the graph of a function f is illustrated to the right. use the graph of f as the first step toward graphing…

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.\n(a) ( f(x)=f(x)+3 ) (b) ( g(x)=f(x + 2) ) (c) ( p(x)=-f(x) )\n(d) ( h(x)=f(x + 2)-2 ) (e) ( q(x)=\frac{1}{2} f(x) ) (f) ( g(x)=f(-x) )\n(g) ( h(x)=f(2x) )\n(c) choose the correct graph of ( p(x)=-f(x) ) below.

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.\n(a) ( f(x)=f(x)+3 ) (b) ( g(x)=f(x + 2) ) (c) ( p(x)=-f(x) )\n(d) ( h(x)=f(x + 2)-2 ) (e) ( q(x)=\frac{1}{2} f(x) ) (f) ( g(x)=f(-x) )\n(g) ( h(x)=f(2x) )\n(c) choose the correct graph of ( p(x)=-f(x) ) below.

Answer

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.