Math Help Homework?

Question

The function g(x) is obtained by shifting the graph of y=x^2 

If g(3) = 25 

(a) g is the result of applying only a horizontal shift to y=x^2 

g(x) = 


(b) g is the result of applying only a vertical shift to y=x^2 

g(x) = 

g left-parenthesis x right-parenthesis equals 

(c) g is the result of applying a horizontal shift right 2 units and an appropriate vertical shift of y=x^2 

g(x) =

 

 

Answer 1

(a).

The parent function is f(x) = x² and the transformation function is g(3) = 25.

 Horizontal shift c units to the right : g(x) = f(x - c), where c is the positive real number.

 Horizontal shift c units to the left : g(x) = f(x + c), where c is the positive real number.

First check right :

g(x) = f(x - c) = (x - c)².

Substitute x = 3 in the above equation.

g(3) = (3 - c)².

Substitute g(3) = 25 in the above equation.

25 = (3 - c)².

± 5 = 3 - c.

3 - c = 5 and 3 - c = - 5.

c = - 2 and c = 8.

The transformation function is g(x) =  (x - 8)² (right) or g(x) =  (x + 2)² (left).

Answer 2

(b).

The parent function is f(x) = x² and the transformation function is g(3) = 25.

 Vertical shift c units upward : g(x) = f(x) + c, where c is the positive real number.

Vertical shift c units downward : g(x) = f(x) - c, where c is the positive real number.

First check upward :

g(x) = f(x) + c = x² + c.

Substitute x = 3 in the above equation.

g(3) = 3² + c.

Substitute g(3) = 25 in the above equation.

25 = 9 + c.

c = 16.

The transformation function is g(x) = x² + 16 (up ward).

 

First check downward :

g(x) = f(x) - c = x² - c.

Substitute x = 3 in the above equation.

g(3) = 3² - c.

Substitute g(3) = 25 in the above equation.

25 = 9 - c.

c = - 16.

The transformation function is g(x) = x² + 16 (up ward).

Answer 3

(c).

The parent function is y = f(x) = x² and the transformation function is g(3) = 25.

Horizontal shift c units to the right : g(x) = f(x - c), where c is the positive real number.

Horizontal shift 2 units to the right : h(x) = f(x - 2) = f(x) = (x - 2)².

Vertical shift c units upward : g(x) = h(x) + c, where c is the positive real number.

Vertical shift c units downward : g(x) = h(x) - c, where c is the positive real number.

First check upward :

g(x) = h(x) + c = (x - 2)² + c.

Substitute x = 3 in the above equation.

g(3) = (3 - 2)² + c

g(3) = 1 + c

Substitute g(3) = 25 in the above equation.

25 = 1 + c

c = 24.

The transformation function is g(x) = (x - 2)² + 24 (up ward).

 

Next check downward :

g(x) = h(x) - c = (x - 2)² - c.

Substitute x = 3 in the above equation.

g(3) = (3 - 2)² - c

g(3) = 1 - c

Substitute g(3) = 25 in the above equation.

25 = 1 - c

c = - 24.

The transformation function is g(x) = (x - 2)² + 24 (up ward).