4.1 Linear Functions

Find an equation of the line given two points.

Find the equation of a line containing the points (−4, −3) and (1, −5)

Studies from the early 2010s indicated that teens sent about 60 texts a day, while more recent data indicates much higher messaging rates among all users, particularly considering the various apps with which people can communicate.http://www.cbsnews.com/8301-501465_162-57400228-501465/teens-are-sending-60-texts-a-day-study-says/. For each of the following scenarios, find the linear function that d…

If $f \left(x\right)$ is a linear function, and $\left(3 , −2\right)$ and $\left(8 , 1\right)$ are points on the line, find the slope. Is this function increasing or decreasing?

If $f \left(x\right)$ is a linear function, and $\left(2 , 3\right)$ and $\left(0 , 4\right)$ are points on the line, find the slope. Is this function increasing or decreasing?

The population of a city increased from 23,400 to 27,800 between 2008 and 2012. Find the change of population per year if we assume the change was constant from 2008 to 2012.

The population of a small town increased from 1,442 to 1,868 between 2009 and 2012. Find the change of population per year if we assume the change was constant from 2009 to 2012.

Suppose Ben starts a company in which he incurs a fixed cost of $1,250 per month for the overhead, which includes his office rent. His production costs are $37.50 per item. Write a linear function $C$ where $C \left(x\right)$ is the cost for $x$ items produced in a given month.

If $f$ is a linear function, with $f \left(3\right) = −2 ,$ and $f \left(8\right) = 1 ,$ find an equation for the function in slope-intercept form.

If $f \left(x\right)$ is a linear function, with $f \left(2\right) = –11 ,$ and $f \left(4\right) = −25 ,$ write an equation for the function in slope-intercept form.

Working as an insurance salesperson, Ilya earns a base salary plus a commission on each new policy. Therefore, Ilya’s weekly income $I ,$ depends on the number of new policies, $n ,$ he sells during the week. Last week he sold 3 new policies, and earned $760 for the week. The week before, he sold 5 new policies and earned $920. Find an equation for $I \left(n\right) ,$ and interpret the meaning of the components of the equa…

the table relates the number of rats in a population to time, in weeks. Use the table to write a linear equation.

A new plant food was introduced to a young tree to test its effect on the height of the tree. the table shows the height of the tree, in feet, $x$ months since the measurements began. Write a linear function, $H \left(x\right) ,$ where $x$ is the number of months since the start of the experiment.

Find a point on the graph we drew in the example that has a negative x-value.

Find the x-intercept of $f \left(x\right) = \frac{1}{2} x - 3.$

Find the x-intercept of $f \left(x\right) = \frac{1}{4} x - 4.$

Given the functions below, identify the functions whose graphs are a pair of parallel lines and a pair of perpendicular lines.

Find a line parallel to the graph of $f \left(x\right) = 3 x + 6$ that passes through the point $\left(3 , \text{0}\right) .$

Find the equation of a line perpendicular to $f \left(x\right) = 3 x + 3$ that passes through the point $\left(3 , \text{0}\right) .$

Given the function $h \left(x\right) = 2 x - 4 ,$ write an equation for the line passing through $\left(0 , 0\right)$ that is

1. ⓐparallel to $h \left(x\right)$
2. ⓑperpendicular to $h \left(x\right)$

A line passes through the points $\left(−2 , \text{6}\right)$ and $\left(4 , 5\right) .$ Find the equation of a perpendicular line that passes through the point $\left(4 , 5\right) .$

A line passes through the points, $\left(−2 , −\text{15}\right)$ and $\left(2 , −3\right) .$ Find the equation of a perpendicular line that passes through the point, $\left(6 , 4\right) .$

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

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.

If a horizontal line has the equation $f \left(x\right) = a$ 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.

Determine whether the equation of the curve can be written as a linear function.

$y = 3 x - 5$

Determine whether the equation of the curve can be written as a linear function.

$3 x + 5 y = 15$

Determine whether the equation of the curve can be written as a linear function.

