Chapter 2.2, Problem 50E

Solution

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

The power function is $y=-2\sqrt{x}$ .

The power of the function is $\frac{1}{2}$ , and the constant of variation is $-2$ .

Given:

  $\begin{array}{cccccc}x& 1& 4& 9& 16& 25\\ y& -2& -4& -6& -8& -10\end{array}$ .

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 $\left(1,-2\right),\left(4,-4\right)$ .

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

  $\begin{array}{l}-2=k\cdot {\left(1\right)}^a\mathrm{...}\left(i\right)\\ -4=k\cdot {\left(4\right)}^a\mathrm{...}\left(ii\right)\end{array}$

Divide equation (i) and (ii)

  $\begin{array}{c}\frac{-2}{-4}=\frac{k\cdot {\left(1\right)}^a}{k\cdot {\left(4\right)}^a}\\ \frac{1}{2}=\frac{1^a}{4^a}\\ \frac{1}{2}={\left(\frac{1}{2}\right)}^{2a}\\ 1=2a\\ a=\frac{1}{2}\end{array}$

Substituting the value of a in equation (i)

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

Therefore, the power function is given by $y=-2x^{\frac{1}{2}}$ .

This can be rewritten as $y=-2\sqrt{x}$ .

The power of the function is $\frac{1}{2}$ , and the constant of variation is $-2$ .