Question

explore - part 2
the scatter plot and line of fit show how joes monthly rent increases over the years. the domain of the model is ( 1leq xleq8 ).

1. we want to predict joes monthly rent after 6 years. is this an example of interpolation or extrapolation?
2. what is the equation of the line of best fit? (hint: choose the two pints that are on the line)
3. predict joes monthly rent after 6 years.
4. predict joes monthly rent after 10 years. is this an example of interpolation or extrapolation?

Explanation:

Step1: Determine Interpolation/Extrapolation for 6 years

Interpolation is predicting within the domain. Since \(1\leq x\leq8\) and \(x = 6\) is within this range.

Step2: Find two points on the line

Let's take \((1,700)\) and \((8,950)\). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{950 - 700}{8 - 1}=\frac{250}{7}\approx35.71\). Using the point - slope form \(y - y_1=m(x - x_1)\), with \((x_1,y_1)=(1,700)\), \(y-700=\frac{250}{7}(x - 1)\), \(y=\frac{250}{7}x-\frac{250}{7}+700=\frac{250}{7}x+\frac{4900 - 250}{7}=\frac{250}{7}x+\frac{4650}{7}\approx35.71x + 664.29\)

Step3: Predict for \(x = 6\)

Substitute \(x = 6\) into \(y=\frac{250}{7}x+\frac{4650}{7}\), \(y=\frac{250\times6+4650}{7}=\frac{1500 + 4650}{7}=\frac{6150}{7}\approx878.57\)

Step4: Determine Interpolation/Extrapolation for 10 years

Extrapolation is predicting outside the domain. Since \(x = 10\) and \(1\leq x\leq8\), \(x = 10\) is outside.

Answer:

1. Interpolation
2. \(y=\frac{250}{7}x+\frac{4650}{7}\) (or approximately \(y = 35.71x+664.29\))
3. Approximately \(\$878.57\)
4. Extrapolation