Download x 2 - Net Start Class

9-4
Functions
Transforming Quadratic
Quadratic Functions
9-4 Transforming
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra
Holt
Algebra
11
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
Objective
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
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.
|3| = 3
|0.05| = 0.05
Step 2 Order the functions.
f(x) = 3x2
g(x) = 0.5x2
Holt Algebra 1
The function with the
narrowest graph has the
greatest |a|.
9-4 Transforming Quadratic Functions
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.
f(x) = 3x2 has the
narrowest graph, and
g(x) = 0.5x2 has the
widest graph
Holt Algebra 1
9-4 Transforming Quadratic Functions
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
f(x) = x2
g(x) =
Holt Algebra 1
x2
The function with the
narrowest graph has the
greatest |a|.
9-4 Transforming Quadratic Functions
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
g(x) = x2 has the
widest graph.
Holt Algebra 1
9-4 Transforming Quadratic Functions
Check It Out! Example 1a
Order the functions from narrowest graph
to widest.
f(x) = –x2, g(x) = x2
Step 1 Find |a| for each function.
|–1| = 1
Step 2 Order the functions.
f(x) = –x2
g(x) =
Holt Algebra 1
x2
The function with the
narrowest graph has the
greatest |a|.
9-4 Transforming Quadratic Functions
Check It Out! Example 1a Continued
Order the functions from narrowest graph
to widest.
f(x) = –x2, g(x) = x2
Check Use a graphing
calculator to compare
the graphs.
f(x) = –x2 has the
narrowest graph and
g(x) = x2 has the
widest graph.
Holt Algebra 1
9-4 Transforming Quadratic Functions
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.
|–4| = 4
|6| = 6 |0.2| = 0.2
Step 2 Order the functions.
g(x) = 6x2
f(x) = –4x2
h(x) = 0.2x2
Holt Algebra 1
The function with the
narrowest graph has the
greatest |a|.
9-4 Transforming Quadratic Functions
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
h(x) = 0.2x2 has
the widest graph.
Holt Algebra 1
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
Example 2A: Comparing Graphs of Quadratic
Functions
Compare the graph of the function with the graph
of f(x) = x2.
g(x) =
x2 + 3
Method 1 Compare the graphs.
• 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 and the graph of
f(x) = x2 opens upward.
Holt Algebra 1
9-4 Transforming Quadratic Functions
Example 2A Continued
Compare the graph of the function with the graph
of f(x) = x2
g(x) =
x2 + 3
• The axis of symmetry is the same.
• The vertex of f(x) = x2 is (0, 0).
The vertex of g(x) =
x2 + 3
is translated 3 units up to (0, 3).
Holt Algebra 1
9-4 Transforming Quadratic Functions
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| > |1|, the graph of g(x) = 3x2 is
narrower than the graph of f(x) = x2.
• Since
for both functions, the axis of
symmetry is the same.
• The vertex of f(x) = x2 is (0, 0). The vertex of
g(x) = 3x2 is also (0, 0).
• Both graphs open upward.
Holt Algebra 1
9-4 Transforming Quadratic Functions
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.
Holt Algebra 1
9-4 Transforming Quadratic Functions
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 graph of g(x) = –x2 – 4
opens downward and the graph
of f(x) = x2 opens upward.
• The axis of symmetry is the same.
• The vertex of f(x) = x2 is (0, 0).
The vertex of g(x) = –x2 – 4
is translated 4 units down to (0, –3).
Holt Algebra 1
9-4 Transforming Quadratic Functions
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 |3|>|1|, the graph of g(x) = 3x2 + 9 is
narrower than the graph of f(x) = x2.
• Since
for both functions, the axis of
symmetry is the same.
• 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.
Holt Algebra 1
9-4 Transforming Quadratic Functions
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.
Holt Algebra 1
9-4 Transforming Quadratic Functions
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.
• The graph of g(x) =
x2 + 2
is wider than the graph of f(x) = x2.
• The graph of g(x) =
x2 + 2
opens upward and the graph
of f(x) = x2 opens upward.
Holt Algebra 1
9-4 Transforming Quadratic Functions
Check It Out! Example 2c Continued
Compare the graph of the function with the
graph of f(x) = x2.
g(x) =
x2 + 2
• The axis of symmetry is the same.
• The vertex of f(x) = x2 is (0, 0).
The vertex of g(x) = x2 + 2
is translated 2 units up to (0, 2).
Holt Algebra 1
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
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.
Holt Algebra 1
9-4 Transforming Quadratic Functions
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 yintercept of h1 is 76 units higher.
Holt Algebra 1
9-4 Transforming Quadratic Functions
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.
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.
Holt Algebra 1
9-4 Transforming Quadratic Functions
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.
Holt Algebra 1
9-4 Transforming Quadratic Functions
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.
Holt Algebra 1
9-4 Transforming Quadratic Functions
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.
Holt Algebra 1