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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.

write a function g(x) that transforms the function f(x)=4 square root x +1 such that-example-1
User Dmehro
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1 Answer

3 votes

Answer:


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

User Daniel Frear
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