aim: how do we totally transform an trigonometric function by manipulating the variables?

Aim: Graphs of y = a sin/cos (bx + d) + c Course: Alg. 2 & Trig.

Do Now:

Aim: How do we totally transform an trigonometric function by manipulating the variables?


Transforming Functions

If k and h are positive numbers and f(x) is a function, then

• f(x) + k shifts f(x) up k units• f(x) – k shifts f(x) down k units

• f(x + h) shifts f(x) left h units• f(x – h) shifts f(x) right h units

f(x) = (x + h)2 + k - parabolic f(x) = |x + h| + k - absolute value

ex. f(x) = (x – 4)2 + 4 is the image of g(x) = x2 after a shift of 4 units to the right and four units up or a translation of T4,4.


Transforming Sine & Cosine Functions

parent function y = sin xy = cos x

y = a sin b(x – h) + k

y = a cos b(x – h) + k

|a| = amplitute (vertical stretch or shrink)

2 period (when is in radians and > 0)x b

b

h = phase shift, or horizontal shift

k = vertical shift

|b| = frequency


2

1

-1

-2

2 4 6

amplitude frequency phase shift vertical shift

Phase Shift

sin4

y x

y = a sin b(x – h) + k

a = 1 b = 1 k = 04

h

4

,12

3,1

4

20

3, 1

2

7, 1

4



Vertical Shift

2cos 3y x y = a cos b(x – h) + k

a = 2 b = 1 k = 30h

6

4

2

-2

-4

5 10

y = cos xy = 2cos xy = 2cos x + 3

0,32cos 2cos 3Ty x y x

y = cos x

max. min.

2a

5 12

2


2

1

-1

-2

-3

2 4 6

2


The Whole Shebang!

3Sketch the graph of 2sin2

3 2

in the interval from 0 to 2 .

y x

a = 2 b = 2 k = -3/23

h

f x = sin x

f x = sin 2x f x = 2sin 2x y = 2 sin 2(x - /3)

y = 2 sin 2(x - /3) - 3/2


Model Problem

Describe any phase and/or vertical shifts.

y = 4 cos (x + 1) – 2

y = .5 sin 3(x - ) – /3

Write an equation for each transformation.

y = sin x; /2 units to right and 3.5 units up


Regents Prep

Which function is a translation of y = sin that is /3 units up and /2 units to the left?

What is the period of the function

A. sin B. sin3 2 2 3

C. sin D. sin2 3 3 2

y y

y y

1sin ?

2 3

xy

1 1 2

A. B. C. D. 62 3 3


Transforming Sine & Cosine Functions

parent function y = sin xy = cos x

y = a sin (bx – h) + ky = a cos (bx – h) + k

|a| = amplitude (vertical stretch or shrink)

2 period (when is in radians and b > 0)x

b

k = vertical shift

|b| = frequency

If is positive, shift right is the phase shift

If is negative, shift left

hh

b h

aim: how do we totally transform an trigonometric function by manipulating the variables?

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