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 + 1)-2 ) (e) ( q(x)=\frac{1}{2}f(x) ) (f) ( g(x)=f(-x) )
(g) ( h(x)=f(2x) )
(e) choose the correct graph of ( q(x)=\frac{1}{2}f(x) ) below.

Explanation:

Step1: Analyze the transformation of \(Q(x)=\frac{1}{2}f(x)\)

For a function \(y = af(x)\), if \(0 < a<1\), it is a vertical compression. Here \(a=\frac{1}{2}\), so the graph of \(y = f(x)\) is vertically compressed by a factor of \(\frac{1}{2}\).

Step2: Check the key points

The original function \(f(x)\) has a maximum value of \(y = 6\) and a minimum value of \(y=-6\). For \(Q(x)=\frac{1}{2}f(x)\), the maximum value becomes \(y = 3\) (\(6\times\frac{1}{2}\)) and the minimum value becomes \(y=-3\) (\(-6\times\frac{1}{2}\))

Answer:

C.