$3 x + 5 y^{2} = 15$

Determine whether the equation of the curve can be written as a linear function.

$- \frac{x - 3}{5} = 2 y$

Determine whether each function is increasing or decreasing.

$g \left(x\right) = 5 x + 6$

Determine whether each function is increasing or decreasing.

$b \left(x\right) = 8 - 3 x$

Determine whether each function is increasing or decreasing.

$k \left(x\right) = −4 x + 1$

Determine whether each function is increasing or decreasing.

$p \left(x\right) = \frac{1}{4} x - 5$

Determine whether each function is increasing or decreasing.

$m \left(x\right) = - \frac{3}{8} x + 3$

Find the slope of the line that passes through the two given points.

$\left(1 , \text{5}\right)$ and $\left(4 , \text{11}\right)$

Find the slope of the line that passes through the two given points.

$\left(8 , –2\right)$ and $\left(4 , 6\right)$

Given each set of information, find a linear equation satisfying the conditions, if possible.

$f \left(- 5\right) = −4 ,$ and $f \left(5\right) = 2$

Given each set of information, find a linear equation satisfying the conditions, if possible.

Passes through $\left(2 , 4\right)$ and $\left(4 , 10\right)$

Given each set of information, find a linear equation satisfying the conditions, if possible.

Passes through $\left(−1 , \text{4}\right)$ and $\left(5 , 2\right)$

Given each set of information, find a linear equation satisfying the conditions, if possible.

x intercept at $\left(−2 , 0\right)$ and y intercept at $\left(0 , −3\right)$

Determine whether the lines given by the equations below are parallel, perpendicular, or neither.

$\begin{matrix}4 x - 7 y = 10 \\ 7 x + 4 y = 1\end{matrix}$

Determine whether the lines given by the equations below are parallel, perpendicular, or neither.

$\begin{matrix}3 y + 4 x = 12 \\ - 6 y = 8 x + 1\end{matrix}$

Find the x- and y-intercepts of each equation.

$f \left(x\right) = - x + 2$

Find the x- and y-intercepts of each equation.

$h \left(x\right) = 3 x - 5$

Find the x- and y-intercepts of each equation.

$- 2 x + 5 y = 20$

Use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

Line 1: Passes through $\left(0 , 6\right)$ and $\left(3 , −24\right)$

Line 2: Passes through $\left(−1 , 19\right)$ and $\left(8 , −71\right)$

Use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

Line 1: Passes through $\left(2 , 3\right)$ and $\left(4 , −1\right)$

Line 2: Passes through $\left(6 , 3\right)$ and $\left(8 , 5\right)$

Use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

Line 1: Passes through $\left(2 , 5\right)$ and $\left(5 , - 1\right)$

Line 2: Passes through $\left(−3 , 7\right)$ and $\left(3 , −5\right)$

Write an equation for the line described.

Write an equation for a line parallel to $g \left(x\right) = 3 x - 1$ and passing through the point $\left(4 , 9\right) .$

Write an equation for the line described.

Write an equation for a line perpendicular to $p \left(t\right) = 3 t + 4$ and passing through the point $\left(3 , 1\right) .$

$f \left(x\right) = −3 x - 1$

$f \left(x\right) = 2$

$f \left(x\right) = 3 x + 2$

Which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data.

Which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data.

Which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data.

Which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data.

Use a calculator or graphing technology to complete the task.

If $f$ is a linear function, $f \left(0.1\right) = 11.5$ , and $f \left(0.4\right) = –5.9$ , find an equation for the function.

