Question

Graph the following function by considering how the function x^2 has been shifted, reflected, stretched or compressed.

Answer 1

Given the quadratic function g(x) defined as:

$g(x)=-\frac{(x-2)^2}{2}-2$

We can go from the function f(x) = x² to g(x) making the transformations:

1) A reflection about the x-axis:

$f(x)\to-f(x)$

2) A horizontal dilation by a factor of 1/2:

$f(x)\to\frac{1}{2}f(x)$

3) A shift of 2 units down:

$f(x)\to f(x)-2$

4) A shift of 2 units right:

$f(x)\to f(x-2)$

Combining all these transformations:

$f(x)\to-\frac{f(x-2)}{2}-2=-\frac{(x-2)^2}{2}-2=g(x)$

Then, the graphs of f(x) (red) and g(x) (green) are: