Chapter 2.2, Problem 49E

Solution

Write an equation for the power function, and state its power and constant of variation.

The power function is $y=\frac{8}{x^2}$ .

The power of the function is $-2$ , and the constant of variation is $8$ .

Given:

  $\begin{array}{cccccc}x& 2& 4& 6& 8& 10\\ y& 2& 0.5& 0.222& 0.125& 0.08\end{array}$ .

Concept Used:

• A power function can be represented in the form $y=k\cdot x^a$ .here k is the constant of variation and a is the power.

Calculation:

From the given table, the two points that satisfy the power function are $\left(2,2\right),\left(4,0.5\right)$ .

Substituting these values in the power function $y=k\cdot x^a$

  $\begin{array}{l}2=k\cdot {\left(2\right)}^a\mathrm{...}\left(i\right)\\ 0.5=k\cdot {\left(4\right)}^a\mathrm{...}\left(ii\right)\end{array}$

Divide equation (i) and (ii)

  $\begin{array}{c}\frac{2}{0.5}=\frac{k\cdot {\left(2\right)}^a}{k\cdot {\left(4\right)}^a}\\ 4=\frac{2^a}{2^{2a}}\\ 2^2=2^{a-2a}\\ 2^2=2^{-a}\\ a=-2\end{array}$

Substituting the value of a in equation (i)

  $\begin{array}{c}2=k\cdot {\left(2\right)}^{-2}\\ k=\frac{2}{2^{-2}}\\ k=2^{1+2}\\ k=8\end{array}$

Therefore, the power function is given by $y=8x^{-2}$ .

This can be rewritten as $y=\frac{8}{x^2}$ .

The power of the function is $-2$ , and the constant of variation is $8$ .