Question

Identify a horizontal or vertical stretch or compression of the function ƒ(x) = x² by observing the equation of the function g(x) = (9x)².

Answer 1

• Let f (x ) is a function and c be a positive real number,  the graph of the function g (x ) = f (cx ) is represented as follows.

1. when, c > 1,  the function compresses it in the x - direction.
2. when, 0 < c < 1, the function stretches it.

The function is g ( x ) = (9x )².

The heighest power of x in g ( x ) = (9x )² is 2, so the parent function is square function.

The equation of parent function is f ( x ) = x ² .

The function is g ( x ) = (9x )² and the parent function is f ( x ) = x ² .

So, the functions f and g have the following relationship.

g ( x ) = f ( 9x ).

Compare the function g ( x ) = f ( 9x ) with g (x ) = f (cx ).

c = 9 ( > 1 ).

So, the function g (x ) = (9x )² compresses it in the x - direction.

Answer 2

Nonrigid Transformations : 

• A nonrigid transformation of the graph of y = f(x) is represented by g(x) = c f(x), where the transformation is a vertical stretch if c > 1 and a vertical shrink if 0 < c < 1.
• Another nonrigid transformation of the graph of y = f(x) is represented by h(x) = f (cx), where the transformation is a horizontal shrink if c > 1 and a horizontal stretch if 0 < c < 1.

The parent function is f(x) = x², and the transformation function is g(x) = (9x)².

 The graph of g(x) = (9x)² is a horizontal shrink or compression ( since c = 9 > 1)  of the graph of f(x) = x² because

g(x) = (9x)²

        = f (9x).