Document 1795837

LESSON
17.1
Name
Connecting Intercepts
and Zeros
Class
Date
17.1 Connecting Intercepts
and Zeros
Essential Question: How can you use the graph of a quadratic function to solve its related
quadratic equation?
Texas Math Standards
Resource
Locker
A1.7.A…graph quadratic functions on the coordinate plane and use the graph to identify key
attributes, if possible, including x-intercept, y-intercept, zeros, maximum value, minimum
values, vertex, and the equation of the axis of symmetry.
The student is expected to:
A1.7.A
Graphing Quadratic Functions
in Standard Form
Explore
Graph quadratic functions on the coordinate plane and use the graph to
identify key attributes, if possible, including x-intercept, y-intercept,
zeros, maximum value, minimum values, vertex, and the equation of the
axis of symmetry.
A parabola can be graphed using its vertex and axis of symmetry. Use these characteristics, the y-intercept, and
symmetry to graph a quadratic function.
Mathematical Processes
Graph y = x 2 - 4x - 5 by completing the steps.
A1.1.C

1.D, 2.E, 3.F, 3.H
PREVIEW: LESSON
PERFORMANCE TASK
View the Engage section online. Discuss what type of
path is made by a diver diving into the water when
her initial height is the height of the diving platform.
Then preview the Lesson Performance Task.
© Houghton Mifflin Harcourt Publishing Company

= 2
= 2
= -9
The vertex is
( 2 , -9 ).
Find the y-intercept.
y = x 2 - 4x - 5
y= 0
2
( )
- 4 0 + -5
(
)
The y-intercept is -5 ; the graph passes through 0, -5 .

Find two more points on the same side of the axis of symmetry as the y-intercept.
b. Find y when x = -1.
a. Find y when x = 1.
y = x - 4x - 5
y = x 2 - 4x - 5
2
2
= 1
= 1 - 4 -5
= 1 -
= -8
= 0
The first point is
Module 17
2
( )
( -4 ) 5
= -1 - 4 -1 - 5
-4· 1 -5
( 1 , -8 ).
-
(
The second point is -1 , 0
ges must
EDIT--Chan
DO NOT Key=TX-B
Correction
823
info”
through “File
be made
).
Lesson 1
Date
Class
ercepts
ecting Int
17.1 ConnZeros
and
Name
d
its relate
on to solve
atic functi
of a quadr
use the graph
y key
can you
ion: How
graph to identif
equation?
and use the
minimum
quadratic
nate plane
um value,
on the coordi cept, zeros, maxim
functions
y-inter
quadratic
x-intercept,
try.
A1.7.A…graph
including
axis of symme
if possible,
on of the
attributes,
equati
the
ns
, and
values, vertex
ic Functio
Resource
Locker
HARDCOVER PAGES 615624
Quest
Essential
A1_MTXESE353879_U7M17L1.indd 823
g Quadrat
cept, and
Graphin
s, the y-inter
Form
characteristic
Use these
in Standard
symmetry.
Explore
and axis of
its vertex
graphed using
function.
la can be
A parabo
a quadratic
to graph
steps.
symmetry
eting the
5 by compl
x2 - 4x Graph y =
etry.
axis of symm
Find the
b
_
x = - 2a

Find the
= 2
The axis
8 -5
= 4 -
Find the

= -9
of symmetry
y
g Compan
Publishin
Harcour t
n Mifflin
+ ( -5 )
(
( )
( )
= -8
The first
Module 17
L1.indd
9_U7M17
SE35387
( 2 , -9 ).
)
4 -5
= 1 -
A1_MTXE
is
.
-4 0
h 0, -5
passes throug
; the graph
y-intercept.
rcept is -5
etry as the
The y-inte
axis of symm
-1.
side of the
when x =
b. Find y
on the same
more points
2
Find two
4x - 5
y=x -2
1.
when x =
-5
a. Find y
- 4 -1
= -1
5
2
4x
y=x -2
-4 - 5
-5
-4· 1
= 1 = 1
y= 0
© Houghto
The vertex
.
is x = 2
y-intercept.
2
4x - 5
y=x -2

Turn to these pages to
find this lesson in the
hardcover student
edition.
vertex.
2
4x - 5
y=x -2
-5
-4· 2
= 2
-4
_
2· 1
=-
Lesson 17.1
-4· 2 -5
= 4 - 8 -5
1

823
2
-4
= -_
The axis of symmetry is x = 2 .
Given a quadratic function modeling a real-world situation, explain to a
partner what the zeros of the function represent.
You can write the equation with one side equal to
0, graph the related function, and find the zeros of
the function.
y = x 2 - 4x - 5
2·
Language Objective
Essential Question: How can you use
the graph of a quadratic function to
solve its related quadratic equation?
Find the vertex.
b
x = -_
2a
Select tools, including real objects, manipulatives, paper and pencil, and
technology as appropriate, and techniques, including mental math,
estimation, and number sense as appropriate, to solve problems.
ENGAGE

