Chapter 2.5, Problem 76AYU

To determine

To graph: Complete the square of the quadratic expression $f\left(x\right)=x^2+2x$, then graph the function using the technique of shifting.

Explanation of Solution

Given:

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

Add +9 and $-9$ in the equation,

$x^2-6x+9-9$

We get, $f\left(x\right)= {\left(x-3\right)}^2-9$

Graph:

Now use the following steps to obtain the graph of $f\left(x\right)$

Step 1: The function $f\left(x\right)=x^2$ is the square function.

$f\left(x\right)=x^2$     square function

Step 2: To obtain the graph of $f\left(x\right)={\left(x-3\right)}^2$, replace $x$ by $x-3$ from each $y\text{-coordinate}$ on the graph of $f\left(x\right)=x^2$, that it is shifted right 3 units.

$f\left(x\right)={\left(x-3\right)}^2$     replace $x$ by $x-3$; horizontal shift right 3 units.

Step 3: To obtain the graph of $f\left(x\right)={\left(x-3\right)}^2-9$, subtract 9 from each $y\text{-coordinate}$ on the graph of $f\left(x\right)={\left(x-3\right)}^2$, that it is shifted down 9 units.

$f\left(x\right)={\left(x-3\right)}^2-9$     subtract 9, vertically shift down 9 units.