Notes 4-8

Section 4-8Translation Images of Circular (Trig) Functions

Warm-upDescribe the transformation for each of the

following equations, as based on the parent equation of y = sin x.

1. y = sin(x + 14) 2. y = sin x + 14

3. y = sin x - 14 4. y = sin(x - 14)

Shifted 14 units left Shifted 14 units up

Shifted 14 units down Shifted 14 units right

Phase Shift

The smallest positive or largest negative number used in a horizontal translation for a sine or cosine

wave

Example 1a. Graph two cycles of the following function

y = cos x − π

2( )b. What is the phase shift of this function when

compared to the parent function y = cos x?

π2

...to the right

c. What is the phase shift when compared to y = sin x?

It IS y = sin x

TheoremWe can rewrite some of these phase shifts quite

easily:

sin: cos:+ + - - + - - +

Just follow these patterns

Example 2Find another possible equation for each function

listed below.

a. y = cos π

2− x( )

b. y = sin x − π( )+ + - -

y = sin x

- - + +

y = -sin x

Example 3Compare the following graphs.

y = sinx

Phase shift:

Vertical Shift: 2

−

5π6

y = sin x + 5π

6( ) + 2

Example 4Compare the following graphs:

y = tanx y = −1 + tanx

There was a vertical shift of -1

In phase:

Out-of-phase:

Inductance:

When voltage and current flow coincide with each other

When the current is behind the voltage

When current flow is out-of-phase

Example 5In question 14 from section 4-7, AC current y was

modeled with the equation

y = 10 sin 120πx( )

where x is time in seconds. Maximum inductance occurs when the voltage leads the current by

seconds. What is an equation for the current in the new model?

π2

y = 10 sin 120π x −

π2

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Homework

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