Find the axis of symmetry.
823
point is
( 1 , -8 ).
= 0
The second
point is
( -1 , 0 ).
Lesson 1
823
11/24/14
4:52 PM
11/24/14 4:52 PM

Graph the axis of symmetry, the vertex, the y-intercept, and the two extra points on the
same coordinate plane. Then reflect the graphed points over the axis of symmetry to create
three more points, and sketch the graph.
8
y
EXPLORE
Graphing Quadratic Functions in
Standard Form
(x = 2)
4
(-1, 0)
-8 -4 0
-4
(0, -5)
-8
(1, -8)
x
4
QUESTIONING STRATEGIES
8
How many points on the graph of a quadratic
function are on the axis of symmetry?
Explain. One; the only point on the graph of a
quadratic function that is on the axis of symmetry is
the vertex of the function.
(2, -9)
Reflect
1.
Discussion Why is it important to find additional points before graphing a quadratic function?
Additional points provide more information about the shape of the parabola, and the
Why is it helpful to find the axis of symmetry
when graphing a quadratic function? After
you find the axis of symmetry and a few points on
one side of it, you can use symmetry to quickly and
easily find an equal number of points on the other
side.
sketch of the quadratic function will be more accurate.
Explain 1
Using Zeros to Solve Quadratic Equations
Graphically
A zero of a function is an x-value that makes the value of the function 0. The zeros of a function are the x-intercepts
of the graph of the function. A quadratic function may have one, two, or no zeros.
Quadratic equations can be solved by graphing the related function of the equation. To write the related function,
rewrite the quadratic equation so that it equals zero on one side. Replace the zero with y.
Example 1

Solve by graphing the related function.
2x 2 - 5 = -3
a. Write the related function. Add 3 to both sides to get 2x 2 - 2 = 0. The related
function is y = 2x 2 - 2.
b. Make a table of values for the related function.
8
x
-2
x-1
0
1
2
y
6
0
-2
0
6
c. Graph the points represented by the table and connect the points.
d. The zeros of the function are -1 and 1, so the solutions of the
equation 2x 2 - 5 = -3 are x = -1 and x = 1.
Module 17
824
(-2, 6)
4
-2
(2, 6)
(1, 0)
(-1, 0)
-4
y
0
-4
2
(0, -2)
EXPLAIN 1
© Houghton Mifflin Harcourt Publishing Company
Graph the related function. Find the x-intercepts of the graph, which are the zeros of the function. The zeros of the
function are the solutions to the original equation.
Using Zeros to Solve Quadratic
Equations Graphically
AVOID COMMON ERRORS
Students may think that the zeros of a quadratic
function can be found by substituting 0 for x in the
function. Make sure students understand that the
zeros of a function are the values of x when y = 0, not
the values of y when x = 0.
x
4
-8
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Critical Thinking
Lesson 1
PROFESSIONAL DEVELOPMENT
A1_MTXESE353879_U7M17L1.indd 824
Math Background
Quadratic equations are often used to model the motion of falling objects. The
general formula for this motion is h = -16t 2 + v 0t + h 0, where h represents the
height of the object in feet, t is the number of seconds the object has been falling,
v 0 is the initial vertical velocity of the object in feet per second, and h 0 is the initial
height of the object in feet. The coefficient -16 is equal to half of the constant
acceleration due to gravity, -32 ft/s 2. Students can compare the quadratic
equations given for falling objects in this lesson to the general formula to
determine the values of v 0 and h 0 in each case.
11/22/14 7:38 PM
For cases in which a quadratic function has two
zeros, discuss with students how to use the zeros to
determine the function’s axis of symmetry. Students
should realize that the axis of symmetry will be the
x-value of the point that is midway between the two
zeros.
Connecting Intercepts and Zeros
824
B
QUESTIONING STRATEGIES
6x + 8 = -x 2
a. Write the related function. Add x 2 to both sides to get
When a quadratic function has two zeros that
are opposites, what must be true about the
function? The axis of symmetry must be x = 0, and
the vertex must not be (0, 0).
8
(-5, 3)
4
x 2 + 6x + 8 = 0 .
The related function is y
= x 2 + 6x + 8.
y
(-1, 3)
x
(-4, 0)
(-2, 0)
-8 -4 0
4
(-3, -1) -4
b. Make a table of values for the related function.
x
-5
x-4
-3
-2
-1
y
3
0
-1
0
3
8
-8
c. Graph the points represented by the table and connect the points.
d. The zeros of the function are -4 and -2 , so the solutions of the equation
6x + 8 = -x 2 are x = -4 and x = -2 .
Reflect
2.
How would the graph of a quadratic equation look if the equation has one zero?
If the quadratic equation has one zero, the graph will intersect the x-axis at its vertex.
Your Turn
3.
x 2 - 4 = -3
8
x 2 - 4 = -3
4
(2, 3)
(-2, 3)
x
(1, 0)
(-1, 0)
-8 -4 0
4
8
(0, -1)
-4
© Houghton Mifflin Harcourt Publishing Company
x2 - 1 = 0
The related function is y = x 2 - 1.
Graph:
The zeros of the function are 1 and -1, so the solutions of
the equation are x = -1 and x = 1.
Module 17
y
825
-8
Lesson 1
COLLABORATIVE LEARNING
A1_MTXESE353879_U7M17L1.indd 825
Peer-to-Peer Activity
Have students work in pairs. Have each student find the solution to the same
quadratic equation by graphing. One student in each pair solves the equation by
finding the zeros of the related function, and the other student uses two points of
intersection of two related functions to find the solution. Students compare their
answers; while the graphs may be different, the solution should be the same
regardless of the method used.
825
Lesson 17.1
11/24/14 4:56 PM
Explain 2
Using Points of Intersection to Solve Quadratic
Equations Graphically
EXPLAIN 2
You can solve a quadratic equation by rewriting the equation in the form ax 2 + bx = c or a(x - h) = k and then
using the expressions on each side of the equal sign to define a function.
2
Using Points of Intersection to Solve
Quadratic Equations Graphically
Graph both functions and find the points of intersection. The x-coordinates are the points of intersection on the
graph. As with using zeros, there may be two, one, or no points of intersection.
Example 2

