Question

the graph of a function f is illustrated to the right. use the graph of f as the first step toward graphing each of the following functions. (a) ( f(x)=f(x)+3 ) (b) ( g(x)=f(x + 2) ) (c) ( p(x)=-f(x) ) (d) ( h(x)=f(x + 2)-2 ) (e) ( q(x)=\frac{1}{2} f(x) ) (f) ( g(x)=f(-x) ) (g) ( h(x)=f(2x) ) (g) choose the correct graph of ( h(x)=f(2x) ) below.

Explanation:

Step1: Recall the horizontal compression rule

For a function \(y = f(kx)\), if \(k>1\), the graph of \(y = f(x)\) is horizontally compressed by a factor of \(\frac{1}{k}\). Here \(k = 2\), so the graph of \(y=f(x)\) is horizontally compressed by a factor of \(\frac{1}{2}\).

Step2: Analyze the key - points

The original function \(y = f(x)\) has key - points at \(x=-6\pi\), \(x = - 3\pi\), \(x=0\), \(x = 3\pi\), \(x=6\pi\). For the function \(y=f(2x)\), when \(2x=-6\pi\), \(x=-3\pi\); when \(2x=-3\pi\), \(x =-\frac{3\pi}{2}\); when \(2x = 0\), \(x = 0\); when \(2x=3\pi\), \(x=\frac{3\pi}{2}\); when \(2x = 6\pi\), \(x = 3\pi\). The \(y\) - values of the function \(y = f(2x)\) are the same as the \(y\) - values of \(y = f(x)\) for the corresponding \(x\) - values.

Step3: Eliminate wrong options

Option A has a vertical compression (since the \(y\) - values are halved), which is for \(y=\frac{1}{2}f(x)\). Option B has a horizontal shift (since the \(x\) - values of key - points are not related by \(x\to2x\)). Option C has a vertical stretch (since the \(y\) - values are doubled). Option D: The graph of \(y = f(x)\) with \(x\) - values of key - points \(x=-6\pi\), \(x=-3\pi\), \(x = 0\), \(x = 3\pi\), \(x=6\pi\) is transformed to \(x=-3\pi\), \(x=-\frac{3\pi}{2}\), \(x = 0\), \(x=\frac{3\pi}{2}\), \(x = 3\pi\) (horizontal compression by a factor of \(\frac{1}{2}\)) while keeping the \(y\) - values the same as in \(y = f(x)\).

Answer:

D.