Warm-Up
 Please fill out the chart to the best of your ability
Assignment
 p. 168
 # 1, 4, 6
Objectives: Students will know how to identify and graph
shifts, reflections, and nonrigid transformations of functions.
Extra Practice
http://www.khanacademy.org/exercise/shifting_and_reflecting_functions
Extra Examples
http://www.khanacademy.org/video/algebra-ii--shifting-quadratic-graphs?topic=californiastandards-test-algebra-2
The Original Six
 Constant Function f(x) = c
y
x
X
Y
-1
-2
0
-2
1
-2
 Identity Function f(x) = x
y
x
X
Y
-1
-1
0
0
1
1
Absolute Value Function f(x) = |x|
y
X Y
-1 1
x
0
0
1
1
Square Root Function
y
X Y
-1
x
0
0
4
2
f ( x) 
x
Quadratic Function f(x) = x2
y
X Y
-2 4
x
0
0
2
4
Cubic Function f(x) = x3
y
X Y
-1 -1
x
0
0
2
8
Vertical and Horizontal Shifts
 Use a graphing utility to graph:
 Y1 = f(x) = x2. Then, on the same viewing screen, graph
Y2 = (x – 4)2.


How did we change the equation?
How did the graph change?
 Y3 = (x + 4)2, Y4 = x2 – 4, and Y5 = x2 + 4.


How did we change the equations?
How did the graphs change?
 Let c be a positive real number. The following changes
in the function y = f(x) will produce the stated shifts in
the graph of y = f(x).
 h(x) =f(x - c)
 Y2 = (x – 4)2
 h(x) =f(x + c)
 Y3 = (x + 4)2
 h(x) =f(x) - c
 Y4 = x2 – 4
 h(x) =f(x) + c
 Y5 = x2 + 4
Horizontal shift c units to the right
Horizontal shift c units to the left
Vertical shift c units downward
Vertical shift c units upward
Example 1. Given f(x) = x3 + x, describe and graph the
shifts in the graph of f generated by the following
functions.
a) g(x) = (x + 1)3 + x + 1.
b) h(x) = (x - 4)3 + x.
 Let f ( x)  . x Write the equation for the function
resulting from a vertical shift of 3 units downward and
a horizontal shift of 2 units to the right of the graph of
f(x) = | x  2 |  3
Warm Up
 Write about what it means to reflect over the y-axis
and x-axis without using the word symmetry?
Assignment
 http://www.khanacademy.org/exercise/shifting_and_r
eflecting_functions
 Register me as your coach and do 10 problems
Reflecting Graphs
Reflecting Graphs
 Use a graphing utility to graph:
 Y1 = f(x) = (x – 2)3. Then, on the same viewing screen,
graph Y2 = -(x – 2)3.
 Y3 = (-x - 2)3.
 The following changes in the function y = f(x) will
produce the stated reflections in the graph of y = f(x).
 h(x) =f(-x)
Reflection in the y-axis
 h(x) = -f(x)
Reflection in the x-axis
 Example 2. Given f(x) = x3 + 3, describe the
reflections in the graph of f generated by the following
functions.

a) g(x) = -x3 + 3. Reflected in the ???-axis.

b) h(x) = -(x3 + 3) = -x3 - 3.
Reflected in the ???-axis.
 Example 3. Below is the graph of
 a) y =
1.
x
2
b) Graph y = -f(x). c) Graph y = f(-x) + 1
2.
3.
Widening and Narrowing
 Distort the shape of the graph
 Is not shifting or reflecting it.
 Come from equations of the form y = cf(x).
 If c > 1, then there is a vertical stretch of the graph of y =
f(x). If 0 < c < 1, then there is a vertical shrink.
Example 4. Given f(x) = 1- x2,
describe the graph of g(x) = 3 – 3x2.
 Because 3 – 3x2 = 3(1- x2), the graph of g is a vertical
stretch (each y-value is multiplied by 3) of the graph of
f.
X
f(x)=1- x2
g(x) = 3 – 3x2
-1
0
0
0
1
3
1
0
0
2
-3
-9
yx
2
y  2x
2
yx
X
Y
X
Y
X
Y
-2
4
-2
8
-2
-1
1
-1
2
0
0
0
1
1
2
4
2
2 2
y x
3
X
Y
4
-2
(8/3)
-1
1
-1
(2/3)
0
0
0
0
0
1
2
1
1
1
(2/3)
2
8
2
4
2
(8/3)
 Please describe the following function
g(x) = -2f(x)
 Reflection?
1
h(x) = 2 f ( x)
 Reflection?
Wider or Narrower?
Wider or Narrower
Warm Up
In the mail, you receive a coupon for $5 off of a pair of
jeans. When you arrive at the store, you find that all
jeans are 25% off. You find a pair of jeans for $36.
 1. If you use the $5 off coupon first, and then you
use the 25% off on the remaining amount, how
much will the jeans cost?
 2. If you use the 25% off first, and then you use the
$5 off on the remaining amount, how much will
the jeans cost?
Jean fiend
 Let the cost of the jeans be represented by a variable x.
Write a function f(x) that represents the cost of the
jeans after the $5 off coupon.
 Write a function g(x) that represents the cost of
the jeans after the 25% discount.
Function Composition
 Write a new function r(x) that represents the cost of
the jeans if the 25% discount is applied first and the $5
off coupon is applied second.
 Write a new function s(x) that represents the cost
of the jeans if the $5 off coupon is applied first and
the 25% discount is applied second.
Compositions of Functions
 The composition of the function f with the function g
is

(f  g)(x) = f(g(x)).
 f(x) = x – 5, g(x) = .75x
 (f  g)(x) = f(g(x)) = [.75x] - 5
 The composition of the function g with the function f
is

(g  f )(x) = g(f(x)).
 g(x) = .75x, f(x) = x – 5
 (g  f )(x) = .75(x - 5)
Welcome to my domain
 The domain of (f  g) is the set of all x in the domain of
g in the domain of f.
Domain of
f
Domain
of g and
domain
of f  g
 Example 2. f(x) = x2 + 2x and g(x) = 2x + 1. Find the
following.
 Find (f  g)(x)
 f gx   f gx 
 f 2x 1
 2x  1  22x 1
2
 4x 2  4x  1 4x  2
2
 4x  8x  3
Find (g  f )(x)
g
f x   g f x 


 2x 2  2 x  1
 g x2  2 x
 2 x 2  4 x 1
Objectives: Students will know how to find arithmetic
combinations and compositions of functions.
Arithmetic Combinations of
Functions
 Let f and g be functions with overlapping domains.
Then for all x common to both domains:
 (f  g)(x) = f(x)  g(x)
 (fg)(x) = f(x) • g(x)
f (x)
 f 
 g x   g(x) ,
 
p
 provided g(x)  0.
Example 1. f(x) = x2 + 2x and g(x) =
2x + 1. Find the following.
 fg x   f x   g(x)
a)
f
 gx   f (x)  g(x)


 x  2x  2x 1
2
 x 2  4x 1
b)
f
 gx   f (x)  g(x)
2

 x  2x  2x 1
 x2  1
c)


 x  2x 2x 1
2
 2x 3  5x 2  2x
 f 

x 


d)  g 
f x 
gx 
x 2  2x

2x  1