Solve each equation by finding points of intersection of two related
functions.
2
2(x - 4) - 2 = 0
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Math Connections
Write in vertex form.
Rewrite as a(x – h) = k. Graph each side as related function.
2(x - 4) = 2
2
2
a. Let ƒ(x) = 2(x - 4) . Let g(x) = 2.
2
b. Graph ƒ(x) and g(x) on the same graph.
The graphs intersect at two locations: (3, 2) and (5, 2).
-8
-4
This means ƒ(x) = g(x) when x = 3 and x = 5.
(5, 2)
x
0
4
8
-4
-8
2
So the solutions of 2(x - 4) - 2 = 0 are x = 3 and x = 5.

f(x)
4
(3, 2)
g(x)
c. Determine the points at which the graphs of ƒ(x) and g(x) intersect.
Remind students that quadratic equations in the form
2
a(x - h) + k will have the vertex at (h, k). After
identifying the vertex, students can use substitution
to find enough points to graph the quadratic
function.
y
8
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Modeling
3(x - 5) -12 = 0
2
2
3(x - 5) = 12
2
a. Let ƒ(x) = 3 (x - 5) . Let g(x) = 12 .
16
b. Graph ƒ(x) and g(x) on the same graph.
The graphs intersect at two locations:
( 3 , 12 ) and ( 7 , 12 ).
(7, 12)
8
(3, 12)
-8
-4
0
x
4
8
-8
-16
This means ƒ(x) = g(x) when x = 3 and x = 7 .
Therefore, the solutions of the equation ƒ(x) = g(x) are 3 and 7 .
2
So the solutions of 3(x - 5) - 12 = 0 are x = 3 and x = 7 .
Module 17
826
© Houghton Mifflin Harcourt Publishing Company
c. Determine the points at which the graphs of ƒ(x) and g(x) intersect.
When using points of intersection to find solutions to
quadratic equations, make sure that students
understand that only the x-values of the points of
intersection are solutions to the equation. The
y-values come from the function that was created to
find the solutions; they are not part of the solution.
y
QUESTIONING STRATEGIES
How can you use graphing to determine that a
quadratic equation has no solutions? After
rewriting the equation in the form ax 2 + bx = c, if
the graphs of the functions f(x) = ax 2 + bx and
g(x) = c do not intersect, then the quadratic
function has no solutions.
Lesson 1
DIFFERENTIATE INSTRUCTION
A1_MTXESE353879_U7M17L1.indd 826
Communicating Math
11/24/14 4:58 PM
Review the parameters of a parabola that students can use when graphing a
function. Students should understand that being able to identify the vertex of a
parabola will make it easier to graph the function, but they may have different
ideas about how to find enough other points in order to make an accurate graph.
Discuss how students can be sure that they have plotted enough points to sketch
the function on a coordinate grid.
Connecting Intercepts and Zeros
826
Reflect
EXPLAIN 3
4.
Modeling a Real-World Problem
In Part B above, why is the x-coordinates the answer to the equation and not the y-coordinates?
The x-coordinates are the solution because the x-values are the unknown amount being
solved for in the original equation. The y-values are used to create a related function to
find the x values, but since they are not part of the original equation, the y values are not
part of the solution.
INTEGRATE TECHNOLOGY
Before students are familiar with using the
quadratic formula, graphing calculators
offer the most accessible way to find the solutions to
quadratic equations that do not have simple
whole-number solutions.
Your Turn
5.
2
3(x - 2) - 3 = 0
4
3(x - 2) - 3 = 0
2
y
(1, 3)
2
3(x - 2) = 3
2
(3, 3)
x
Let f(x) = 3(x - 2) and let g(x) = 3.
2
The graphs intersect at two locations: (1, 3) and (3, 3).
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Modeling
This means f(x) = g(x) when x = 1 and x = 3.
The functions used to determine the height of a
thrown object can be difficult to understand. Discuss
different ways to rewrite the functions used to
determine height. Students may wish to rewrite a
function like h(t) = -16t2 + 100 as
h(t) = 100 − 16t 2 to make it more evident that the
object loses height for each second of time.
Explain 3
-4
-2
0
2
4
-2
-4
