Question

What transformations are needed to change the parent cosine function to y=3cos(10(x-pi))?

Answer 1

The graph of
`y=cos(x)` is:

*Stretched vertically by a factor of 3

*Compressed horizontally by a factor
`(1)/(10)`

*Moves horizontally
`\pi` units to the rigth

The transformation is:

`y=3f(10(x-\pi))`

Explanation:

If the function
`y=cf(h(x+b))` represents the transformations made to the graph of
`y= f(x)` then, by definition:

If
`0 <c <1` then the graph is compressed vertically by a factor c.

If
`|c| > 1` then the graph is stretched vertically by a factor c

If
`c <0` then the graph is reflected on the x axis.

If
`b> 0` The graph moves horizontally b units to the left

If
`b <0` The graph moves horizontally b units to the rigth

If
`0 <h <1` the graph is stretched horizontally by a factor
`(1)/(h)`

If
`h> 1` the graph is compressed horizontally by a factor
`(1)/(h)`

In this problem we have the function
`y=3cos(10(x-pi))` and our parent function is
`y = cos(x)`

The transformation is:

`y=3f(10(x-\pi))`

Then
`c =3>1` and
`b =-\pi < 0` and
`h=10 > 1`

Therefore the graph of
`y=cos(x)` is:

Stretched vertically by a factor of 3.

Also as
`h=10` the graph is compressed horizontally by a factor
`(1)/(10)` .

Also, as
`b =-\pi < 0` The graph moves horizontally
`\pi` units to the rigth