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)+2 ) (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) )
(d) choose the correct graph of ( h(x)=f(x + 2)-2 ) below.
a. b. c. d.

Explanation:

Step1: Horizontal Shift

For the function \(H(x)=f(x + 2)-2\), the transformation \(y = f(x + c)\) (where \(c=2>0\)) shifts the graph of \(y = f(x)\) to the left by \(2\) units.

Step2: Vertical Shift

The transformation \(y=f(x)+d\) (where \(d=-2\)) shifts the graph of \(y = f(x)\) (after the horizontal shift) down by \(2\) units.

Answer:

Based on the rules of function transformations (horizontal shift left by \(2\) units and vertical shift down by \(2\) units), the correct graph can be identified among the options (assuming the options are analyzed with these transformation rules in mind). Since the original graph of \(y = f(x)\) has key - points, after shifting left \(2\) units (so \(x\) values of key - points decrease by \(2\)) and down \(2\) units (so \(y\) values of key - points decrease by \(2\)), the graph that matches these new key - point positions is the correct one.