Use a calculator or graphing technology to complete the task.

the table shows the input, $p ,$ and output, $q ,$ for a linear function $q .$

1. ⓐFill in the missing values of the table.
2. ⓑWrite the linear function

Find the value of $x$ if a linear function goes through the following points and has the following slope: $\left(x , 2\right) , \left(−4 , 6\right) , m = 3$

Find the equation of the line that passes through the following points:

$\left(a , b\right)$ and $\left(a , b + 1\right)$

Find the equation of the line that passes through the following points:

$\left(a , 0\right)$ and $\left(c , d\right)$

Find the equation of the line perpendicular to the line $g \left(x\right) = −0. \text{01} x +\text{2} .\text{01}$ through the point $\left(1 , \text{2}\right) .$

Use the functions $f \left(x\right) = −0. \text{1} x +\text{200}\textrm{ }\text{and}\textrm{ } g \left(x\right) = 20 x + 0.1.$

Where is $f \left(x\right)$ greater than $g \left(x\right) ?$ Where is $g \left(x\right)$ greater than $f \left(x\right) ?$

A gym membership with two personal training sessions costs $125, while gym membership with five personal training sessions costs $260. What is cost per session?

A phone company charges for service according to the formula: $C \left(n\right) = 24 + 0.1 n ,$ where $n$ is the number of minutes talked, and $C \left(n\right)$ is the monthly charge, in dollars. Find and interpret the rate of change and initial value.

A city’s population in the year 1960 was 287,500. In 1989 the population was 275,900. Compute the rate of growth of the population and make a statement about the population rate of change in people per year.

Suppose that average annual income (in dollars) for the years 1990 through 1999 is given by the linear function: $I \left(x\right) = 1054 x + 23,286$, where $x$ is the number of years after 1990. Which of the following interprets the slope in the context of the problem?

1. ⓐAs of 1990, average annual income was $23,286.
2. ⓑIn the ten-year period from 1990–1999, average annual income increased by a total of $1,054.
3. ⓒEach year in the decad…

Studies from the early 2010s indicated that teens sent about 60 texts a day, while more recent data indicates much higher messaging rates among all users, particularly considering the various apps with which people can communicate. For each of the following scenarios, find the linear function that describes the relationship between the input value and the output value. Then, determine whether the…

If $f \left(x\right)$ is a linear function, and $\left(3,−2\right)$ and $\left(8,1\right)$ are points on the line, find the slope. Is this function increasing or decreasing?

If $f \left(x\right)$ is a linear function, and $\left(2 , 3\right)$ and $\left(0 , 4\right)$ are points on the line, find the slope. Is this function increasing or decreasing?

The population of a city increased from 23,400 to 27,800 between 2008 and 2012. Find the change of population per year if we assume the change was constant from 2008 to 2012.

The population of a small town increased from 1,442 to 1,868 between 2009 and 2012. Find the change of population per year if we assume the change was constant from 2009 to 2012.

Write the point-slope form of an equation of a line with a slope of 3 that passes through the point $\left(6 , –1\right) .$ Then rewrite it in the slope-intercept form.

Write the point-slope form of an equation of a line with a slope of $–2$ that passes through the point $\left(–2 , 2\right) .$ Then rewrite it in the slope-intercept form.

Write the point-slope form of an equation of a line that passes through the points (5, 1) and (8, 7). Then rewrite it in the slope-intercept form.

Write the point-slope form of an equation of a line that passes through the points $\left(–1 , 3\right)$ and $\left(0 , 0\right) .$ Then rewrite it in the slope-intercept form.

Suppose Ben starts a company in which he incurs a fixed cost of $1,250 per month for the overhead, which includes his office rent. His production costs are $37.50 per item. Write a linear function $C$ where $C \left(x\right)$ is the cost for $x$ items produced in a given month.

If $f$ is a linear function, with $f \left(3\right) = −2$, and $f \left(8\right) = 1$, find an equation for the function in slope-intercept form.

If $f \left(x\right)$ is a linear function, with $f \left(2\right) = – 11 ,$ and $f \left(4\right) = - 25 ,$ find an equation for the function in slope-intercept form.

Working as an insurance salesperson, Ilya earns a base salary plus a commission on each new policy. Therefore, Ilya’s weekly income, $I ,$ depends on the number of new policies, $n ,$ he sells during the week. Last week he sold 3 new policies, and earned $760 for the week. The week before, he sold 5 new policies and earned $920. Find an equation for $I \left(n\right) ,$ and interpret the meaning of the components of the equ…

the table relates the number of rats in a population to time, in weeks. Use the table to write a linear equation.

