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{\frac{1}{4} x$$-2$

B. $g(x) =4\sqrt{x}$$-2$

C. $g(x)=\sqrt{4x}$$-2$

D. $g(x)=\sqrt{4x-2}$

Answer 1

Answer:

$\textsf{A.} \quad g(x)=\sqrt{4x}-2$

Step-by-step explanation:

Transformations

For a > 0

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

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

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

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

Given parent function:

$f(x) = \sqrt{x}$

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.

$f\left(\frac{1}{4}x\right)\implies g(x)=\sqrt{\frac{1}{4}x}$

2. Vertical shift by 2 units down:

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