Chapter 4

Explain how to find the input variable in a word problem that uses a linear function.

Describe what it means if there is a model breakdown when using a linear model.

Terry is skiing down a steep hill. Terry's elevation, \(E(t)\), in feet after \(t\) seconds is given by \(E(t)=3000-70 t\). Write a complete sentence describing Terry's starting elevation and how it is changing over time.

Terry is kking down a steep hill. Terry's elevation, \(E(t)\) in feet after \(t\) seconds is given by \(E(t)=3000-70 t .\) Write a complete sentence describing Terrys starting elevation and how it is changing over time.

Explain how to find the output variable in a word problem that uses a linear function.

What is interpolation when using a linear model?

Jessica is walking home from a friend's house. After 2 minutes she is 1.4 miles from home. Twelve minutes after leaving, she is 0.9 miles from home. What is her rate in miles per hour?

Explain how to interpret the initial value in a word problem that uses a linear function.

What is extrapolation when using a linear model?

A boat is 100 miles away from the marina, sailing directly toward it at 10 miles per hour. Write an equation for the distance of the boat from the marina after \(t\) hours.

Explain the difference between a positive and a negative correlation coeffici \(\mathrm{t}\)

If the graphs of two linear functions are perpendicular, describe the relationship between the slopes and the \(y\) -intercepts.

Find the area of a parallelogram bounded by the \(y\) -axis, the line \(x=3,\) the line \(f(x)=1+2 x,\) and the line parallel to \(f(x)\) passing through \((2,7)\)

If a horizontal line has the equation \(f(x)=a\) and a vertical line has the equation \(x=a,\) what is the point of intersection? Explain why what you found is the point of intersection.

Explain how to interpret the absolute value of a correlation coefficient.

Find the area of a triangle bounded by the \(x\) -axis, the line \(f(x)=12-\frac{1}{3} x,\) and the line perpendicular to \(f(x)\) that passes through the origin.

A regression was run to determine whether there is a relationship between hours of TV watched per day \((x)\) and number of sit-ups a person can do \((y)\). The results of the regression are given below. Use this to predict the number of situps a person who watches 11 hours of TV can do. $$ \begin{array}{l} y=a x+b \\ a=-1.341 \\ b=32.234 \\ r=-0.896 \end{array} $$

For the following exercises, determine whether the equation of the curve can be written as a linear function. $$ y=\frac{1}{4} x+6 $$

Find the area of a triangle bounded by the \(y\) -axis, the line \(f(x)=9-\frac{6}{7} x\) , and the line perpendicular to \(f(x)\) that passes through the origin.

A regression was run to determine whether there is a relationship between the diameter of a tree \((x,\) in inches \()\) and the tree's age \((y,\) in years). The results of the regression are given below. Use this to predict the age of a tree with diameter 10 inches. $$ \begin{array}{l} y=a x+b \\ a=6.301 \\ b=-1.044 \\ r=-0.970 \end{array} $$

For the following exercises, determine whether the equation of the curve can be written as a linear function. $$ y=3 x-5 $$

Find the area of a parallelogram bounded by the \(x\) -axis, the line \(g(x)=2,\) the line \(f(x)=3 x,\) and the line parallel to \(f(x)\) passing through \((6,1)\) .

For the following exercises, draw a scatter plot for the data provided. Does the data appear to be linearly related? $$ \begin{array}{|c|c|c|c|c|c|} \hline 0 & 2 & 4 & 6 & 8 & 10 \\ \hline-22 & -19 & -15 & -11 & -6 & -2 \\ \hline \end{array} $$

For the following exercises, determine whether the equation of the curve can be written as a linear function. $$ y=3 x^{2}-2 $$

For the following exercises, consider this scenario: A town's population has been decreasing at a constant rate. In 2010 the population was \(5,900.5 \mathrm{y} 2012\) the population had dropped \(4,700 .\) Assume this trend continues. Predict the population in 2016

For the following exercises, draw a scatter plot for the data provided. Does the data appear to be linearly related? $$ \begin{array}{c|c|c|c|c|c|} \hline 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 46 & 50 & 59 & 75 & 100 & 136 \\ \hline \end{array} $$

For the following exercises, determine whether the equation of the curve can be written as a linear function. $$ 3 x+5 y=15 $$

For the following exercises, consider this scenario: A town's population has been decreasing at a constant rate. In 2010 the population was \(5,900.5 \mathrm{y} 2012\) the population had dropped \(4,700 .\) Assume this trend continues. Identify the year in which the population will reach 0.

