Chapter 2.5, Problem 82AYU

To determine

To graph: Complete the square of the quadratic expression $f\left(x\right)= -2x^2-12x-13$, then graph the function using the technique of shifting.

Explanation of Solution

Given:

$f\left(x\right)= -2x^2-12x-13$

Add +5 and $-5$

$f\left(x\right)= -2x^2-12x-13+5-5$

$=-2x^2-12x-18+5$

$=-2\left(x^2+6x+9\right)+5$

$=-2(x+3{)}^2+5$

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 $f\left(x\right)=-x^2$, multiply by $-1$ in the graph of $f\left(x\right)=x^2$, that it is reflection about the $x\text{-axis}$.

$f\left(x\right)=-x^2$     multiply by $-1$ reflection about the $x\text{-axis}$.

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

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

Step 4: To obtain the graph of $f(x)=-2{\left(x+3\right)}^2$, multiply each $y\text{-coordinate}$ of the graph of $f(x)=-{\left(x+3\right)}^2$ 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 5: To obtain the graph of $f(x)=-2{\left(x+3\right)}^2+5$, add 5 from each $y\text{-coordinate}$ on the graph of $f(x)=-2{\left(x+3\right)}^2$, that it is shifted up 5 units.

$f(x)=-2{\left(x+3\right)}^2+5$     add 5, vertically shift up 5 units.