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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
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-2-4 -2 42
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x
y
A
B
C
Chapter 8 437 Lesson 5
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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
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x
y
Chapter 8 438 Lesson 5
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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
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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
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
= − + −
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