Question

Which equation represents the transformation formed by horizontally stretching the graph of f(x) = √x by a factor of 4 and then vertically shifting the graph 2 units down?

Responses

A.
`g(x)=\sqrt{(1)/(4) x`
`-2`

B.
`g(x) =4√(x)`
`-2`

C.
`g(x)=√(4x)`
`-2`

D.
`g(x)=√(4x-2)`

Answer 1

`\textsf{A.} \quad g(x)=√(4x)-2`

Explanation:

Transformations

For a > 0

`f(x)+a \implies f(x) \: \textsf{translated $a$ units up}`

`f(x)-a \implies f(x) \: \textsf{translated $a$ units down}`

`a\:f(x) \implies f(x) \: \textsf{stretched parallel to the $y$-axis (vertically) by a factor of $a$}`

`f(ax) \implies f(x) \: \textsf{stretched parallel to the $x$-axis (horizontally) by a factor of $(1)/(a)$}`

Given parent function:

`f(x) = √(x)`

1. Horizontal stretch by a factor of 4:

f(4x) is a horizontal stretch by a factor of 1/4.

So f(1/4x) is a horizontal stretch by a factor of 4.

`f\left((1)/(4)x\right)\implies g(x)=\sqrt{(1)/(4)x}`

2. Vertical shift by 2 units down:

`f\left((1)/(4)x\right)-2\implies g(x)=\sqrt{(1)/(4)x}-2`