Q. 81 - Precalculus Enhanced with Graphing Utilities

Question

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

$f\left(x\right)=-3x^2-12x-17$

Step-by-Step Solution

Answer

The graph of the given equation is

Step 1. Given information

Given expression:

$f\left(x\right)=-3x^2-12x-17$

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's add and subtract $2^2$ on right side therefore,

$$\begin{gathered} f\left(x\right)=-3x^2-12x-17 \\ rewrite the expression as \\ =-3(x^2+4x)-17 \\ Adding \& Subtracting 2^2 inside parenthesis we get, \\ =-3(x^2+4x+2^2-2^2)-17 \\ =-3{(x+2)}^2+12-17 \\ =-3{(x+2)}^2-5 \end{gathered}$$

Step 4. Graphing using shifting technique

In order to graph the given expression, we need to draw the graph of $f\left(x\right)=x^2$ i.e.

Step 5. Graphing the actual expression

Replace $x$ by $x+2$ in the expression $$\begin{gathered} f\left(x\right)=x^2 \\ ={(x+2)}^2 \end{gathered}$$

Now multiply $-3$which results in the inverted graph i.e,

Now subtracting $-5$ from the given expression gives us the result,

$f\left(x\right)=-3{(x+2)}^2-5$