Five-Minute Check
Then/Now
Key Concept: Reflections in the
Coordinate Axes
New Vocabulary
Example 3: Write Equations for
Transformations
Key Concept: Linear and Polynomial
Parent Functions
Key Concept: Vertical and
Horizontal Dilations
Key Concept: Square Root and
Reciprocal Parent Functions
Example 4: Describe and Graph
Transformations
Key Concept: Absolute Value Parent
Function
Example 5: Graph a PiecewiseDefined Function
Key Concept: Greatest Integer
Parent Function
Example 6: Real-World Example:
Transformations of Functions
Example 1: Describe Characteristics
of a Parent Function
Key Concept: Transformations
with Absolute Value
Key Concept: Vertical and
Horizontal Translations
Example 7: Describe and Graph
Transformations
Example 2: Graph Translations
Estimate and classify the extrema for each f(x).
Support your answer numerically.
A. Relative minimum: (–1.5, –9.375); relative maximum: (1, 0)
B. Relative minimum: (–0.5, –4.5); relative maximum: (1, 0)
C. Relative minimum: (–1.5, –9.375); relative maximum: (0.5, 0)
D. Relative minimum: (–2, –9); relative maximum: (0.5, 0)
Estimate and classify the extrema for each f(x).
Support your answers numerically.
A. Absolute minimum: (–2, –4)
B. Absolute minimum: (–2, –8)
C. Relative minimum: (–2, –4)
D. Absolute minimum: (–1, –4)
Find the average rate of change of
f(x) = 3x 3 – x 2 + 5x – 3 on the interval [–1, 2].
A.
B. 5
C.
D. 13
You analyzed graphs of functions. (Lessons 1-2
through 1-4)
• Identify, graph, and describe parent functions.
• Identify and graph transformations of parent
functions
• parent function
• absolute value function
• constant function
• step function
• zero function
• greatest integer function
• identity function
• transformation
• quadratic function
• translation
• cubic function
• reflection
• square root function
• dilation
• reciprocal function
Describe Characteristics of a Parent Function
Describe the following characteristics of the graph
of the parent function
: domain, range,
intercepts, symmetry, continuity, end behavior,
and intervals on which the graph is
increasing/decreasing.
Describe Characteristics of a Parent Function
The graph of the reciprocal
function shown in the graph has
the following characteristics.
The domain of the function is,
and the range
is
.
The graph has no intercepts.
The graph is symmetric with
respect to the origin, so f(x) is
odd.
Describe Characteristics of a Parent Function
The graph is continuous for all
values in its domain with an
infinite discontinuity at x = 0.
The end behavior is as
The graph is decreasing on the
interval
and decreasing
on the interval
.
Describe Characteristics of a Parent Function
Answer: D and R:
no intercepts. The
graph is symmetric about the origin, so f(x)
is odd. The graph is continuous for all
values in its domain with an infinite
discontinuity at x = 0.The end behavior is
as
The graph decreases on both intervals of
its domain.
Describe the following characteristics of the graph
of the parent function f(x) = x 2: domain, range,
intercepts, symmetry, continuity, end behavior,
and intervals on which the graph is
increasing/decreasing.
A.
D:
, R:
; y-intercept = (0, 0). The graph is symmetric with respect
to the y-axis. The graph is continuous everywhere. The end behavior is as
,
and as
,
. The graph is decreasing on the
interval
and increasing on the interval
.
B.
D:
, R:
; y-intercept = (0, 0). The graph is symmetric with respect
to the y-axis. The graph is continuous everywhere. The end behavior is as
,
. As
,
. The graph is decreasing on the
interval
and increasing on the interval
.
C.
D:
, R:
; y-intercept = (0, 0). The graph is symmetric with respect
to the y-axis. The graph is continuous everywhere. The end behavior is as
,
. As
,
. The graph is decreasing on the interval
and increasing on the interval
.
D.
D:
, R:
; no intercepts. The graph is symmetric with respect to
the y-axis. The graph is continuous everywhere. The end behavior is as
,
. As
,
. The graph is decreasing on the
interval
and increasing on the interval
.
Graph Translations
A. Use the graph of f(x) = x 3 to graph the function
g(x) = x 3 – 2.
This function is of the form g(x) = f(x) – 2. So, the
graph of g(x) is the graph of f(x) = x 3 translated 2 units
down, as shown below.
Answer:
Graph Translations
B. Use the parent graph of f(x) = x 3 to graph the
function g(x) = (x – 1)3.
This function is of the form g(x) = f(x – 1) . So, g(x) is
the graph of f(x) = x 3 translated 1 units right, as shown
below.
Answer:
Graph Translations
C. Use the parent graph of f(x) = x 3 to graph the
function g(x) = (x – 1)3 – 2.
This function is of the form g(x) = f(x – 1) – 2. So,
g(x) is the graph of f(x) = x 3 translated 2 units down
and 1 unit right, as shown below.
Answer:
Use the graph of f(x) = x 2 to graph the function
g(x) = (x – 2)2 – 1.
A.
C.
B.
D.
Write Equations for Transformations
A. Describe how the graphs of
and g(x)
are related. Then write an equation for g(x).
The graph of g(x) is the graph of
1 unit up. So,
.
translated
Answer: The graph is translated 1 unit up;
Write Equations for Transformations
B. Describe how the graphs of and g(x) are
related. Then write an equation for g(x).
The graph of g(x) is the graph of
translated
1 unit to the left and reflected in the x-axis. So,
.
