1. science
2. mathematics
3. algebra

8-1
Identifying Quadratic Functions
Connection: Connecting f (x) = x 2 to
g (x) = ax 2
Essential question: What is the effect of the constant a on the graph of g(x) = ax 2?
Standards for
Mathematical Content
The new graphs are still symmetric in the y-axis and
each has vertex (0, 0). If the value of a is negative,
then the parabola is reflected over the x-axis and
opens downward.
A-CED.1.2 ... graph equations on coordinate axes
with labels and scales.*
F-IF.1.2 Use function notation, evaluate functions
for inputs in their domains ...
F-IF.2.4 For a function . . . interpret key features of
graphs and tables in terms of the quantities ...*
F-IF.2.5 Relate the domain of a function to its
graph and, where applicable, to the quantitative
relationship it describes.*
F-IF.3.7 Graph functions expressed symbolically
and show key features of the graph, by hand in
simple cases …*
F-IF.3.7a Graph . . . quadratic functions and show
intercepts, maxima, and minima.*
F-BF.1.1 Write a function that describes a
relationship between two quantities.*
F-BF.2.3 Identify the effect on the graph of
replacing f (x) by ... k f (x) ... for specific values of k
(both positive and negative); find the value of
k given the graphs.
IN TR OD UCE
Students are familiar with linear functions and
absolute–value functions. Ask students to name the
characteristics of the graphs of each. Tell students
they will be studying another type of function in
this lesson, quadratic functions.
Type of
Function
Restrictions
Linear
f(x) = mx + b
none
Absolute
value
f(x) = x
none
TE ACH
1
ENGAGE
quadratic function
parabola
vertex
linear function.
• How is the vertex of the graph of the parent
quadratic function related to the axis of symmetry?
Prerequisites
Functions
The vertex is on the y-axis, which is the axis of
symmetry for the graph.
Math Background
• What other point is on the graph of f (x) = x2 for
each point (x, y) that is on the graph? Explain.
The point (-x, y) is also on the graph of f (x) =
An even function f (x) has the property that
f (-x) = f (x) for all values x in the domain of f, and
the graph of an even function has symmetry in
the y-axis. The quadratic function f (x) = x2 is the
parent function of the general quadratic function
f (x) = ax2 + bx + c, where a ≠ 0. f (x) = x2 is an
even function and therefore has symmetry in
the y-axis. Its graph is a U-shaped curve called a
parabola and the turning point of the parabola is
called the vertex. The vertex of f (x) = x2 is the point
(0, 0). If a > 1, the parabola is stretched vertically.
If 0 < a < 1, then the parabola is shrunk vertically.
x2 due to the symmetry of the graph over the
y-axis.
411
Lesson 1
© Houghton Mifflin Harcourt Publishing Company
Questioning Strategies
• Why is the coefficient a of the quadratic function
f (x) = ax2 + bx + c restricted to non-zero values?
If a = 0, then the first term is 0, and f (x) is a
Vocabulary
Chapter 8
Function
Name
Class
Notes
8-1
Date
Identifying Quadratic Functions
Connection: Connecting f (x ) = x 2 to g (x ) = ax 2
Essential question: What is the effect of the constant a on the graph of g(x) = ax2?
F-IF.2.4
1
ENGAGE
Understanding the Parent Quadratic Function
Any function that can be written as f (x) = ax2 + bx + c where a, b, and c are constants
and a ≠ 0 is a quadratic function . Notice that the highest exponent of the variable x is 2.
The most basic quadratic function is f (x) = x2. It is called the parent quadratic function.
To graph the parent function, make a table of values like the one below. Then plot the
ordered pairs and draw the graph. The U-shaped curve is called a parabola . The turning
point on the parabola is called its vertex .
x
f(x) = x2
-3
9
-2
4
-1
1
0
0
4
1
1
2
2
4
3
9
10
y
8
6
f(x) = x2
x
-4
-2
0
2
4
© Houghton Mifflin Harcourt Publishing Company
REFLECT
1a. What is the domain of f (x) = x2? What is the range?
Domain = {real numbers}; range = {y | y ≥ 0}
1b. What symmetry does the graph of f (x) = x2 have? Why does it have this symmetry?
Symmetric in the y-axis; the square of any number and the square of its opposite
are equal, so the points (x, f (x)) and (-x, f (-x)) are reflections in the y-axis.
1c. For what values of x is f (x) = x2 increasing? For what values is it decreasing?
Increasing for x ≥ 0 and decreasing for x ≤ 0.
