Q. 82 - Precalculus Enhanced with Graphing Utilities

Question

complete the square of each quadratic expression. Then graph each function using the technique of shifting.

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

Step-by-Step Solution

Answer

The graph of the given equation is

Step 1. Given information

Given expression:

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

Step 2. Write the concept

In order to complete the square, we will add or subtract some numbers in order to simplify it.

Step 3. Calculation

Let us take out $-2$ from the given expression therefore,

$$\begin{gathered} -2x^2-12x-13=-2\left(x^2+6x+\frac{13}{2}\right) \\ Now we subtract 9 from it \\ f\left(x\right)=-2\left(x^2+6x+9+\frac{13}{2}-9\right) \\ =-2\left[{(x+3)}^2+\frac{13}{2}-9\right] \\ =-2\left[{(x+3)}^2-\frac{5}{2}\right] \end{gathered}$$

Step 4. Graphing using shifting technique

In order to graph the given expression we need to draw the graph of $x^2$ i.e.

Step 5. Graphing the actual expression

Replace $y=f\left(x\right) with y=f(x-h)$ here h=3 therefore,

Step 6. Shifting vertically

The function $$\begin{gathered} y=f\left(x\right)-k will be shifted vertically downward by k units \\ f\left(x\right)={(x+3)}^2 by \frac{5}{2}units. \end{gathered}$$

Now, the the actual graph will be the reflection of above graph multiplied by 2.