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.

Explanation:

Step1: Horizontal Shift

For the function \(y = f(x + 2)\), according to the rule of horizontal shift \(y=f(x - h)\) (where \(h\) is the shift amount), when \(h=- 2\), the graph of \(y = f(x)\) is shifted 2 units to the left.

Step2: Vertical Shift

For the function \(y=f(x + 2)-2\), according to the rule of vertical shift \(y = f(x)+k\) (where \(k\) is the shift amount), when \(k=-2\), the graph of \(y = f(x + 2)\) (which was already shifted left by 2 units) is shifted 2 units down.

Answer:

The graph of \(H(x)=f(x + 2)-2\) is obtained by first shifting the graph of \(y = f(x)\) 2 units to the left and then 2 units down.