Warm-UpPlease fill out the chart to the best of your

ability

Assignmentp. 168# 1, 4, 6

Section 1.5 Shifting, Reflecting, and Stretching Graphs

Objectives: Students will know how to identify and graph shifts, reflections, and nonrigid

transformations of functions.

Extra Practicehttp://www.khanacademy.org/exercise/shifting_and_reflecting_functions

Extra Exampleshttp://www.khanacademy.org/video/algebra-ii--shifting-quadratic-graphs?topic=california-standards-test-algebra-2

The Original SixConstant Function f(x) = c

Identity Function f(x) = x

   

            

y

x

y

x

X Y

-1 -2

0 -2

1 -2

X Y

-1 -1

0 0

1 1

  Absolute Value Function f(x) = |x|     Square Root Function   

y

x

X Y

-1

1

0 0

1 1

y

x

X Y

-1

0 0

4 2

f ( x) x

Quadratic Function f(x) = x2

Cubic Function f(x) = x3

y

x

y

x

X Y

-1

-1

0 0

2 8

X Y

-2

4

0 0

2 4

Vertical and Horizontal ShiftsUse 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) Horizontal shift c units to the rightY2 = (x – 4)2

h(x) =f(x + c) Horizontal shift c units to the leftY3 = (x + 4)2

h(x) =f(x) - c Vertical shift c units downwardY4 = x2 – 4

h(x) =f(x) + c Vertical shift c units upwardY5 = x2 + 4

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 . 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

xxf )(

f(x) = | x 2 | 3

Warm UpWrite about what it means to reflect over the

y-axis and x-axis without using the word symmetry?

Assignmenthttp://www.khanacademy.org/exercise/shiftin

g_and_reflecting_functionsRegister 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 ofa) y = b) Graph y = -f(x). c) Graph y

= f(-x) + 1

2x

1. 2. 3.

Widening and NarrowingDistort the shape of the graphIs 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

X Y

-2 (8/3)

-1 (2/3)

0 0

1 (2/3)

2 (8/3)

X Y

-2 8

-1 2

0 0

1 2

2 8

X Y

-2 4

-1 1

0 0

1 1

2 4

2xy 2

3

2xy 22xy 2xy

X Y

-2 4

-1 1

0 0

1 1

2 4

Please describe the following function

g(x) = -2f(x)Reflection? Wider or Narrower?

h(x) = Reflection? Wider or Narrower

)(2

1xf

Warm UpIn 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 CompositionWrite 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 FunctionsThe 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 domainThe 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) Find (g f )(x)

f g x f g x f 2x 1

2x 1 2 2 2x 1 4x 2 4x 1 4x 2

4x 2 8x 3

g f x g f x g x2 2x 2 x 2 2x 12x2 4x 1

1.5 Combinations of Functions

Objectives: Students will know how to find arithmetic combinations and compositions of

functions.

Arithmetic Combinations of FunctionsLet 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)

provided g(x) 0.

f

g

x f (x)

g(x),

p

Example 1. f(x) = x2 + 2x and g(x) = 2x + 1. Find the following.a)  

b)  

c)  

d)  

f g x f (x) g(x)

x 2 2x 2x 1

x2 4x 1

f g x f (x) g(x)

x 2 2x 2x 1

x2 1

fg x f x g(x)

x2 2x 2x 1

2x3 5x 2 2x

f

g

x

f x g x

x2 2x2x 1