Question

What is the function g(x) created from f(x) = x^2 by moving the graph left 9 units, vertically stretching it by a factor of 4, and shifting the graph up 5 units?

Answer 1

Given:

The parent function is $f(x)=x^2$

We need to determine the function g(x) created from the function f(x) by moving the graph left 9 units, vertically stretching it by a factor of 4, and shifting the graph up 5 units.

Shift left 9 units:

The rule to shift the graph c units to the left is given by

$y=f(x+c)$

where c is the positive real number.

Hence, to shift the graph 9 units to the left is given by

$f(x)=(x+9)^2$

Vertical stretch by a factor of 4:

The rule to stretch the graph vertically by a factor of c is given by

$y=cf(x)$

where $c>1$

Hence, to stretch the graph vertically by a factor of 4, we have;

$f(x)=4(x+9)^2$

Shifting 5 units up:

The rule to shift the graph c units upward is given by

$y=f(x)+c$

where c is the positive real number.

Hence, to shift the graph 5 units upward is given by

$f(x)=4(x+9)^2+5$

Therefore, the function $g(x)=4(x+9)^2+5$ is created from the graph $f(x)=x^2$ by moving the graph left 9 units, vertically stretching it by a factor of 4 and shifting up 5 units.

Hence, the graph g(x) is $g(x)=4(x+9)^2+5$