Chapter 8.6, Problem 21E

To determine

To determine if the given data represents a linear, an exponential or a quadratic function and then write the function.

Answer to Problem 21E

  $f\left(x\right)=2x^2-2x-4$

Explanation of Solution

Given information :

First we shall calculate the difference.

We get,

x	-2	-1	0	1	2	3
y	8	0	-4	-4	0	8
First difference		(0-8)=-8	(-4-0)=-4	(-4-4)=0	(0-(-4))=4	(8-0)=8
Second difference			4	4	4	4

Hence we can observe that the second difference is equal.

Hence the function is a quadratic function.

Now we shall consider any point on the line. Let us take (1, -4)

Now we know that the intercept form of a quadratic equation is $y=a\left(x+1\right)\left(x-2\right)$

Put the values of the point we get,

  $\begin{array}{l}-4=a\left(1+1\right)\left(1-2\right)\\ -4=-2a\\ a=2\end{array}$

Hence the function is,

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