For the following exercises, draw a scatter plot for the data provided. Does the data appear to be linearly related? $$ \begin{array}{|c|c|c|c|c|c|} \hline 100 & 250 & 300 & 450 & 600 & 750 \\ \hline 12 & 12.6 & 13.1 & 14 & 14.5 & 15.2 \\ \hline \end{array} $$

For the following exercises, determine whether the equation of the curve can be written as a linear function. $$ 3 x^{2}+5 y=15 $$

For the following exercises, consider this scenario: A town's population has been increased at a constant rate. In 2010 the population was \(4,020 .\) By 2012 the population had increased to \(52,070 .\) Assume this trend continues. Predict the population in 2016

For the following exercises, determine whether the equation of the curve can be written as a linear function. $$ 3 x+5 y^{2}=15 $$

For the following exercises, consider this scenario: A town's population has been increased at a constant rate. In 2010 the population was \(4,020 .\) By 2012 the population had increased to \(52,070 .\) Assume this trend continues. Identify the year in which the population will reach \(75,000\) .

For the following exercises, determine whether the equation of the curve can be written as a linear function. $$ -2 x^{2}+3 y^{2}=6 $$

For the following data, draw a scatter plot. If we wanted to know when the population would reach \(15,000\) , would the answer involve interpolation or extrapolation? Eyeball the line, and estimate the answer. $$\begin{array}{ccccc}{\text { Year }} & {1990} & {1995} & {2000} & {2005} & {2010} \\ {\text { Population }} & {11,500} & {12,100} & {12,700} & {13,000} & {13,750}\end{array}$$

For the following exercises, consider this scenario: A town has an initial population of \(75,000 .\) It grows at a constant rate of \(2,500\) per year for 5 years. Find the linear function that models the town's population \(P\) as a function of the year, \(t,\) where \(t\) is the number of years since the model began.

For the following exercises, determine whether the equation of the curve can be written as a linear function. $$ -\frac{x-3}{5}=2 y $$

For the following data, draw a scatter plot. If we wanted to know when the temperature would reach \(28^{\circ} \mathrm{F}\) , would the answer involve interpolation or extrapolation? Eyeball the line and estimate the answer. $$\begin{array}{llllll}{\text { Temperature, }^{\circ} \mathrm{F}} & {16} & {18} & {20} & {25} & {30} \\ {\text { Time, seconds }} & {46} & {50} & {54} & {55} & {62}\end{array}$$

For the following exercises, consider this scenario: A town has an initial population of \(75,000 .\) It grows at a constant rate of \(2,500\) per year for 5 years. Find a reasonable domain and range for the function \(P .\)

For the following exercises, determine whether each function is increasing or decreasing. $$ f(x)=4 x+3 $$

For the following exercises, consider this scenario: A town has an initial population of \(75,000 .\) It grows at a constant rate of \(2,500\) per year for 5 years. If the function \(P\) is graphed, find and interpret the \(x\) -and \(y\) -intercepts.

For the following exercises, determine whether each function is increasing or decreasing. $$ g(x)=5 x+6 $$

For the following exercises, consider this scenario: A town has an initial population of \(75,000 .\) It grows at a constant rate of \(2,500\) per year for 5 years. If the function \(P\) is graphed, find and interpret the slope of the function.

For the following exercises, determine whether each function is increasing or decreasing. $$ a(x)=5-2 x $$

For the following exercises, determine whether each function is increasing or decreasing. $$ b(x)=8-3 x $$

For the following exercises, determine whether each function is increasing or decreasing. $$ h(x)=-2 x+4 $$

For the following exercises, consider this scenario: The weight of a newborn is 7.5 pounds. The baby gained one-half pound a month for its first year. Find the linear function that models the baby's weight, \(W,\) as a function of the age of the baby, in months,

For the following exercises, determine whether each function is increasing or decreasing. $$ k(x)=-4 x+1 $$

For the following exercises, consider this scenario: The weight of a newborn is 7.5 pounds. The baby gained one-half pound a month for its first year. Find a reasonable domain and range for the function \(W .\)

For the following exercises, determine whether each function is increasing or decreasing. $$ j(x)=\frac{1}{2} x-3 $$