9-4 Transforming Quadratic Functions
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 x = 0; (0, 3); opens upward
2. y = 2x2 x = 0; (0, 0); opens upward
3. y = –0.5x2 – 4 x = 0; (0, –4); opens
downward
Holt Algebra 1
9-4 Transforming Quadratic Functions
Learning Target
Students will be able to: Graph and
transform quadratic functions.
Holt Algebra 1
9-4 Transforming Quadratic Functions
Remember!
You saw in Lesson 5-9 that the graphs of all
linear functions are transformations of the linear
parent function y = x.
Holt Algebra 1
9-4 Transforming Quadratic Functions
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:
• The axis of symmetry
is x = 0, or the y-axis.
• The vertex is (0, 0)
• The function has only
one zero, 0.
Holt Algebra 1
9-4 Transforming Quadratic Functions
Holt Algebra 1
9-4 Transforming Quadratic Functions
The value of a in a quadratic function determines
not only the direction a parabola opens, but also
the width of the parabola.
Holt Algebra 1
9-4 Transforming Quadratic Functions
Order the functions from narrowest graph to
widest.
f(x) = 3x2, g(x) = 0.5x2
Find |A| for each function.
|3| = 3
|0.05| = 0.05
1.
f(x) = 3x2
2.
g(x) = 0.5x2
Holt Algebra 1
The function with the
narrowest graph has the
greatest |A|.
9-4 Transforming Quadratic Functions
Order the functions from narrowest graph to
widest.
f(x) = x2, g(x) =
x2, h(x) = –2x2
|1| = 1
|–2| = 2
1.
2.
3.
h(x) = –2x2
f(x) = x2
g(x) =
Holt Algebra 1
x2
The function with the
narrowest graph has the
greatest |A|.
9-4 Transforming Quadratic Functions
Order the functions from narrowest graph
to widest.
f(x) = –x2, g(x) = x2
The function with the
narrowest graph has the
|–1| = 1
greatest |A|.
1.
f(x) = –x2
2.
g(x) =
Holt Algebra 1
x2
9-4 Transforming Quadratic Functions
Holt Algebra 1
9-4 Transforming Quadratic Functions
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.
Holt Algebra 1
9-4 Transforming Quadratic Functions
Holt Algebra 1
9-4 Transforming Quadratic Functions
Helpful Hint
When comparing graphs, it is helpful to draw
them on the same coordinate plane.
Holt Algebra 1
9-4 Transforming Quadratic Functions
Compare the graph of the function with the graph
of f(x) = x2.
2+ 3
g(x)
=
x
• The graph of g(x) =
x2 + 3
is wider than the graph of f(x) = x2.
• The graph of g(x) =
x2 + 3
opens downward.
f  x   x2
1 2
g  x   x  3
4
Holt Algebra 1
9-4 Transforming Quadratic Functions
Compare the graph of the function with the graph
of f(x) = x2
g(x) = 3x2
g  x   3x2
f  x   x2
Holt Algebra 1
9-4 Transforming Quadratic Functions
Compare the graph of each the graph of
f(x) = x2.
g(x) = –x2 – 4
f  x   x2
g  x    x2  4
Holt Algebra 1
9-4 Transforming Quadratic Functions
Compare the graph of the function with the
graph of f(x) = x2.
g(x) = 3x2 + 9
g  x   3x2  9
f  x   x2
3
Holt Algebra 1
9-4 Transforming Quadratic Functions
Compare the graph of the function with the
graph of f(x) = x2.
g(x) =
x2 + 2
 Wider
x
g  x 
2
Holt Algebra 1
1 2
x 2
2
4
g  x    x2  4
f  x   x2
9-4 Transforming Quadratic Functions
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.
Holt Algebra 1
9-4 Transforming Quadratic Functions
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.
h1(t) = –16t2 + 400 Dropped from 400 feet.
h2(t) = –16t2 + 324 Dropped from 324 feet.
16t  400  0
16t  324  0
16t  400
16
16
t 2  25
t 5
16t 2  324
16
16
t 2  81/ 4
t  9/2
2
2
Holt Algebra 1
h t 
2
50
t
9-4 Transforming Quadratic Functions
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 yintercept of h1 is 76 units higher.
h t 
h1 t   16t 2  400
h2 t   16t 2  324
50
t
b. Use the graphs to tell when each
4.5 seconds
softball reaches the ground.
5 seconds
Holt Algebra 1
9-4 Transforming Quadratic Functions
Caution!
Remember that the graphs show here represent
the height of the objects over time, not the paths
of the objects.
HW pp. 617-619/10-42, 44-49
Holt Algebra 1