Question

Describe how the graph of the given function is related to the graph of $f\left(x\right)=x^2$

$g\left(x\right)=x^2-6$

Short answer

The graph of$g\left(x\right)=x^2-6$ is the graph of $f\left(x\right)=x^2$, vertically translated 6 units down.

Step-by-step solution

Step 1. Define the standard form of the quadratic function.

A quadratic function, which is written in the form, $y=a{x}^2+bx+c$, where,$a\ne 0$ is called the standard form of the quadratic function.

Step 2. Define the transformations of the graph of the function.

(1) The graph of $f\left(x\right)+c$ shifts the graph of $f\left(x\right),c$ units up.

(2) The graph of $f\left(x\right)+c$ shifts the graph of $f\left(x\right),c$ units down.

Step 3. Describe how the graph of gx=x2−6 is related to the graph of fx=x2.

Observe the equation $g\left(x\right)=x^2-6$.

The constant term $c$is $-16$, so the graph is translated 6 units down.

Therefore, the graph of $g\left(x\right)=x^2-6$is the graph of $f\left(x\right)=x^2$, vertically translated 6 units down.