So the solutions of 3(x- 2) - 3 = 0 are 1 and 3.
2
Modeling a Real-World Problem
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©USBFCO/
Shutterstock
Many real-world problems can be modeled by quadratic functions.
Example 3
Create a quadratic function for each problem and then solve it
by using a graphing calculator.
Nature A squirrel is in a tree holding a chestnut at a height of 46 feet above the
ground. It drops the chestnut, which lands on top of a bush that is 36 feet below
the squirrel. The function h(t) = -16t 2 + 46 gives the height in feet of the chestnut
as it falls, where t represents time. When will the chestnut reach the top of the bush?
Analyze Information
Identify the important information.
• The chestnut is 46 feet above the ground, and the top of the bush is 36 feet
below the squirrel.
• The chestnut’s height as a function of time can be represented by
h(t) = -16 t 2 + 46 , where (h)t is the height of the chestnut in feet as it is
falling.
Formulate a Plan
Create a related quadratic equation to find the height of the chestnut in relation to
time. Use h(t) = -16t 2 + 46 and insert the known value for h.
Module 17
827
Lesson 1
LANGUAGE SUPPORT
A1_MTXESE353879_U7M17L1.indd 827
Connect Vocabulary
Students may expect that problems involving points of intersection will always
have solutions. Relate the word intersection to its uses outside the math classroom.
Just as two streets in a town may or may not intersect, the graphs of two functions
may or may not intersect. When dealing with the intersection of a quadratic
function and a linear function, remind students that there may be 0, 1, or 2 points
of intersection.
827
Lesson 17.1
11/22/14 7:58 PM
Solve
QUESTIONING STRATEGIES
Write the equation that needs to be solved. Since the top of the bush is 36 feet below
the squirrel, it is 10 feet above the ground.
-16t 2 + 46 = 10
Separate the function into y = f (t) and y = g (t). f (t) = -16 t
g (t) = 10 .
2
When finding the time it takes for an object to
fall to the ground, why is only the positive
zero of the function used as an answer? Since the
value for time will always be a positive number, the
negative zero of the function can be ignored as a
solution.
+ 46 and
To graph each function on a graphing calculator, rewrite them in terms
of x and y.
y = -16 x
2
+ 46 and y = 10
Graph both functions. Use the intersect feature to find the amount of
time it takes for the chestnut to hit the top of the bush.
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Technology
The chestnut will reach the top of the bush in 1.5 seconds.
Justify and Evaluate
-16
Students who have the minimum and
maximum x- and y-values set incorrectly on
their graphing calculators may not be able to see the
point of intersection when they graph two functions.
Discuss how to change the dimensions of the graph
by accessing the WINDOW menu.
( 1.5 ) + 46 = 10
2
-36 + 46 = 10
10 = 10
When t is replaced by
1.5
in the original equation, -16t 2 + 46 = 10 is true.
Reflect
6.
Your Turn
7.
Nature An egg falls from a nest in a tree 25 feet off the ground and lands on a potted plant that is 20 feet
below the nest. The function h(t) = -16t 2 + 25 gives the height in feet of the egg as it drops, where
t represents time. When will the egg land on the plant?
-16x 2 + 25 = 5
y = -16x 2 + 25 and y = 5.
The egg will hit the plant after about 1.12 seconds.
Module 17
A1_MTXESE353879_U7M17L1.indd 828
828
© Houghton Mifflin Harcourt Publishing Company
In Example 3 above, the graphs also intersect to the left of the y-axis. Why is that point irrelevant to
the problem?
That point is irrelevant to the problem since negative time has no meaning in this problem.
Lesson 1
11/22/14 8:33 PM
Connecting Intercepts and Zeros
828
Explain 4
EXPLAIN 4
Interpreting a Quadratic Model
The solutions of a quadratic equation can be used to find other information about the situation modeled by the
related function.
Interpreting a Quadratic Model
Example 4

AVOID COMMON ERRORS
When viewing graphs of quadratic functions
modeling height, students may believe that the shape
of the graph represents the path an object takes while
in the air. Remind students that while the y-axis
represents the height of the object, the x-axis does not
show distance, but rather time.
Use the given quadratic function model to answer questions about the
situation it models.
Nature A dolphin jumps out of the water. The quadratic function h(t) = -16t 2 + 20t
models the dolphin’s height above the water in feet after t seconds. How long is the dolphin
out of the water?
Use the level of the water as a height of 0 feet. When the dolphin leaves and then reenters
the water again, its height is 0 feet.
Solve 0 = -16t 2 + 20t to find the times when the dolphin both leaves the water and then
reenters. The difference between the times is the amount of time the dolphin is out of the
water.
a. Write the related function for 0 = -16x 2 + 20x.
y = -16x 2 + 20x
b. Graph the function on a graphing calculator. Use the trace feature to
estimate the zeros.
QUESTIONING STRATEGIES
For a quadratic function modeling the height
of a thrown object, when one of the zeros of
the function is 0, what must be true about the
object? The object must have been thrown from
ground level, so that the height of the object equals
0 at time 0.
The zeros appear to be 0 and 1.25.
When x = 0, the equation reduces to 0 = 0, which is true. So x = 0
is a solution.
Check x = 1.25.
2
-16(1.25) + 20(1.25) = -16(1.5625) + 25 = -25 + 25 = 0
so 1.25 is a solution.
Since 1.25 - 0 = 1.25, the dolphin is out of the water for 1.25 seconds.
© Houghton Mifflin Harcourt Publishing Company
For quadratic functions that model the height
of a thrown object, how can you use the zeros
of the function to determine when the object is at its
maximum height? Explain. Since the axis of
symmetry for a quadratic function lies midway
between the two zeros of the function, the
maximum height is reached at a time halfway
between the two zeros.

