Q. 79 - Precalculus Enhanced with Graphing Utilities

Question

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

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

Step-by-Step Solution

Answer

The graph of the given equation is

Step 1. Given information

Given expression:

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

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 $3^2$ inside the parenthesis, therefore

$$\begin{gathered} f\left(x\right)=2x^2-12x+19 \\ =2(x^2-6x)+19 \\ Adding \& Subtracting 3^2 inside parenthesis we get, \\ =2(x^2-6x+3^2-3^2)+19 \\ =2{(x-3)}^2-18+19 \\ =2{(x-3)}^2+1 \end{gathered}$$

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 $x-3$ in the expression

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

Multiply it with $2$ and add it with $1$ we get,

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