Question

The parent function `f\left(x\right)=\sqrt[3]{x}` is compressed vertically by a factor of `\frac{1}{3}` and then translated 3 units left and 7 units down. What is the transformed function `g\left(x\right)`?

Answer 1

Answer:

The answer is below

Explanation:

Given that f(x) = x√3.

A function can be vertically stretched or compressed by multiplying it by a positive constant. If the constant is greater than 1, it is vertically stretched and if the constant is less than 1 it is vertically compressed.

If a function f(x) = x is compressed or stretched by a constant a, then the new function g(x) = a f(x)

If a function f(x) = x is translated a units down, then the new function g(x) =  f(x) - a

If a function f(x) = x is translated a units left, then the new function g(x) =  f(x-a)

If f(x) = x√3 is compressed vertically by a factor of 1/3. The new function is

$f(x)'=x\sqrt{3} *\frac{1}{3} \\\\f(x)'=\frac{x}{3} \sqrt{3}$

If it is then translated 3 units left and 7 units down, the transformed function g(x) is:

$g(x)=(\frac{x-3}{3}\sqrt{3} )-7$