9) P = π 10) P = π/2 11) P = π/5; ± π/10

12) P = (2 π)/3; ± π/3 13) P = π/4; ± π/8

14) P = (3π2)/2; ±(3π2)/4

15) 16)

17) 18)

13.6 Homework

Section 13.7

Translating Sine and Cosine Functions

The same translating rules apply to all functions

Each constant in the equation does the same job

y = a • f (b (x – h)) + k

a – vertical stretch

b – horizontal stretch

h – horizontal shift

k – vertical shift

Translating Functions

In periodic functions the horizontal shift is also called the “phase shift”

The phase shift tells us how far around the unit circle we need to start to have the same results

What is the value of h in each function? Describe the phase shift in terms of left or right.

g (x) = f (x + 1)

h = –1; left 1 m (x) = f (x – 3)

h = 3; right 3 y = sin (x + π)

h = –π; left π

Phase Shift

A phase shift moves the graph sideways k moves the graph up or down

Graphing Translations

Translate the graph f (x) to be f (x – 1)

Translate a Function

2

4

Parent Functions: y = a sin bx y = a cos bx

Translated Functions y = a sin b (x – h) + k y = a cos b (x – h) + k

Parent Functions

Translating Rules

|a| = amplitude

= period (x is in radians and b > 0)

h = phase shift, or horizontal shift

k = vertical shift

2πb

For the next class complete #3 – 30 every 3rd, starting on page 746.

Homework (part 1)

3) h = 1.6; right 1.6 6) h = 5π/7; right 5π/7

9) 12) 15)

18) Amp: 4, Per: π, Left 1, down 2 21)

24) 27) 30)

Homework (part 1) Answers

We can use the values for period, amplitude, phase shift, and vertical shift

Begin with the parent function and place the values for a, b, h, and k in their appropriate places

Write an equation for each translation:

1) y = sin (x), 4 units down

y = sin (x) – 4

2) y = cos (x), π units left

y = cos (x + π)

3) y = sin (x), period of 3, amp of 2, right π/2, down 1

y = 2 sin (x – π/2) – 1

Writing Equations

2π3

Month Average High

January 42

February 45

March 52

April 59

May 68

June 79

July 84

August 82

September 74

October 63

November 50

December 42

Plot a graph of the data (in degrees) and write a cosine function to model the information. Let a > 0.

Weather Cycles

J J JF M MA A S O N D

50˚F

J

y = 21 cos (x – 180) + 63

For the next class complete #34 – 43 starting on page 746.

Homework (part 2)