Sports A baseball coach uses a pitching machine to simulate pop flies during practice.
The quadratic function y = −16t 2 + 80t + 5 models the height in feet of the baseball after
x seconds. The ball leaves the pitching machine and is caught at a height of 5 feet. How long
is the baseball in the air?
Solve 0 = −16t 2 + 80t + 5 to find the times when the baseball enters
the air and when it is caught.
a. Write the related function for 0 = −16t + 80t + 5.
2
y = -16x 2 + 80x + 5
b. Graph the function on a graphing calculator. Use the trace feature
to find the zeros.
The zeros appear to be
0 and
5.
Since x = 0 makes the right side of the equation equal to 5, which is the height
Module 17
A1_MTXESE353879_U7M17L1.indd 829
829
Lesson 17.1
829
Lesson 1
11/22/14 8:34 PM
of the baseball when it is released by the pitching machine, it is a solution. Check to
see if 5 is a solution.
-16x + 80x + 5 = -16
2
so 5 is a solution.
( 5 ) + 80( 5 ) + 5 = - 400 + 400 + 5 =
ELABORATE
2
5 ,
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Communication
The ball is in the air for 5 seconds.
Discuss with students what kinds of real-world
problems can be solved by identifying the zeros of a
quadratic function. Students should understand that
the zeros of a quadratic function will not be the
solution for every real-world problem.
Your Turn
8.
Nature The quadratic function y = −16x + 5x models the height, in feet, of a flying fish above the water
after x seconds. How long is the flying fish out of the water?
The graph of y = -16x 2 + 5x shows a zero at about 0.3125.
2
The fish is out of the water for 0.3125 second.
SUMMARIZE THE LESSON
Elaborate
9.
How do you solve equations graphically using
zeros and points of intersection? To solve
quadratic equations graphically using zeros, rewrite
the equation so one side is equal to zero, then graph
the other side of the equation and identify the
zeros. To solve quadratic equations graphically
using points of intersection, rewrite the equation in
the form ax 2 + bx = c, graph both sides of the
equation, then find the x-values of the points of
intersection.
How is graphing quadratic functions in standard form similar to using zeros to solve quadratic equations
graphically?
If there are two solutions for a quadratic function, the reflection of the point representing
one solution across the axis of symmetry will be the other point, which represents the
other solution. Both methods use the value of the function at 0 to find the second point.
10. How can graphing calculators be used to solve real-world problems represented by quadratic equations?
Graphing calculators can be used to solve real-world quadratic equations by writing the
equation and then finding either an intersection or the zeros of the equation’s graph.
intersection. The x-coordinates of the intersection(s) will be the solutions to the quadratic
equation.
Module 17
A1_MTXESE353879_U7M17L1.indd 830
830
© Houghton Mifflin Harcourt Publishing Company
11. Essential Question Check-In How can you use the graph of a quadratic function to solve a related
quadratic equation by way of intersection?
You can graph the two sides of the equation as two functions and find their points of
Lesson 1
11/22/14 8:38 PM
Connecting Intercepts and Zeros
830
Evaluate: Homework and Practice
EVALUATE
• Online Homework
• Hints and Help
• Extra Practice
Solve each equation by graphing the related function and finding its zeros.
3x 2 - 9 = -6
1.
20
Concepts and Skills
Practice
Explore
Graphing Quadratic Functions in
Standard Form
Exercises 21–22
Example 1
Using Zeros to Solve Quadratic
Equations Graphically
Exercises 1–6, 25
Example 2
Using Points of Intersection to Solve
Quadratic Equations Graphically
Exercises 7–12,
23
Exercises 17–20
The zeros y = 2x 2 - 8 are 2 and -2,
so x = -2 or x = 2.
4.
8
(-6, 4)
(2, 12)
8
0
-8
The zeros of y = 4x 2 - 4 are 1 and -1,
so x = -1 or x = 1.
8
6.
(-1, -4)
-8
(-2, 3)
4
Lesson 17.1
(2, 3)
2
(1, 0) x
-4
8
(0, -3)
-2
0
-2
2
(0, -1)
4
-4
The zeros of y = x 2 - 1 are -1 and
1, x = -1 or x = 1.
831
Lesson 1
Depth of Knowledge (D.O.K.)
Mathematical Processes
2/23/14 1:47 AM
1–12
1 Recall
1.E Create and use representations
13–20
1 Recall
1.A Everyday life
21
1 Recall
1.C Select tools
22
1 Recall
1.E Create and use representations
2 Skills/Concepts
1.G Explain and justify arguments
23–25
831
y
(-1, 0)
x
Module 17
Exercise
-8
4
The zeros of y = x 2 + 2x - 3 are -3 and
-1, so x = -3 or x = 1.
A1_MTXESE353879_U7M17L1 831
-4
x
4
8
(-3, -2)
-1 = -x 2
y
(1, 0)
-8 -4 0
(-2, -3)
-4
0
The zeros of y = x 2 + 7x + 10 are -5 and
-2, so x = -5 or x = -2.
4
(-3, 0)
-4
(-4, -2)
2x - 3 = -x 2
(-2, 0)
(-5, 0)
2
4
(0, -4)
-8
y
4
x
-16
5.
7x + 10 = -x 2
y
(1, 0)
© Houghton Mifflin Harcourt Publishing Company
Example 4
Interpreting a Quadratic Model
x
8
-32
4x 2 - 7 = -3
-4 -2
(-1, 0)
(4, 24)
16
(2, 0)
The zeros of y = 3x 2 - 3 are 1 and -1,
so x = -1 or x = 1.
16
y
-8 -4 0
4
(-2, 0)
(0, -8)
-16
-20
(-2, 12)
Exercises 13–16,
24
32
(-4, 24)
(1, 0) x
-4 -2 0
2
4
(-1, 0) -10 (0, -3)
3.
Example 3
Modeling a Real-World Problem
2x 2 - 9 = -1
y
10 (2, 9)
(-2, 9)
ASSIGNMENT GUIDE
2.
Solve each equation by finding points of intersection of two related
functions.
7.
2(x - 3) - 4 = 0
2
8.
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Communication
