Chapter 2.5, Problem 77AYU

To determine

To graph: Complete the square of the quadratic expression $f\left(x\right)=x^2-8x+1$, then graph the function using the technique of shifting.

Explanation of Solution

Given:

$f\left(x\right)=x^2-8x+1$

Add +15 and $-15$ in the equation,

$f\left(x\right)=x^2-8x+1+15-15$

We get, $f\left(x\right)= {\left(x-4\right)}^2-15$

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 $f\left(x\right)={\left(x-4\right)}^2$, replace $x$ by $x-4$ from each $y\text{-coordinate}$ on the graph of $f\left(x\right)=x^2$, that it is shifted right 4 units.

$f\left(x\right)={\left(x-4\right)}^2$     replace $x$ by $x-4$; horizontal shift right 4 units.

Step 3: To obtain the graph of $f\left(x\right)={\left(x-4\right)}^2-15$, subtract 15 from each $y\text{-coordinate}$ on the graph of $f\left(x\right)={\left(x-4\right)}^2$, that it is shifted down 15 units.

$f\left(x\right)={\left(x-4\right)}^2-15$     subtract 15, vertically shift down 15 units.