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Functions
Essential Question: What is the effect of the constant a on the graph of f​(x)​= a​x ​2​?
Resource
Locker
Explore Understanding the Parent Quadratic Function
A function that can be represented in the form of ƒ​(x)​ = a​x  2​ ​+ bx + c is called a quadratic function. The terms a,
b, and c, are constants where a ≠ 0. The greatest exponent of the variable x is 2. The most basic quadratic function is
ƒ​(x)​= x​  ​2​, which is the parent quadratic function.
A
Here is an incomplete table of values for
the parent quadratic function. Complete it.
© Houghton Mifflin Harcourt Publishing Company
x
B
Plot the ordered pairs as points on the graph,
and connect the points to sketch a curve.
y
f​(x)​= x​  ​2​
−3
ƒ​(x)​= ​x2​ ​= (​​ −3)​​ ​ = 9
8
−2
4
6
−1
1
4
0
0
1
1
2
4
3
9
2
2
x
-4
-2
0
2
4
The curve is called a parabola. The point through
which the parabola turns direction is called its vertex.
The vertex occurs at (​ 0, 0)​for this function. A vertical
line that passes through the vertex and divides the
parabola into two symmetrical halves is called the axis of
symmetry. For this function, the axis of symmetry is the
y-axis.
Reflect
1.
Discussion What is the domain of ƒ​(x)​= ​x2​ ​?
The domain is the set of all real numbers.
2.
Discussion What is the range of ƒ​(x)​= ​x2​ ​?
The range is the set y ≥ 0.
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Graphing g(x) = ax 2 when a > 0
Explain 1
The graph g(x) = ax 2 , is a vertical stretch or compression of its parent functionƒ(x) = x2. The graph opens
upward when a > 0.
Vertical Compression
Vertical Stretch
g(x) = ax 2 with |a| > 1.
g(x) = ax 2 with 0< |a| < 1.
The graph of
g(x) is narrower than the parent
function ƒ(x).
The graph of
g(x) is wider than the parent
function ƒ(x).
y
8 g(x)
8
6
f(x)
y
6
f(x)
4
4
g(x)
2
x
-4
-2
0
2
x
-4
4
-2
0
2
4
The domain of a quadratic function is all real numbers. When
a > 0, the graph of g(x) = ax 2opens
upward, and the function has a minimum value that occurs at the vertex of the parabola. So, the range of
g(x) = ax 2,where a > 0, is the set of real numbers greater than or equal to the minimum value.
Example 1 Graph each quadratic function by plotting points and sketching the curve.
State the domain and range.

g(x) = 2x2
y
x
g(x) = 2x2
-3
18
-2
8
6
-1
2
4
0
0
2
1
2
8
3
18
Domain: all real numbers
© Houghton Mifflin Harcourt Publishing Company
2
8
x
-4
-2
0
2
4
x
Range: y ≥ 0
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B
g​(x)​= __
​ 12 ​​x
2​ ​
x
g​(x)​= __
​ 12 ​ ​ x2​ ​
-3
1
4 ​ __
​
2
-2
2
0
0
2
2
3
1
4 ​ _
​
2
y
8
6
4
Domain:
all
real numbers
2
x
-4
0
-2
2
Range:
y ≥ 0
4
Reflect
3.
For a graph that has a vertical compression or stretch, does the axis of symmetry change?
No,
the axis of symmetry does not change.
Graph each quadratic function. State the domain and range.
4. g​(x)​= 3​x 2​ ​ D: all real; R: y ≥ 0
5. g​(x)​= _​ 13 ​​x 2​ ​ D: all real; R: y ≥ 0
4
y
4
2
y
2
x
-4
© Houghton Mifflin Harcourt Publishing Company
-2
0
2
x
-4
4
-2
0
2
4
Explain 2 Graphing g​(x)​ = a​x  ​2​when a < 0
The graph of y = −​x 2​ ​opens downward. It is a reflection of the graph of y = ​x 2​ ​across the x-axis. So, When a < 0,
the graph of g​(x)​= a​x  2​ ​opens downward, and the function has a maximum value that occurs at the vertex of the
parabola. In this case, the range is the set of real numbers less than or equal to the maximum value.
Vertical Stretch
Vertical Compression
(x)​= a​x  2​ ​with |a| > 1.
g​
(x)​= a​x  2​ ​with 0 < |a| < 1.
g​
The graph of g​(x)​is narrower than the parent
function f(x).
The graph of g​(x)​is wider than the parent
function f ​(x)​.
4
2
y
4
f(x)
2
y
f(x)
x
-4
-2
0
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-4
4
-2
-4
Module 19
2
x
g(x)
g(x)
-2
0
2
4
-2
-4
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Example 2 Graph each quadratic function by plotting points and sketching the curve.
State the domain and range.


