Q23. - Algebra 1

Question

Describe how the graph of the function $f\left(x\right)=x^2+8$ is related to the graph $f\left(x\right)=x^2$.

Step-by-Step Solution

Answer

The graph of the parent function is shifted up to 8 units.

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 vertical translation of the graph.

The graph $g\left(x\right)=x^2+c$ is the graph $f\left(x\right)=x^2$ translated vertically.

If $c>0$, the graph $f\left(x\right)=x^2$ is translated $\left|c\right|$ units up.

 If $c<0$, the graph $f\left(x\right)=x^2$ is translated $\left|c\right|$ units down.

Step 3. Determine the relationship of the graph of the function f x = x 2 + 8 with the graph of the function f x = x 2 .

Observe the function $f\left(x\right)=x^2+8$

Here, $c=8$

So, the graph of the parent function is translated 8 units up to form the graph of the function $f\left(x\right)=x^2+5$.

Therefore, the graph of the parent function is shifted to 8 units.