MTH 112Elementary Functions

Chapter 5The Trigonometric Functions

Section 6 – Graphs of Transformed Sine

and Cosine Functions

Graphs of Sine & Cosine

-1

-0.8

-0.6

-0.4

-0.20

0.2

0.4

0.6

0.8

1

-6 -4 -2 2 4 6x

-1

-0.8

-0.6

-0.4

-0.20

0.2

0.4

0.6

0.8

1

-6 -4 -2 2 4 6x

y = sin x y = cos x

Period = 2

Amplitude = 1

What will different constants in different locations do to these functions?

Review of Transformations of Functions

Compare: y = f(x) and y = -f(x) Reflection wrt the x-axis.

Review of Transformations of Functions

Compare: y = f(x) and y = f(-x) Reflection wrt the y-axis

Review of Transformations of Functions

Compare: y = f(x) and y = a f(x), a > 1 Vertical Stretch

32 xy 32 2 xy

Review of Transformations of Functions

Compare: y = f(x) and y = a f(x), 0 < a < 1 Vertical Compression

32 xy 32

1 2 xy

Review of Transformations of Functions

Compare: y = f(x) and y = f(bx), b > 1 Horizontal Compression

32 xy 32 2 xy

Review of Transformations of Functions

Compare: y = f(x) and y = f(bx), 0 < b < 1 Horizontal Stretch

32 xy 32

12

xy

32 xy

Review of Transformations of Functions

Compare: y = f(x) and y = f(x - c), c > 0 Horizontal Shift Right

32 2 xy

32 xy

Review of Transformations of Functions

Compare: y = f(x) and y = f(x - c), c < 0 Horizontal Shift Left

32 2 xy

xy

Review of Transformations of Functions

Compare: y = f(x) and y = f(x) + d, d > 0 Vertical Shift Up

2 xy

xy

Review of Transformations of Functions

Compare: y = f(x) and y = f(x) + d, d < 0 Vertical Shift Down

2 xy

Graphs of Sine & Cosine

-1

-0.8

-0.6

-0.4

-0.20

0.2

0.4

0.6

0.8

1

-6 -4 -2 2 4 6x

-1

-0.8

-0.6

-0.4

-0.20

0.2

0.4

0.6

0.8

1

-6 -4 -2 2 4 6x

y = sin x y = cos x

Period = 2

Amplitude = 1

What will different constants in different locations do to these functions?

y = A sin x

Amplitude = |A| A < 0 reflex wrt x-axis.

xy sin

xy sin3

xy sin3

y = sin Bx

period = 2 / B

It will be assumed that B > 0 since …

y = sin (-Bx) = -sin Bx

y = cos (-Bx) = cos Bx

xy sin

xy 3sin

xy3

1sin

y = sin (x - C)

C > 0 phase shift right C < 0 phase shift left

xy sin

2sin xy

2sin xy

y = sin (Bx - C)

Could rewrite as …y = sin [B(x – C/B)]

period = 2/B phase shift C/B left (+) or right(-)

As before, assume that B > 0. Otherwise modify the equation.

xy sin

23sin xy

y = sin x + D

D > 0 translate up D < 0 translate down

xy sin

3sin xy

3sin xy

y = A sin (Bx - C) + Dy = A cos (Bx - C) + D

Everything in the previous slides

applies in the same way to

y = cos x.

y = A sin (Bx - C) + Dy = A cos (Bx - C) + D

cos x is just a phase shift of sin x

xy sin

2sin

xy

2sin

xy

y = A sin (Bx - C) + Dy = A cos (Bx - C) + D

Any number of these constants can be included resulting in a combination of results. They may … stretch compress reflect phase shift translate

Which constants affect which

characteristics?

It can be assumed that B is positive.

y = A sin (Bx - C) + Dy = A cos (Bx - C) + D

Any number of these constants can be included resulting in a combination of results. They may … stretch |A| > 1 & B < 1 compress |A| < 1 & B > 1 reflect wrt the x-axis: A < 0 phase shift C/B right (+) or left (-) translate D up (+) or down (-)

Amplitude = |A| Period = 2 / B

It can be assumed that B is positive.

Always determine the result in this order. {

One more problem …

What would the graph of

y = x + sin x

look like?

xxy sin