Question

How do transformations affect the graph of y = sin (x) and y= cos(x)?​

Answer 1

The general form for a sine function is

y = A*sin(B(x-C)) + D

where,

• |A| = amplitude
• B = 2pi/T, with T being the period, so T = 2pi/B
• C = phase shift
• D = midline

Put another way,

• A affects how vertically tall the sine graph is
• B tells when the graph will repeat itself, or how long one cycle is
• C determines the left and right shift of the graph
• D determines the up and down shift of the graph

For example,

y = 3sin(2(x-5)) + 10

has

• A = 3, so the amplitude is |A| = |3| = 3, meaning the original parent function has been vertically stretched by a factor of 3.
• B = 2 which leads to T = 2pi/B = 2pi/2 = pi as the period. The period is half as much as the parent function's period of 2pi radians, so the graph is horizontally compressed by a factor of 2.
• C = 5 is the phase shift telling us to move the parent function 5 units to the right
• D = 10 tells us to move the parent function 10 units up

The same basic idea applies to cosine graphs as well because cosine is a cofunction of sine.

Answer 2

The same way they affect the graphs of any other functions.

Explanation:

The transformations we're generally concerned with are ones that re-scale the graph vertically or horizontally, or shift it vertically or horizontally. Such transformations have the same effects on any function.

The given functions are periodic, so certain reflections and horizontal translations will appear to have no effect. For example, the functions ...

y = sin(x)

y = sin(x +2π)

will have identical graphs. (That is the meaning of "has a period of 2π". A horizontal shift by any multiple of the period gives the same graph.)