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) )
(a) choose the correct graph of ( f(x)=f(x)+3 ) below.

Explanation:

Step1: Recall vertical shift rule

For a function \(y = f(x)+k\), if \(k>0\), the graph of \(y = f(x)\) is shifted vertically upwards by \(k\) units. Here \(k = 3\) for \(F(x)=f(x)+3\).

Step2: Analyze the original function's key - points

The original function \(y = f(x)\) has a minimum value of \(- 6\) and a maximum value of \(6\).

Step3: Apply the vertical shift to key - points

For the function \(F(x)=f(x)+3\), the minimum value becomes \(-6 + 3=-3\) and the maximum value becomes \(6+3 = 9\).

Answer:

C.