Chapter 9, Problem 7CT

To determine

To write: The description of using the method of completing the square to determine whether the function can be represented by graph.

Answer to Problem 7CT

The function cannot be represented by the given graph.

Explanation of Solution

Given information:

The function: $f\left(x\right)=2x^2+4x-6$

The given function is $f\left(x\right)=2x^2+4x-6$ .

It is known that a quadratic equation $f\left(x\right)=a{x}^2+bx+c$ can be solved using completing the square method by rewriting the equation in the vertex form as,

  $f\left(x\right)=a{\left(x-h\right)}^2+k$

In the above equation the vertex is $\left(h,k\right)$ .

To solve the given equation, add 6 on each side of the equation.

  $\begin{array}{l}f\left(x\right)+6=2x^2+4x-\cancel{6}+\cancel{6}\\ f\left(x\right)+6=2x^2+4x\end{array}$

Add 2 on each side of the above expression.

  $\begin{array}{l}f\left(x\right)+6+2=2x^2+4x+2\\ f\left(x\right)+8=2\left(x^2+2x+1\right)\\ f\left(x\right)+8=2{\left(x+1\right)}^2\\ f\left(x\right)=2{\left(x+1\right)}^2-8\end{array}$

The above equation is n vertex form where the vertex is $\left(-1,-8\right)$ .

The vertex lies on Quadrant III.

But the graph of equation is in Quadrant IV.

Hence, the function cannot be represented by the given graph.