Question

1. the percent of voters between the ages of 18 and 29 that participated in each united states presidential election between the years 1988 to 2016 are shown in the table. the function p gives the percent of voters between 18 and 29 years old that participated in the election in the year t. a. determine the average rate of change between 1992 and 2000. b. pick two different values of t so that the function has a negative average rate of change betw the two values. determine the average rate of change. c. pick two different values of t so that the function has a positive average rate of change be the two values. determine the average rate of change.

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) over the interval \([x_1,x_2]\) is \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}\). Here, \(x\) represents the year \(t\) and \(y\) represents \(P(t)\) (the percentage of voters).

Step2: Solve part (a)

For the years \(t_1=1992\) and \(t_2 = 2000\), \(P(1992)=42.7\) and \(P(2000)=34.5\).

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Step3: Solve part (b)

Let's choose \(t_1 = 1992\) (\(P(1992) = 42.7\)) and \(t_2=1996\) (\(P(1996)=33.1\))

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Step4: Solve part (c)

Let's choose \(t_1=1996\) (\(P(1996)=33.1\)) and \(t_2 = 2004\) (\(P(2004)=45.0\))

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Answer:

a. The average rate of change between 1992 and 2000 is \(-1.025\).
b. For \(t = 1992\) and \(t = 1996\), the average rate of change is \(-2.4\).
c. For \(t = 1996\) and \(t = 2004\), the average rate of change is \(1.4875\).