Transforming Quadratic Functions

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Transforming Quadratic Functions
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 1
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Warm Up For each quadratic function, find the
axis of symmetry and vertex, and state whether
the function opens upward or downward. 1. y x2
3 2. y 2x2 3. y 0.5x2 4
x 0 (0, 3) opens upward
x 0 (0, 0) opens upward
x 0 (0, 4) opens downward
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Objective
Graph and transform quadratic functions.
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The quadratic parent function is f(x) x2. The
graph of all other quadratic functions are
transformations of the graph of f(x) x2.
For the parent function f(x) x2
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The value of a in a quadratic function determines
not only the direction a parabola opens, but also
the width of the parabola.
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Example 1A Comparing Widths of Parabolas
Order the functions from narrowest graph to
widest.
f(x) 3x2, g(x) 0.5x2
Step 1 Find a for each function.
0.05 0.05
3 3
Step 2 Order the functions.
The function with the narrowest graph has the
greatest a.
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Example 1A Continued
Order the functions from narrowest graph to
widest.
f(x) 3x2, g(x) 0.5x2
Check Use a graphing calculator to compare the
graphs.
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Example 1B Comparing Widths of Parabolas
Order the functions from narrowest graph to
widest.
f(x) x2, g(x) x2, h(x) 2x2
Step 1 Find a for each function.
1 1
2 2
Step 2 Order the functions.
h(x) 2x2
The function with the narrowest graph has the
greatest a.
f(x) x2
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Example 1B Continued
Order the functions from narrowest graph to
widest.
f(x) x2, g(x) x2, h(x) 2x2
Check Use a graphing calculator to compare the
graphs.
h(x) 2x2 has the narrowest graph and
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Check It Out! Example 1a
Order the functions from narrowest graph to
widest.
Step 1 Find a for each function.
1 1
Step 2 Order the functions.
f(x) x2
The function with the narrowest graph has the
greatest a.
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Check It Out! Example 1a Continued
Order the functions from narrowest graph to
widest.
Check Use a graphing calculator to compare the
graphs.
f(x) x2 has the narrowest graph and
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Check It Out! Example 1b
Order the functions from narrowest graph to
widest.
f(x) 4x2, g(x) 6x2, h(x) 0.2x2
Step 1 Find a for each function.
Step 2 Order the functions.
The function with the narrowest graph has the
greatest a.
g(x) 6x2
f(x) 4x2
h(x) 0.2x2
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Check It Out! Example 1b Continued
Order the functions from narrowest graph to
widest.
f(x) 4x2, g(x) 6x2, h(x) 0.2x2
Check Use a graphing calculator to compare the
graphs.
g(x) 6x2 has the narrowest graph and
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The value of c makes these graphs look different.
The value of c in a quadratic function determines
not only the value of the y-intercept but also a
vertical translation of the graph of f(x) ax2
up or down the y-axis.
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Example 2A Comparing Graphs of Quadratic
Functions
Compare the graph of the function with the graph
of f(x) x2.
Method 1 Compare the graphs.
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Example 2A Continued
Compare the graph of the function with the graph
of f(x) x2
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Example 2B Comparing Graphs of Quadratic
Functions
Compare the graph of the function with the graph
of f(x) x2
g(x) 3x2
Method 2 Use the functions.

• Since 3 gt 1, the graph of g(x) 3x2 is
  narrower than the graph of f(x) x2.

• The vertex of f(x) x2 is (0, 0). The vertex of
  g(x) 3x2 is also (0, 0).

• Both graphs open upward.

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Example 2B Continued
Compare the graph of the function with the graph
of f(x) x2
g(x) 3x2
Check Use a graph to verify all comparisons.
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Check It Out! Example 2a
Compare the graph of each the graph of f(x) x2.
g(x) x2 4
Method 1 Compare the graphs.

• The axis of symmetry is the same.

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Check It Out! Example 2b
Compare the graph of the function with the graph
of f(x) x2.
g(x) 3x2 9
Method 2 Use the functions.

• Since 3gt1, the graph of g(x) 3x2 9 is
  narrower than the graph of f(x) x2.