(x + 2) - 4 = 0
2
y
4
(1.59, 4) (4.41, 4)
(-4, 4)
2
4
y
(0, 4)
x
-4
-2
0
2
x
-4
4
-2
-2
-4
4
-2
2
10. -(x + 2) - 2 = 0
2
-(x - 3) + 4 = 0
4
2
The graphs of f(x) = (x + 2) and g(x) = 4
intersect at (-4, 4) and (0, 4). So x = 0 or
x = -4.
2
8
0
-4
The graphs of f(x) = 2(x - 3) and
g(x) = 4 intersect at (1.59, 4) and
(4.41, 4). So x = 1.59 or x = 4.41.
9.
2
y
4
(1, 4) (5, 4)
y
2
x
x
-8
-4
0
4
-4
8
-4
-2
0
2
4
-2
-4
-8
The graphs of f(x) = (x - 3) and g(x) = 4
intersect at (5, 4) and (1, 4). So x = 5 or x = 1.
2
The graphs of f(x) = (x + 2) and g(x) = -2
do not intersect. So there is no solution.
2
12. (x + 2) - 2 = 0
2
2
y
4
≈(–3.5. 2)
(-2, 1)
2
(0, 1)
-4
0
2
2
y
≈(–0.5, 2)
x
-2
© Houghton Mifflin Harcourt Publishing Company
11. (x + 1) - 1 = 0
4
Students may feel that certain equations must be
solved using certain methods. Discuss the fact that
most problems can be solved using more than one of
the methods outlined in this lesson, and that they will
learn additional methods as they continue in algebra.
2
x
-4
4
-2
-2
0
2
4
-2
-4
-4
The graphs of f(x) = (x + 1) and g(x) = 1
intersect at (-2, 1) and (0, 1). So x = -2
or x = 0.
The graphs of f(x) = (x + 2) and g(x) = 2
intersect at about (-3.5, 2) and about
(-0.5, 2).
2
2
So x ≈ -3.5 or x ≈ -0.5.
―
Note that exact values are x = -2 ± √2 .
Module 17
A1_MTXESE353879_U7M17L1.indd 832
832
Lesson 1
2/14/15 7:22 PM
Connecting Intercepts and Zeros
832
Create a quadratic equation for each problem and then solve the
equation with a related function using a graphing calculator.
AVOID COMMON ERRORS
When solving quadratic equations graphically,
students may mistake the zeros of a quadratic
function for the places where the function intersects
the y-axis. Remind students that the zeros of a
function are the x-values at the points where the
function intersects the x-axis.
13. Nature A bird is in a tree 30 feet off the ground and drops a twig that lands on a
rosebush 25 feet below. The function h(t) = -16t 2 + 30, where t represents the time
in seconds, h gives the height, in feet, of the twig above the ground as it falls. When
will the twig land on the bush?
The graphs of y = -16x 2 + 30 and y = 5 intersect at about (1.25, 5). The twig will hit the
rosebush after about 1.25 seconds.
14. Nature A monkey is in a tree 50 feet off the ground and
drops a banana, which lands on a shrub 30 feet below. The
function h(t) = -16t 2 + 50, where t represents the time in
seconds, h gives the height, in feet, of the banana above the
ground as it falls. When will the banana land on the shrub?
The graphs of y = -16x 2 + 50 and y = 20 intersect at
about (1.4, 20). The banana will hit the shrub after
about 1.4 seconds.
© Houghton Mifflin Harcourt Publishing Company • ©Dmitry Laudin/
Shutterstock
15. Sports A trampolinist jumps 60 inches in the air off a trampoline 2 inches off the
ground. The function h(t) = -16t 2 + 60, where t represents the time in seconds, h
gives the height, in inches, of the trampolinist above the ground as he falls. When will
the trampolinist land on the trampoline?
The graphs of y = -16x 2 + 60 and y = 58 intersect at about (0.4, 58). The
trampolinist will land back on the trampoline after about 0.4 seconds.
16. Physics A ball is dropped from 10 feet above the ground. The function
h(t) = -16t 2 + 10, where t represents the time in seconds, h gives the height, in
feet, of the ball above the ground. When will the ball be 4 feet above the ground?
The graphs of y = -16x 2 + 10 and y = 4 intersect at about (0.6, 4). The ball
will land on the roof of the trash shed in about 0.6 seconds.
Use the given quadratic function model to answer questions about
the situation it models.
17. Nature A shark jumps out of the water. The quadratic function ƒ(x) = -16x 2 + 18x
models the shark’s height, in feet, above the water after x seconds. How long is the
shark out of the water?
The graph of f(x) = -16x 2 + 18x shows a zero of 1.125. The shark is out
of the water for about 1.125 seconds.
Module 17
A1_MTXESE353879_U7M17L1.indd 833
833
Lesson 17.1
833
Lesson 1
11/22/14 8:51 PM
18. Sports A baseball coach uses a pitching machine to simulate pop flies during
practice. The quadratic function ƒ(x) = -16x 2 + 70x + 10 models the height in feet
of the baseball after x seconds. How long is the baseball in the air?
CRITICAL THINKING
Challenge students to solve a quadratic equation
graphically using two different methods: by finding
the zeros of the function and by finding points of
intersection. They should find that the graphs look
similar, but that one is translated vertically relative to
the other. Students should understand that when
solving one problem using two different methods, the
graphs may look different, but the final solutions will
be the same.
The graph of f(x) = -16x 2 + 70x + 10 shows a zero of about 4.5. The ball is
in the air for 4.5 seconds.
19. The quadratic function ƒ(x) = -16x 2 + 11x models the height, in feet, of a fish above
the water after x seconds. How long is the fish out of the water?
The graph of f(x) = -16x 2 + 11x shows a zero of about 0.7.
The fish is in the air for about 0.7 seconds.
© Houghton Mifflin Harcourt Publishing Company • ©Radharc Images/Alamy
20. A football coach uses a passing machine to simulate 50-yard passes
