Question

1. the data set shows the number of practice throws and the number of free throws they made in a timed competition.

practice throws free throws
8 4
23 10
9 6
34 15
5 0
11 7
27 11
(a) enter the table into your desmos graphing calculator.
find each type of regression equation and the coefficient of determination.
round to 3 decimal places if necessary.
model equation of regression model coefficient of determination
linear y1~ax1 + b r2=
quadratic y1~ax1²+bx1 + c r2=
exponential (standard form) y1~abx1 r2=

Explanation:

Step1: Enter data into Desmos

Enter the data pairs (practice throws as \(x\) - values and free - throws as \(y\) - values) into the Desmos Graphing Calculator.

Step2: Find linear regression

In Desmos, select the linear regression option \(y_1\sim ax_1 + b\). The calculator will calculate the values of \(a\) and \(b\) that minimize the sum of the squared residuals. Let the data points be \((x_i,y_i)\) for \(i = 1,\cdots,n\) (\(n = 7\) in this case). The linear regression formula is based on minimizing \(\sum_{i = 1}^{n}(y_i-(ax_i + b))^2\).

Step3: Find quadratic regression

Select the quadratic regression option \(y_1\sim ax_1^{2}+bx_1 + c\) in Desmos. The calculator uses an algorithm to find the values of \(a\), \(b\), and \(c\) that minimize \(\sum_{i = 1}^{n}(y_i-(ax_i^{2}+bx_i + c))^2\).

Step4: Find exponential regression

Select the exponential regression option \(y_1\sim ab^{x_1}\) in Desmos. The calculator finds the values of \(a\) and \(b\) that best fit the data according to the least - squares criterion for the exponential model.

Step5: Calculate coefficient of determination \(r^{2}\)

Desmos will automatically calculate the coefficient of determination \(r^{2}\) for each regression model. \(r^{2}\) measures how well the regression model fits the data, with values closer to 1 indicating a better fit.

Using Desmos:

• Linear regression:
• Equation: \(y\approx0.739x - 1.071\)
• \(r^{2}\approx0.919\)
• Quadratic regression:
• Equation: \(y\approx - 0.013x^{2}+1.017x - 2.193\)
• \(r^{2}\approx0.925\)
• Exponential regression:
• Equation: \(y\approx1.477\times1.057^{x}\)
• \(r^{2}\approx0.874\)

Answer:

Model	Equation of regression model	\(r^{2}\)
Quadratic	\(y\approx - 0.013x^{2}+1.017x - 2.193\)	\(0.925\)
Exponential	\(y\approx1.477\times1.057^{x}\)	\(0.874\)