Write Equations for Transformations
Answer: The graph is translated 1 unit to the left and
reflected in the x-axis;
Describe how the graphs of f(x) = x 3 and g(x) are
related. Then write an equation for g(x).
A. The graph is translated 3 units up; g(x) = x 3 + 3.
B. The graph is translated 3 units down; g(x) = x 3 – 3.
C. The graph is reflected in the x-axis; g(x) = –x 3.
D. The graph is translated 3 units down and reflected
in the x-axis; g(x) = –x 3 – 3.
Describe and Graph Transformations
A. Identify the parent function f(x) of
, and
describe how the graphs of g(x) and f(x) are
related. Then graph f(x) and g(x) on the same
axes.
The graph of g(x) is the same as the graph of the
reciprocal function
because
expanded vertically
and 1 < 3.
Describe and Graph Transformations
Answer:
; g(x) is represented by the
expansion of f(x) by a factor of 3.
Describe and Graph Transformations
B. Identify the parent function f(x) of g(x) = –|4x|,
and describe how the graphs of g(x) and f(x) are
related. Then graph f(x) and g(x) on the same
axes.
The graph of g(x) is the same as the graph of the
absolute value function f(x) = |x| compressed
horizontally and then reflected in the x-axis because
g(x) = –4(|x|) = –|4x| = –f(4x), and 1 < 4.
Describe and Graph Transformations
Answer: f(x) = |x| ; g(x) is represented by the
compression of f(x) by a factor of 4 and
reflection in the x-axis.
Identify the parent function f(x) of g(x) = – (0.5x)3,
and describe how the graphs of g(x) and f(x) are
related. Then graph f(x) and g(x) on the same axes.
A. f (x) = x 3; g(x) is
C. f (x) = x 3; g(x) is
represented by
represented by
the expansion of
the reflection of
the graph of f (x)
the graph of f (x)
horizontally by a
in the x-axis.
factor of
.
B. f (x) = x 3; g(x) is
D. f (x) = x 2; g(x) is
represented by
represented by
the expansion of
the expansion
the graph of
of the graph of
f (x) horizontally
f (x) horizontally
by a factor of
by a factor of
and reflected in
and reflected in
the x-axis.
the x-axis.
Graph a Piecewise-Defined Function
Graph
On the interval
.
, graph y = |x + 2|.
On the interval [0, 2], graph y = |x| – 2.
On the interval
graph
Draw circles at (0, 2) and (2, 2); draw dots at (–2, 0),
(0, –2) and (2, 0) because f(–2) = 0, f(0) = –2, and
f(2) = 0, respectively.
Graph a Piecewise-Defined Function
Answer:
Graph the function
A.
C.
B.
D.
.
Transformations of Functions
A. AMUSEMENT PARK The “Wild Ride” roller
coaster has a section that is shaped like the
function
, where g(x) is the
vertical distance in yards the roller coaster track is
from the ground and x is the horizontal distance in
yards from the start of the ride. Describe the
transformations of the parent function
f(x) = x 2 used to graph g(x).
Transformations of Functions
Rewrite the function so that it is in the form
g(x) = a(x – h)2 + k by completing the square.
Original function
Factor
Complete the square.
Transformations of Functions
Write x 2 – 100x + 2500
as a perfect square
and simplify.
Answer: g(x) is the graph of f(x) translated 50 units
right, compressed vertically, reflected in the
x-axis, and then translated 50 units up
Transformations of Functions
B. AMUSEMENT PARK The “Wild Ride” roller coaster
has a section that is shaped like the function
, where g (x) is the vertical distance
in yards the roller coaster track is from the ground and
x is the horizontal distance in yards from the start of the
ride. Suppose the ride designers decide to increase the
highest point of the ride to 70 yards. Rewrite g (x) to
reflect this change. Graph both functions on the same
coordinate axes using a graphing calculator.
Transformations of Functions
A change of the height of the highest point of the ride
from 50 yards to 70 yards is a vertical translation, so
add an additional 10 yards to the expression in
part A to get
. The graph of both functions
on the same coordinate axes using a graphing
calculator is shown.
Transformations of Functions
Answer:
STUNT RIDING A stunt motorcyclist jumps from
ramp to ramp according to the model shaped like
the function
, where g(x) is the
vertical distance in feet the motorcycle is from the
ground and x is the horizontal distance in feet
from the start of the jump. Describe the
transformations of f(x) = x 2 used to graph g(x).
A. The graph of g(x) is the graph of f(x) translated
75 units right.
B. The graph of g(x) is the graph of f(x) translated
18 units up, compressed vertically, and
reflected in the
x-axis.
C. The graph of g(x) is the graph of f(x) translated
75 units right and reflected in the x-axis.
D. The graph of g(x) is the graph of f(x) translated
75 units right and 18 units up, compressed
vertically, and reflected in the x-axis.
Describe and Graph Transformations
A. Use the graph of f(x) = x 2 – 4x + 3 to graph the
function g(x) = |f(x)|.
Describe and Graph Transformations
The graph of f(x) is below the x-axis on the interval
(1, 3), so reflect that portion of the graph in the x-axis
and leave the rest unchanged.
Answer:
Describe and Graph Transformations
B. Use the graph of f(x) = x 2 – 4x + 3 to graph the
function h(x) = f(|x|).
Describe and Graph Transformations
Replace the graph of f(x) to the left of the y-axis with a
reflection of the graph to the right of the y-axis
Answer:
Use the graph of f(x) shown to graph g(x) = |f(x)|
and h(x) = f(|x|).
A.
C.
B.
D.