Chapter 2.5, Problem 79AYU

To determine

To graph: Complete the square of the quadratic expression $f\left(x\right)= 2x^2-12x+19$, then graph the function using the technique of shifting.

Explanation of Solution

Given:

$f\left(x\right)= 2x^2-12x+19$

$f\left(x\right)= 2x^2-12x+18+1$

$=2\left(x^2-6x+9\right)+1$

$=2\left(x-3\right){}^2+1$

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 $f(x)=2{\left(x-3\right)}^2$, multiply each $y\text{-coordinate}$ of the graph of $f\left(x\right)={\left(x-3\right)}^2$ by 2, that it is vertically stretched by the factor of 2.

$f(x)=2{\left(x-3\right)}^2$     multiply by 2, vertically stretched by a factor of 2.

Step 4: To obtain the graph of $f(x)=2{\left(x-3\right)}^2+1$, add 1 from each $y\text{-coordinate}$ on the graph of $f(x)=2{\left(x-3\right)}^2$ by 2, that it is shifted up 1 unit.

$f(x)=2{\left(x-3\right)}^2+1$     add 1, vertically shift up 1 unit.