Question

State the various transformations applied to the base function ƒ(x) = x² to obtain a graph of the function g(x) = − (2x)² − 1.

Answer 1

The function is g ( x ) = - (2x)² - 1.

Rewrite the function as g ( x ) = - 4x² - 1.

The heighest power of x in g ( x ) = - 4x² - 1 is 2, so the parent function is square function.

The equation of parent function is f ( x ) = x² .

The transformation function is g ( x ) = - 4 x² - 1 and the parent function is f ( x ) = x² .

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

g ( x ) = - 4f ( x ) - 1.

To transform the graph of f ( x ) = x² into the graph of  g(x) follow the steps :

Step 1 :

The graph of g (x) is the reflection of the graph of f ( x ) in the x - axis.

Then,  the function is g (x) = - x² .

Step 2 :

The graph of g (x) is the vertical stretch (each y - value is multiplied by 4) of the graph of f (x).

Then,  the function is g (x) = - 4x² .

Step 3 :

The graph of g (x) is the vertical shift(down ward) of one unit of the graph of f ( x ).

Then, the function is g (x) = - 4x² - 1.

The functions f and g are related as follows :