Q. 80 - Precalculus Enhanced with Graphing Utilities

Question

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

$f\left(x\right)=3x^2+6x+1$

Step-by-Step Solution

Answer

The graph of the given equation is

Step 1. Given information

Given expression:

$f\left(x\right)=3x^2+6x+1$

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 1 inside the bracket, therefore

$$\begin{gathered} f\left(x\right)=3x^2+6x+1 \\ =3(x^2+2x)+1 \\ =3(x^2+2x+1-1)+1 \\ =3{(x+1)}^2-3+1 \\ =3{(x+1)}^2-2 \end{gathered}$$

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 $x+1$ in the expression

$$\begin{gathered} f\left(x\right)=x^2 \\ ={(x+1)}^2 \end{gathered}$$

Now multiply it with $3$ and subtracting $2$ we get,