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

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

Answer

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.