during practice. The quadratic function ƒ(x) = -16x 2 + 60x + 5
models the height in feet of the football after x seconds. How long is
the football in the air?
The graph of f(x) = -16x 2 + 60x + 5 shows a zero of about
3.8. The football is in the air for about 3.8 seconds.
Module 17
A1_MTXESE353879_U7M17L1.indd 834
834
Lesson 1
11/22/14 8:52 PM
Connecting Intercepts and Zeros
834
21. In each polynomial function in standard form, identify a, b, and c.
JOURNAL
Have students write a paragraph outlining the steps
they take when graphing a quadratic function by
hand. Students should be sure to describe how to find
enough points so that the function can be accurately
graphed on a coordinate grid.
a. y = 3x 2 + 2x + 4
a.
a = 3, b = 2, c = 4
b. y = 2x + 1
b.
a = 0, b = 2, c = 1
c. y = x
c.
a = 1, b = 0, c = 0
d.
a = 0, b = 0, c = 5
e.
a = 3, b = 8, c = 11
2
d. y = 5
e. y = 3x + 8x + 11
2
22. Identify the axis of symmetry, y-intercept, and vertex of the quadratic function
y = x 2 + x - 6 and then graph the function on a graphing calculator to confirm.
b
1
_
= -_
Axis of symmetry: x = -
2a
2
y-Intercept: y = 0 2 + 0 - 6 = -6
1
Vertex: - 1 , -6
because y = - 1
4
2
2
(
_
_)
( _) - _12 - 6 = -6 _41
2
H.O.T. Focus on Higher Order Thinking
23. Counterexamples Pamela says that if the graph of a function opens upward, then
the related quadratic equation has two solutions. Provide a counterexample to refute
Pamela’s claim.
Sample answer: The graph of f(x) = x 2 opens upward, but the related equation,
x 2 = 0, has only one solution.
© Houghton Mifflin Harcourt Publishing Company
24. Explain the Error Rodney was given the function h(t) = -16t 2 + 50 representing
the height above the ground (in feet) of a water balloon t seconds after being dropped
from a roof 50 feet above the ground. He was asked to find how long it took the
balloon to fall 20 feet. Rodney used the equation -16t 2 + 50 = 20 to solve the
problem. What was his error?
Falling 20 feet means that the balloon would have been 50 - 20 = 30 feet above
the ground, so Rodney should have solved the equation -16t 2 + 50 = 30 instead.
25. Critical Thinking If Jamie is given the graph of a quadratic equation with only
the x-intercepts and a random point labeled, can she determine an equation for the
function? Explain.
Yes; Jamie can find the zeros of the function using the x-intercepts, and
write the equation in the factored form f(x) = k(x - a) (x - b), where a
and b are the zeros. She can then substitute the coordinates of the given
point to find the value of k.
Module 17
A1_MTXESE353879_U7M17L1.indd 835
835
Lesson 17.1
835
Lesson 1
11/24/14 5:03 PM
Lesson Performance Task
QUESTIONING STRATEGIES
The graph crosses the x-axis at (2, 0) so the x-intercept is 2.
So, one solution of the equation y = -16x 2 + 8x + 48 is
x = 2. This solution means that Stella reaches the surface of
the water 2 seconds after she starts the dive.
If the graph is extended on the left, it will cross the
x-intercept once more. Since parabolas are symmetric, use
the axis of symmetry to find the second x-intercept.
b
= 0.25.
The equation of the axis of symmetry is x = -___
2a
Since (2, 0) is 1.75 units to the right of the axis of symmetry,
the second x-intercept will be 1.75 units to the left. The
point 1.75 units to the left of the axis of symmetry is
(-1.5, 0). Since time cannot be negative, this solution does
not make sense.
How can you find the height at which the
diver starts her dive? Find the value of the
function when t = 0.
Height of Dive
y
48
Height (ft)
Stella is competing in a diving competition. Her height in feet above the
water is modeled by the function ƒ(x) = -16x 2 + 8x + 48, where x is
the time in seconds after she jumps from the diving board. Graph the
function and solve. What do the solutions mean in the context of the
problem? Are there solutions that do not make sense? Explain.
How can you find the maximum height that
the diver reaches? Use the graph to find the
y-value of the vertex of the parabola, or find the
x-value halfway between the two zeros and
calculate the corresponding y-value.
36
24
12
x
0
1
2
3
Time (s)
4
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Reasoning
Have students examine how the zeros, intercepts,
function, and graph modeling the dive would be
different if the diver had started the dive at the level
of the water, that is, at height 0.
© Houghton Mifflin Harcourt Publishing Company
Module 17
836
Lesson 1
EXTENSION ACTIVITY
A1_MTXESE353879_U7M17L1 836
Give students the functions g(t) = -16t 2 + 8t + 24 and h(t) = -16t 2 + 16 and
explain that they represent the height in feet of two different divers, where t is
time in seconds from the start of each dive. Have students compare the initial
height, maximum height, and time to reach the water for the two divers.
Students should find that g(t) represents a dive starting at 24 feet, reaching a
maximum of 25 feet, and hitting the water after 1.5 seconds, and h(t) represents a
dive starting at 16 feet, which is also its maximum height, and hitting the water
after 1 second.
2/14/15 2:04 PM
Scoring Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain his/her reasoning.
0 points: Student does not demonstrate understanding of the problem.
Connecting Intercepts and Zeros
836