• The vertex of f(x) x2 is (0, 0). The vertex of
  
• g(x) 3x2 9 is translated 9 units up to (0, 9).

• Both graphs open upward.

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Check It Out! Example 2b Continued
Compare the graph of the function with the graph
of f(x) x2.
g(x) 3x2 9
Check Use a graph to verify all comparisons.
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Check It Out! Example 2c
Compare the graph of the function with the graph
of f(x) x2.
g(x) x2 2
Method 1 Compare the graphs.
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Check It Out! Example 2c Continued
Compare the graph of the function with the graph
of f(x) x2.
g(x) x2 2
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The quadratic function h(t) 16t2 c can be
used to approximate the height h in feet above
the ground of a falling object t seconds after it
is dropped from a height of c feet. This model is
used only to approximate the height of falling
objects because it does not account for air
resistance, wind, and other real-world factors.
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Example 3 Application
Two identical softballs are dropped. The first is
dropped from a height of 400 feet and the second
is dropped from a height of 324 feet.
a. Write the two height functions and compare
their graphs.
Step 1 Write the height functions. The
y-intercept c represents the original height.
h1(t) 16t2 400 Dropped from 400 feet.
h2(t) 16t2 324 Dropped from 324 feet.
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Example 3 Continued
Step 2 Use a graphing calculator. Since time and
height cannot be negative, set the window for
nonnegative values.
The graph of h2 is a vertical translation of the
graph of h1. Since the softball in h1 is dropped
from 76 feet higher than the one in h2, the
y-intercept of h1 is 76 units higher.
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Example 3 Continued
b. Use the graphs to tell when each softball
reaches the ground.
The zeros of each function are when the softballs
reach the ground.
The softball dropped from 400 feet reaches the
ground in 5 seconds. The ball dropped from 324
feet reaches the ground in 4.5 seconds
Check These answers seem reasonable because the
softball dropped from a greater height should
take longer to reach the ground.
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Check It Out! Example 3
Two tennis balls are dropped, one from a height
of 16 feet and the other from a height of 100
feet.
a. Write the two height functions and compare
their graphs.
Step 1 Write the height functions. The
y-intercept c represents the original height.
h1(t) 16t2 16 Dropped from 16 feet.
h2(t) 16t2 100 Dropped from 100 feet.
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Check It Out! Example 3 Continued
Step 2 Use a graphing calculator. Since time and
height cannot be negative, set the window for
nonnegative values.
The graph of h2 is a vertical translation of the
graph of h1. Since the ball in h2 is dropped from
84 feet higher than the one in h1, the
y-intercept of h2 is 84 units higher.
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Check It Out! Example 3 Continued
b. Use the graphs to tell when each tennis ball
reaches the ground.
The zeros of each function are when the tennis
balls reach the ground.
The tennis ball dropped from 16 feet reaches the
ground in 1 second. The ball dropped from 100
feet reaches the ground in 2.5 seconds.
Check These answers seem reasonable because the
tennis ball dropped from a greater height should
take longer to reach the ground.
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Lesson Quiz Part I

• 1. Order the function f(x) 4x2, g(x) 5x2,
  and h(x) 0.8x2 from narrowest graph to widest.
• 2. Compare the graph of g(x) 0.5x2 2 with the
  graph of f(x) x2.

g(x) 5x2, f(x) 4x2, h(x) 0.8x2

• The graph of g(x) is wider.
• Both graphs open upward.
• Both have the axis of symmetry x 0.
• The vertex of g(x) is (0, 2) the vertex of f(x)
  is (0, 0).

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Lesson Quiz Part II
Two identical soccer balls are dropped. The first
is dropped from a height of 100 feet and the
second is dropped from a height of 196 feet. 3.
Write the two height functions and compare their
graphs.
The graph of h1(t) 16t2 100 is a vertical
translation of the graph of h2(t) 16t2 196
the y-intercept of h1 is 96 units lower than that
of h2.
4. Use the graphs to tell when each soccer ball
reaches the ground.
2.5 s from 100 ft 3.5 from 196 ft