g(x) = −2x 2
y
x
g(x) = 2x2
-3
-18
-2
-8
-1
-2
0
0
-6
1
-2
-8
2
-8
3
-18
1 x2
g(x) = −_
2
x
1 2
g(x) = − __
x
2
1
−4 _
2
-2
−2
-1
1
−_
2
0
−0
1
1
−_
2
2
−2
3
1
−4 _
2
0
-2
2
4
-4
Domain: all real numbers
y≤0
Range:
y
-4
-2
0
x
2
4
-2
-4
-6
-8
Domain: all real numbers
y≤0
Range:
Reflect
6.
Does reflecting the parabola across the x-axis (a < 0) change the axis of symmetry?
No, the axis of symmetry is a line that extends both up and down and does not change
© Houghton Mifflin Harcourt Publishing Company
-3
-4
x
upon reflection.
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Graph each function. State the domain and range.
7.
g(x) = −3x 2 D: all real; R: y ≤ 0
y
-4
-2
0
8.
g(x) = −__13 x 2 D: all real; R: y ≤ 0
x
2
y
-4
4
-2
-2
x
0
2
4
-2
-4
-4
-6
-6
-8
-8
Writing a Quadratic Function Given a Graph
Explain 3
You can determine a function rule for a parabola with its vertex at the origin by substitutingx and y values for any
other point on the parabola into g(x) = ax 2 and solving fora.
Example 3 Write the rule for the quadratic functions shown on the graph.

8
Use the point (2, 2).
y
6
2
(2, 2)
x
© Houghton Mifflin Harcourt Publishing Company
-2
0

2
-2
-4
(-2, -8) -6
-8
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x
2
-2
2
4
Evaluate x 2.
2 = 4a
Divide both sides by 4 to isolate a.
__1 = a
(
2
g(x) = __21 x 2
Write the function rule.
4
y
-4
= a(2)
Replace x and g(x) with point values. 2
4
-4
g(x) = ax 2
)
Use the point -2, −8 .
g(x) = ax 2
Replace x and g(x) with point values.
−8 = a −2
Evaluate x 2.
( )
−8 = 4 a
Divide both sides by 4 to isolate a.
−2 = a
Write the function rule.
g(x) = −2x 2
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2
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9.
10.
y
8
y
-4
6
4
-2
0
2
-2
4
(-1, 1)
-4
(1, 4)
x
-6
x
-4
0
-2
2
Use the point (1, 4).
g(x) = ax
-8
4
Use the point (1, −1).
g(x) = ax 2
2
4 = a (1)
−1 = a (1)
2
4=a
g(x) = 4x
2
−1 = a
g(x) = −x 2
2
Explain 4
Real-world situations can be modeled by parabolas.
Example 3

For each model, describe what the vertex, y-intercept, and endpoint(s) represent in the
situation it models, and then determine the equation of the function.
Depth (yards)
-4
-4
-2
0
y
x
2
-8
-16
-24
-32
(-4, -32)
(4, -32)
Time (seconds)
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4
The y-intercept occurs at the vertex of the parabola at (0, 0), where
the dolphin is at the surface to breathe.
The endpoint (-4, -32) represents a depth of 32 yards below
the surface at 4 seconds before the dolphin reaches the surface to
breathe.
The endpoint (4, -32) represents a depth of 32 yards below the
surface at 4 seconds after the dolphin reaches the surface to breathe.
The graph is symmetric about the y-axis with the vertex at the
origin, so the function will be of the form y = ax 2, or d(t) = at 2.
Use a point to determine the equation.
d(t) = at 2
2
-32 = a(4)
-32 = a ∙ 16
-2 = a0
The function is d(t) = -2t 2.
894
Schuyl/Alamy
This graph models the depth in
yards below the water’s surface of
a dolphin before and after it rises
to take a breath and descends again.
The depth d is relative to time t,
in seconds, and t = 0 is when
dolphin reaches a depth of 0 yards
at the surface.
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B
Satellite dishes reflect radio waves onto a collector
by using a reflector (the dish) shaped like a parabola.
The graph shows the height h in feet of the reflector
relative to the distance x in feet from the center of the
satellite dish.
Height (feet)
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y
(60, 12)
10
x
0
10
20
30
40
50
60
Distance from Center (feet)
The y-intercept occurs at the vertex, which represents
the distance x = 0 feet from the center of the dish.
The left end-point represents the height h = 0 feet at the center of the dish.
The right end-point represents the height h = 12 feet at the distance x = 60 feet from the center of
the dish.
h(x) = ax 2
The function will be of the form
. Use 60 , 12 to determine the equation.
h(x) = ax 2
( )
12 = a 60
12 =
2
3600
a
1
a =_
300
1
___
h(x) = 300 x 2
11. The graph shows the height h in feet of a rock dropped down a deep well as a function of time t in seconds.
© Houghton Mifflin Harcourt Publishing Company
Using the point (2, -64) to determine the equation:
h(t) = at 2
-64 = a(2)
2
a = -16
0.5
1
1.5
2
x
-16
-32
-48
(2, -64)
Time Seconds
d(t) = -16t 2
A1_MNLESE368187_U8M19L1.indd 895
y
0
-64
-64 = 4a
Module 19
16
Height (feet)
The y-intercept occurs at the left end-point, which is
also the vertex, and represents the height h = 0 at
which the rock was released at ground level.
The right end point represents the height h = -64
feet at which the rock hits the bottom of the well
2
seconds after it was released.
t=
895
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