Question

a. find a power function that models the data.
b. find a linear function that models the data.
c. visually determine which function is the better fit for the data.
a. the power function is ( y = 4.287x^{1.204} )
(use integers or decimals for any numbers in the expression. round to the nearest thousandth as needed.)
b. the linear function is ( y=square x+(square) )
(use integers or decimals for any numbers in the expression. round to the nearest thousandth as needed.)

Explanation:

Step1: Recall the formula for linear function

The general form of a linear function is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. The formula for the slope \(m=\frac{n\sum(xy)-\sum x\sum y}{n\sum(x^{2})-(\sum x)^{2}}\), and \(b=\frac{\sum y - m\sum x}{n}\), with \(n = 6\) (the number of data points).

First, calculate \(\sum x\), \(\sum y\), \(\sum(xy)\) and \(\sum(x^{2})\):
\(\sum x=1 + 2+3 + 4+5 + 6=\frac{6\times(6 + 1)}{2}=21\)
\(\sum y=5 + 9+13 + 21+31 + 45=124\)
\(\sum(xy)=1\times5+2\times9 + 3\times13+4\times21+5\times31+6\times45=5+18+39+84+155+270 = 571\)
\(\sum(x^{2})=1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}=1 + 4+9+16+25+36 = 91\)

Step2: Calculate the slope \(m\)

Substitute the values into the slope formula:
\(m=\frac{6\times571-21\times124}{6\times91-(21)^{2}}=\frac{3426-2604}{546 - 441}=\frac{822}{105}\approx7.829\)

Step3: Calculate the y - intercept \(b\)

Substitute \(m\approx7.829\) and \(\sum x = 21\), \(\sum y=124\) into the formula for \(b\):
\(b=\frac{124-7.829\times21}{6}=\frac{124 - 164.409}{6}=\frac{- 40.409}{6}\approx - 6.735\)

Answer:

The linear function is \(y = 7.829x-6.735\)