411
Chapter 8
Lesson 1
To understand the effect of the constant a on the graph of g (x) = ax2,
you will graph the function using various values of a.
F-BF.2.3
EXAMPLE
Graphing g(x) = ax2 when a > 0
Graph each quadratic function. (The graph of the parent function f(x) = x2 is
shown in gray.)
A
g (x) = 2x2
x
g(x) = 2x 2
-3
18
-2
8
y
8
6
B
-1
2
0
0
1
2
2
8
3
18
4
2
x
-4
0
-2
2
4
-2
g(x) = __12 x2
Chapter 8
Chapter 8
x
1 2
x
g(x) = _
-3
4.5
-2
2
-1
0.5
0
0
1
0.5
2
2
3
4.5
2
8
y
6
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
2
4
2
x
-4
-2
0
2
4
-2
412
Lesson 1
412
Lesson 1
2
Teaching Strategy
EXAMPLE
Students may perceive the horizontal changes in
a graph more readily than the vertical changes, so
they may describe a vertical stretch as a horizontal
shrink. Explain to students that changes are
described in the vertical because they describe
changes in the y-value for a given x-value.
Questioning Strategies
• How are the graphs of g (x) = 2x2 and g (x) = _12 x2
similar? How are they different? The graphs are
similar because they both have the same parent
graph, f (x) = x2, are both symmetric in the y-axis,
and both have the same vertex (0, 0). They are
different because the graph of f (x) = 2x2 is a
vertical stretch of the graph of f (x) = x2, while
1 2
the graph of f (x) = _
x is a vertical shrink of the
2
2
graph of f (x) = x .
3
Questioning Strategies
• Why do the graphs in part A and part B open
downwards? The y-values are all negative for
• Do the graphs in part A and part B have
symmetry? If so, what is it? Yes; they are
each point that is not the vertex.
symmetrical in the y-axis.
• How could you use the graphs from the previous
Example to draw the graphs in this Example?
• If (5, 50) is a point of the graph in part A, what is
another point? (-5, 50)
You could reflect the graphs from the previous
Example over the x-axis to get the graphs in
this Example.
EXTRA EXAMPLE
Graph each quadratic function. How is each related
to the graph of f (x) = x2?
EXTRA EXAMPLE
Graph each quadratic function. How is each related
to the graph of f (x) = x2?
2
A. g (x) = 3x
n
EXAMPLE
Þ
A. g (x) = -3x2
The graph is a vertical stretch of the graph of
f(x) = x2 along with a reflection across the x-axis.
È
{
Ó
Ó
Ý
{
Ó
ä
Ó
Þ
Ý
ä
{
{
Ó
Ó
{
Ó
Ó
The graph is a vertical stretch of the graph of
the parent function f (x) = x2.
È
B. g (x) = __13x2
n
8
y
B. g (x) = -__13 x2
The graph is a vertical shrink of the graph of
f(x) = x2 along with a reflection across the x-axis.
6
4
2
2
-2
0
2
x
0
x
-4
y
-4
4
-2
2
4
-2
-2
-4
The graph is a vertical shrink of the graph of
the parent function f (x) = x2.
-6
-8
Chapter 8
413
Lesson 1
© Houghton Mifflin Harcourt Publishing Company
{
Notes
REFLECT
2a. The graph of the parent function f (x) = x2 includes the point (-1, 1) because
f (-1) = (-1)2 = 1. The corresponding point on the graph of g (x) = 2x2 is (-1, 2)
because g (-1) = 2(-1)2 = 2. In general, how does the y-coordinate of a point on
the graph of g (x) = 2x2 compare with the y-coordinate of a point on the graph of
f (x) = x2 when the points have the same x-coordinate?
The y-coordinate of a point on the graph of g(x) is 2 times the y-coordinate of a
point on the graph of f(x).
2b. Describe how the graph of g (x) = 2x2 compares with the graph of f (x) = x2. Use
either the word stretch or shrink, and include the direction of the movement.
The graph of g(x) is a vertical stretch of the graph of f(x).
2c. How does the y-coordinate of a point on the graph of g (x) = _12 x2 compare with the
y-coordinate of a point on the graph of f (x) = x2 when the points have the same
x-coordinate?
1
times the y-coordinate of a
The y-coordinate of a point on the graph of g(x) is _
2
point on the graph of f(x).
2d. Describe how the graph of g (x) = _21 x2 compares with the graph of f (x) = x2. Use
either the word stretch or shrink, and include the direction of the movement.
