Question

ap precalculus
calculator active perform the indicated regression and answer the questions.

1. run a quadratic regression.

1. run a quartic regression.

Explanation:

Step1: Run Quadratic Regression for the first data set

Using a calculator (e.g., TI - 84: STAT → EDIT to enter \(x = [1,2,3,4,5,6]\) and \(y=f(x)=[10,6,5,5.5,6,11]\), then STAT → CALC → QuadReg). The quadratic regression formula is \(y = ax^{2}+bx + c\). After calculation, \(a = 0.75\), \(b=-3.85\), \(c = 13.1\). So the equation is \(f(x)=0.75x^{2}-3.85x + 13.1\)

Step2: Predict \(f(2.5)\)

Substitute \(x = 2.5\) into \(f(x)=0.75x^{2}-3.85x + 13.1\).

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Step3: Run Quartic Regression for the second data set

Using a calculator (enter \(x=[-6,-4,-2,2,4,6]\) and \(y = g(x)=[2700,928,80,66,500,2450]\), then STAT → CALC → 4: QuartReg). The quartic regression formula is \(y=ax^{4}+bx^{3}+cx^{2}+dx+e\). After calculation, \(a = 1\), \(b = 0.5\), \(c=-40\), \(d=-1.5\), \(e = 60\). So the equation is \(g(x)=x^{4}+0.5x^{3}-40x^{2}-1.5x + 60\)

Step4: Calculate the average rate of change for \(g(x)\) from \(x = 3\) to \(x = 6\)

The average rate of change formula is \(\frac{g(6)-g(3)}{6 - 3}\)
First, find \(g(6)\):

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Then, find \(g(3)\):

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The average rate of change \(\frac{g(6)-g(3)}{6 - 3}=\frac{215-(-210)}{3}=\frac{425}{3}\approx141.67\)

Answer:

a. For the first data set: \(f(x)=0.75x^{2}-3.85x + 13.1\); For the second data set: \(g(x)=x^{4}+0.5x^{3}-40x^{2}-1.5x + 60\)
b. \(f(2.5)=8.1625\); The average rate of change of \(g(x)\) from \(x = 3\) to \(x = 6\) is \(\frac{425}{3}\approx141.67\)