Transforming Quadratic Functions

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Transforming 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 1 Comparing Widths of Parabolas
Order the functions from narrowest graph to
widest.
f(x) 3x2, g(x) 0.5x2
The function with the narrowest graph has the
greatest a.
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Example 2
Order the functions from narrowest graph to
widest.
f(x) x2, g(x) x2, h(x) 2x2
h(x) 2x2
f(x) x2
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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 3
Compare the graph of the function with the graph
of f(x) x2.
Method 1 Compare the graphs.
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Example 3 Continued
Compare the graph of the function with the graph
of f(x) x2
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Example 4
Compare the graph of the function with the graph
of f(x) x2
g(x) 3x2
Both graphs open upward. G(x) is narrower
than f(x).
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Example 5
Compare the graph of each the graph of f(x) x2.
g(x) x2 4
The graph of g(x) opens downward, while the
graph of f(x) opens upward. The graph of g(x) is
translated down four units. both graphs have the
same width.
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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 6
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 6 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 6 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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Try these

• 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.
• It is translated down two units.

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Try these
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