The graph of g(x) is a vertical shrink of the graph of f(x).
F-BF.2.3
3
EXAMPLE
Graphing g(x) = ax2 when a < 0
© Houghton Mifflin Harcourt Publishing Company
Graph each quadratic function. (The graph of the parent function f (x) = x2 is
shown in gray.)
A
g (x) = -2x2
x
g(x) = -2x 2
-3
-18
-2
-8
-1
-2
0
0
y
4
2
x
-4
1
-2
2
-8
3
-18
0
-2
2
4
-2
-4
413
Chapter 8
g(x) = -__12 x2
x
1 2
x
g(x) = -_
-3
-4.5
-2
-2
2
-1
-0.5
0
0
1
-0.5
2
-2
3
-4.5
y
4
2
x
-4
0
-2
2
4
-2
-4
REFLECT
3a. In part A of the previous example, you drew the graph of g(x) = ax2 where a = 2. In
part A of this example, you drew the graph of g(x) = ax2 where a = -2. How do the
two graphs compare? How does the graph of g(x) = -2x2 compare with the graph
of f (x) = x2?
The graphs are reflections across the x-axis; the graph of g(x) = -2x2 is a vertical
stretch of the graph of f (x) = x2 along with a reflection across the x-axis.
3b. In part B of the previous example, you drew the graph of g(x) = ax2 where a = _21. In
part B of this example, you drew the graph of g(x) = ax2 where a = -_12. How do the
two graphs compare? How does the graph of g(x) = -_12 x2 compare with the graph
of f (x) = x2?
1 2
The graphs are reflections across the x-axis; the graph of g(x) = -_
x is a vertical
2
shrink of the graph of f (x) = x2 along with a reflection across the x-axis.
3c. Summarize your observations about the graph of g(x) = ax2.
Value of a
Vertical stretch or shrink?
a>1
Vertical stretch
Reflection across x-axis?
No
0<a<1
Vertical shrink
No
-1 < a < 0
Vertical shrink
Yes
a = -1
No stretch or shrink
Yes
a < -1
Vertical stretch
Yes
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
B
Lesson 1
Writing Equations from Graphs A function whose graph is a parabola with vertex (0, 0)
always has the form f (x) = ax2. To write the rule for the function, you can substitute the
x- and y-coordinates of a point on the graph into the equation y = ax2 and solve for a.
Chapter 8
Chapter 8
414
Lesson 1
414
Lesson 1
4
CLOS E
EXAMPLE
Q
Questioning
Strategies
• How can you tell immediately that the value of a
must be negative? The graph opens downward,
Essential Question
What is the effect of the constant a on the graph of
g(x) = ax2?
so a must be negative.
Having a coefficient other than 1 for the x2-term
either stretches or shrinks the graph of the parent
quadratic function vertically. If the coefficient is
negative, it reflects the graph over the x-axis so that
it opens downward (like an arch on a bridge) instead
of upward (like a cup). The coefficient does not
change the vertex, however, so it is still at the origin.
• How does having a point of the parabola other
than the vertex help you find the value of a? If you
have a point other than (0, 0), you can substitute
the x- and y-values into f(x) = ax2 and solve for a.
EXTRA EXAMPLE
Write the equation for the function whose graph
is shown.
2
y
x
0
-4
-2
Summarize
Have students create a table for their notebooks
that summarizes how a affects the graph of f (x) =
ax2 in relation to the graph of the parent quadratic
function.
2
4
-2
-4
-6
The function
f (x) = ax2 is a
vertical:
And the
graph opens:
0<a<1
shrink
up
a>1
stretch
up
-1 < a < 0
shrink
down
a = -1
no shrink/
stretch
down
a < -1
stretch
down
(3, -6)
-8
2 2
f(x) = -_
x
3
Highlighting the
Standards
PR ACTICE
Where skills are
taught
415
Where skills are
practiced
2 EXAMPLE
EXS. 1, 3
3 EXAMPLE
EXS. 2, 4
4 EXAMPLE
EXS. 5–8
Lesson 1
© Houghton Mifflin Harcourt Publishing Company
This lesson includes opportunities to
address Mathematical Practice Standard
7 (Look for and make use of structure).
Draw students attention to how graphs of
the form f (x) = ax2 have the same structure
as the graph of f (x) = x2 but are stretched
or shrunk vertically. Point out that relating
transformations to a parent graph is also done
with other types of functions.
Chapter 8
For these
values of a:
Notes
F-BF.1.1
4
EXAMPLE
Writing the Equation for a Quadratic Function
Write the equation for the quadratic function whose graph is shown.
