Question

The graph of y = cos⁡ x is transformed to y = a cos ⁡(x − c) + d by a vertical compression by a factor of 1/2 and a translation 3 units up. The new equation is:

Y = ½ cos (x + 3)

Y = 2 cos x + 3

Y = 2 cos x - 3

Y = ½ cos x + 3

Answer 1

Answer: Last Option

`Y =(1)/(2)cosx + 3`

Explanation:

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

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

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

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

If
`d> 0` the graph moves vertically upwards d units.

If
`d <0` the graph moves vertically down d units.

In this problem we have the function
`y = cos⁡x`

And we know that The graph of
`y = cos⁡x` is transformed with a vertical compression by a factor of 1/2 and a translation 3 units up

therefore it is true that
`0 <k <1` and
`k=(1)/(2)` and
`d =3> 0`

Therefore the new equation is:

`Y =(1)/(2)cosx + 3`