A new plant food was introduced to a young tree to test its effect on the height of the tree. the table shows the height of the tree, in feet, $x$ months since the measurements began. Write a linear function, $H \left(x\right) ,$ where $x$ is the number of months since the start of the experiment.

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

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?

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.

Determine whether the equation of the curve can be written as a linear function.

$y = \frac{1}{4} x + 6$

Determine whether the equation of the curve can be written as a linear function.

$y = 3 x^{2} - 2$

Determine whether the equation of the curve can be written as a linear function.

$3 x^{2} + 5 y = 15$

Determine whether the equation of the curve can be written as a linear function.

$- 2 x^{2} + 3 y^{2} = 6$

Determine whether each function is increasing or decreasing.

$f \left(x\right) = 4 x + 3$

Determine whether each function is increasing or decreasing.

$a \left(x\right) = 5 - 2 x$

Determine whether each function is increasing or decreasing.

$h \left(x\right) = - 2 x + 4$

Determine whether each function is increasing or decreasing.

$j \left(x\right) = \frac{1}{2} x - 3$

Determine whether each function is increasing or decreasing.

$n \left(x\right) = - \frac{1}{3} x - 2$

Find the slope of the line that passes through the two given points.

$\left(2 , 4\right)$ and $\left(4 , \textrm{ }\text{10}\right)$

Find the slope of the line that passes through the two given points.

$\left(−1 , \text{4}\right)$ and $\left(5 , \text{2}\right)$

Find the slope of the line that passes through the two given points.

$\left(6 , 11\right)$ and $\left(- 4 , 3\right)$

Given each set of information, find a linear equation satisfying the conditions, if possible.

$f \left(−1\right) = 4$ and $f \left(5\right) = 1$

Given each set of information, find a linear equation satisfying the conditions, if possible.

Passes through $\left(1 , 5\right)$ and $\left(4 , 11\right)$

Given each set of information, find a linear equation satisfying the conditions, if possible.

Passes through $\left(- 2 , \textrm{ }\text{8}\right)$ and $\left(4 , \textrm{ }\text{6}\right)$

Given each set of information, find a linear equation satisfying the conditions, if possible.

x intercept at $\left(- 5 , \textrm{ }\text{0}\right)$ and y intercept at $\left(0 , \textrm{ }\text{4}\right)$

Which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data.

Which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data.

Which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data.

Which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data.

If $f$ is a linear function, $f \left(0.1\right) = 11.5 , \textrm{ }\text{and} f \left(0.4\right) = – 5.9 ,$ find an equation for the function.

the table shows the input, $p ,$ and output, $q ,$ for a linear function $q .$ a. Fill in the missing values of the table. b. Write the linear function $k .$

Find the value of $x$ if a linear function goes through the following points and has the following slope: $\left(x , 2\right) , \left(- 4 , 6\right) , m = 3$

Find the equation of the line that passes through the following points: $\left(a , b\right)$ and $\left(a , b + 1\right)$

Find the equation of the line that passes through the following points: $\left(a , 0\right)$ and $\left(c , d\right)$

A gym membership with two personal training sessions costs $125, while gym membership with five personal training sessions costs $260. What is cost per session?

A phone company charges for service according to the formula: $C \left(n\right) = 24 + 0.1 n ,$ where $n$ is the number of minutes talked, and $C \left(n\right)$ is the monthly charge, in dollars. Find and interpret the rate of change and initial value.

A city’s population in the year 1960 was 287,500. In 1989 the population was 275,900. Compute the rate of growth of the population and make a statement about the population rate of change in people per year.

Suppose that average annual income (in dollars) for the years 1990 through 1999 is given by the linear function: $I \left(x\right) = 1054 x + 23 , 286 ,$ where $x$ is the number of years after 1990. Which of the following interprets the slope in the context of the problem?

1. As of 1990, average annual income was $23,286.
2. In the ten-year period from 1990–1999, average annual income increased by a total of $1,054.
3. Each year in the decade of…