation by graphing? Finding

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Solving a Real-World Problem

While practicing a tightrope walk at a height of 20 feet, a circus performer slips and falls

into a safety net 15 feet below. The function h(t) = -16 t 2 + 20, where t represents time

measured in seconds, gives the performer’s height above the ground (in feet) as he falls.

Write and solve an equation to find the elapsed time until the performer lands in the net.

A Write the equation that you need to solve.

B You will solve the equation using a graphing calculator. Because the

calculator requires that you enter functions in terms of x and y, use x

and y to write the equations for the two functions that you will graph.

C When setting a viewing window, you need to decide what portion of each axis to use

for graphing. What interval on the x-axis and what interval on the y-axis are reasonable

for this problem? Explain.

D Graph the two functions, and use the calculator’s trace or intersect feature to find the

elapsed time until the performer lands in the net. Is your answer exact or an approximation?

REFLECT

3a. Although the graphs also intersect to the left of the y-axis, why is that point

irrelevant to the problem?

3b. The distance d (in feet) that a falling object travels as a function of time t (in seconds)

is given by d(t) = 16 t 2 . Use this fact to explain the model given in the problem,

h(t) = -16 t 2 + 20. In particular, explain why the model includes the constant 20 and

why -16 t 2 includes a negative sign.

3c. At what height would the circus performer have to be for his fall to last exactly

1 second? Explain.

E X AM P L E3A-CED.1.2

Chapter 8 439 Lesson 5

Page 4: Solving Quadratic Equations by Graphing · PDF fileSolving Quadratic Equations by Graphing Going Deeper Essential question: How can you solve a quadratic equation by graphing? Finding

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P R A C T I C E

Solve each quadratic equation by graphing. Indicate whether the solutions are exact or approximate.

5. As part of an engineering contest, a student who has designed a protective crate for an egg

drops the crate from a window 18 feet above the ground. The height (in feet) of the crate as it

falls is given by h(t) = -16 t 2 + 18 where t is the time (in seconds) since the crate was dropped.

a. Write and solve an equation to find the elapsed time until the crate passes a window

10 feet directly below the window from which it was dropped.

b. Write and solve an equation to find the elapsed time until the crate hits the ground.

c. Is the crate’s rate of fall constant? Explain.

1. (x + 2 ) 2 - 1 = 3

3. -1 __

2 x 2 + 2 = -4

2. 2(x - 3 ) 2 + 1 = 5

4. -(x - 3 ) 2 - 2 = -6

Chapter 8 440 Lesson 5

Page 5: Solving Quadratic Equations by Graphing · PDF fileSolving Quadratic Equations by Graphing Going Deeper Essential question: How can you solve a quadratic equation by graphing? Finding

Name ________________________________________ Date __________________ Class__________________

Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.

Holt McDougal Algebra 1

Practice Solving Quadratic Equations by Graphing

Solve each equation by graphing the related function. 1. x2 − 6x + 9 = 0 2. x2 = 4

_________________________________________ ________________________________________

3. 2x2 + 4x = 6 4. x2 = 5x − 10

_________________________________________ ________________________________________

5. Water is shot straight up out of a water soaker toy. The quadratic function y = −16x2 + 32x models the height in feet of a water droplet after x seconds. How long is the water droplet in the air?

__________________________________

56

LESSON

8-5

CS10_A1_MEPS709963_C08PWBL05.indd 56 4/21/11 10:36:22 PM

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8-5Name Class Date

Additional Practice

Chapter 8 441 Lesson 5

Page 6: Solving Quadratic Equations by Graphing · PDF fileSolving Quadratic Equations by Graphing Going Deeper Essential question: How can you solve a quadratic equation by graphing? Finding

= − + −

1. Graph the function on the grid below.

3. Based on the graph of the firework, what

are the two zeros of this function?

_________________________________________

_________________________________________

2. The firework will explode when it reaches its highest point. How long after the fuse is lit will the firework explode and how high will the firework be?

________________________________________

________________________________________

4. What is the meaning of each of the zeros you found in problem 3?

________________________________________

________________________________________

________________________________________

5. The quadratic function ( ) = −16 2 + 90 models the height of a baseball in feet after seconds. How long is the baseball in the air? A 2.8125 s C 11.25 s B 5.625 s D 126.5625 s

7. The function = −0.04 2 + 2 models the height of an arch support for a bridge, where is the distance in feet from where the arch supports enter the water. How many real solutions does this function have? F 0 H 2 G 1 J 3

6. The height of a football in feet is given by the function = −16 2 + 56 + 2 where is the time in seconds after the ball was kicked. This function is graphed below. How long was the football in the air?

A 0.5 seconds C 2 seconds B 1.75 seconds D 3.5 seconds

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Problem Solving

Chapter 8 442 Lesson 5