Question

write a function g(x) that transforms the function f(x)=4 square root x +1 such that f(x) is horizontally stretched by a factor of 5, reflected across the x-axis, then translated down 5.

Answer 1

`g(x)=-4\sqrt{(x)/(5)}-6`

Explanation:

Given function:

`f(x)=4√(x)+1`

1. Horizontal stretch

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

Therefore, if f(x) is horizontally stretched by a factor of 5:

`\implies f\left((1)/(5)x\right)=4\sqrt{(x)/(5)}+1`

2. Reflection across the x-axis

`-f(x)\implies f(x) \: \textsf{reflected in the $x$-axis}.`

Therefore, if f(x/5) is reflected in the x-axis:

`\begin{aligned}\implies -f\left((1)/(5)x\right)&=-\left(4\sqrt{(x)/(5)}+1\right)\\\\&=-4\sqrt{(x)/(5)}-1 \end{aligned}`

3. Translation

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

Therefore, if -f(x/5) is translated 5 units down:

`\begin{aligned}\implies -f\left((1)/(5)x\right)-5&=-4\sqrt{(x)/(5)}-1 -5\\\\&=-4\sqrt{(x)/(5)}-6\end{aligned}`

Therefore:

`g(x)=-4\sqrt{(x)/(5)}-6`