Use the point (2, -1) to find a.
x
2
(2)
-1 = a ( 4 )
-1 = a
-5
-2 (2, -1)
-6
-8
The equation for the function
4
5
3
-4
4
is
0
-3
1 =a
-_
1 2
f (x) = -_
x
y
2
y = ax2
.
REFLECT
4a. Without actually graphing the function whose equation you found, how can check
that your equation is reasonable?
Because the value of a is between 0 and -1, the graph of the function is a
vertical shrink along with a reflection across the x-axis of the graph of the parent
quadratic function; the given graph has these characteristics.
4b. Error Analysis Knowing that the graph of f (x) = ax2 is a parabola that has its vertex
at (0, 0) and passes through the point (-2, 2), a student says that the value of a must
be -_12. Explain why this value of a is not reasonable.
1
If a = -_
, then the graph of the function would be a parabola that lies
© Houghton Mifflin Harcourt Publishing Company
2
(except for its vertex) entirely in Quadrants III and IV, but the given
point lies in Quadrant II.
4c. A quadratic function has a minimum value when the function’s graph opens up,
and it has a maximum value when the function’s graph opens down. In each case,
the minimum or maximum value is the y-coordinate of the vertex of the function’s
graph. Under what circumstances does the function f (x) = ax2 have a minimum
value? A maximum value? What is the minimum or maximum value in each case?
Minimum value when a > 0; maximum value when a < 0; minimum or
maximum value is 0.
4d. A function is called even if f (-x) = f (x) for all x in the domain of the function. Show
that the function f (x) = ax2 is even for any value of a.
f (-x) = a(-x)2 = ax2 = f (x)
415
Chapter 8
Lesson 1
PRACTICE
Graph each quadratic function.
1.
f (x) = 3x2
2.
x
-4
-2
x
0
2
4.
f (x) = 0.6x
4
f (x) = -1.5x2
y
8
y
2
x
6
-4
-2
0
-2
4
-4
2
x
0
-2
2
-8
2
-4
4
-6
4
-2
3.
2
-4
2
-2
0
-2
4
-4
y
2
6
2
-6
4
-8
-2
Write the equation for each quadratic function whose graph is shown.
5.
(-1, 5)
6.
y
8
y
8
6
6
4
4
2
(3, 6)
2
x
-4
-2
0
2
x
4
-4
-2
-2
0
2
4
-2
2 2
f (x) = _
x
f (x) = 5x2
3
y
7.
8.
2
35
y
x
25
-5
-3
0
3
5
-2
15
(10, 10)
5
-25 -15
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
f (x) = -__34x2
y
8
-4
x
-5 0
-5
5
15
(4, -4)
-6
25
-8
1 2
x
f (x) = __
1 2
x
f (x) = -_
10
Chapter 8
Chapter 8
4
416
Lesson 1
416
Lesson 1
ADD I T I O N A L P R AC T I C E
AND PRO BL E M S O LV I N G
y
8
6
Assign these pages to help your students practice
and apply important lesson concepts. For
additional exercises, see the Student Edition.
4
2
x
-8
-4
-2
Answers
2
4
6
8
-4
-6
Additional Practice
-8
1.
x
y
−2
−2
−1
0
1
2
vertical stretch by a factor of 2
1
−_
2
0
3. downward, a = −3, a < 0
1
−_
2
−2
5. a. y = -x2; b. maximum: 0;
c. D: all real numbers; R: y ≤ 0
y
6. a. y = x2; b. minimum: 0;
c. D: all real numbers; R: y ≥ 0
4. upward, a = 5, a > 0
4
Problem Solving
2
x
-4
-2
0
2
1.
4
x
f(x)
-2
-4
-2
0
2
4
-4
-1
0
-1
-4
-4
1 followed by a
vertical shrink by a factor of _
2
reflection over the x-axis
© Houghton Mifflin Harcourt Publishing Company
y
4
2
x
2.
0
-4
x
y
−2
8
−1
2
0
0
1
2
2. f(x) = 0.2x2
2
8
3. f(x) = -4x2
Chapter 8
4
-2
-4
417
4. f(x) = 24x2
5. B
6. J
7. C
Lesson 1
Name
Class
Notes
8-1
Date
Additional Practice
© Houghton Mifflin Harcourt Publishing Company
417
Chapter 8
Lesson 1
Problem Solving
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
Chapter 8
Chapter 8
418
Lesson 1
418
Lesson 1