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) )
(c) choose the correct graph of ( p(x)=-f(x) ) below.

Explanation:

Step1: Recall the transformation rule

The transformation \(y = -f(x)\) is a reflection of the graph of \(y = f(x)\) about the \(x\) - axis.

Step2: Analyze the key points

For the original function \(y = f(x)\), if \((x_0,y_0)\) is on the graph of \(y = f(x)\), then \((x_0,-y_0)\) is on the graph of \(y=-f(x)\). For example, if the original function has a maximum point \((a,b)\), the transformed function \(P(x)=-f(x)\) will have a minimum point \((a, - b)\) and vice - versa.

Step3: Eliminate wrong options

• Option A: The shape of the graph is not a simple reflection about the \(x\) - axis.
• Option B: The graph has an incorrect symmetry (it looks like a combination of other transformations rather than just a reflection about the \(x\) - axis).
• Option D: The graph has an incorrect orientation and does not follow the reflection rule \(y=-f(x)\) properly.

Answer:

C.