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Modeling with Linear Functions

Identify steps for modeling and solving.

Identifying Steps to Model and Solve Problems

When modeling scenarios with linear functions and solving problems involving quantities with a constant rate of change, we typically follow the same problem strategies that we would use for any type of function. Let’s briefly review them:

Identify changing quantities, and then define descriptive variables to represent those quantities. When appropriate, sketch a picture or define a coordinate system.
Carefully read the problem to identify important information. Look for information that provides values for the variables or values for parts of the functional model, such as slope and initial value.
Carefully read the problem to determine what we are trying to find, identify, solve, or interpret.
Identify a solution pathway from the provided information to what we are trying to find. Often this will involve checking and tracking units, building a table, or even finding a formula for the function being used to model the problem.
When needed, write a formula for the function.
Solve or evaluate the function using the formula.
Reflect on whether your answer is reasonable for the given situation and whether it makes sense mathematically.
Clearly convey your result using appropriate units, and answer in full sentences when necessary.

Building Linear Models

Now let’s take a look at the student in Seattle. In Elan's situation, there are two changing quantities: time and money. The amount of money they have remaining while on vacation depends on how long they stay. We can use this information to define our variables, including units.

Output: $M,$ money remaining, in dollars
Input: $t,$ time, in weeks

So, the amount of money remaining depends on the number of weeks: $M(t)$

We can also identify the initial value and the rate of change.

Initial Value: They saved $3,500, so $3,500 is the initial value for $M.$
Rate of Change: They anticipate spending $400 each week, so –$400 per week is the rate of change, or slope.

Notice that the unit of dollars per week matches the unit of our output variable divided by our input variable. Also, because the slope is negative, the linear function is decreasing. This should make sense because they are spending money each week.

The rate of change is constant, so we can start with the linear model $M(t)=mt+b.$ Then we can substitute the intercept and slope provided.

To find the $x\text{-}$ intercept, we set the output to zero, and solve for the input.

\[\begin{array}{l}0=-400t+3500 \\ t=\frac{3500}{400} \\ =8.75\end{array}\]

The $x\text{-}$ intercept is 8.75 weeks. Because this represents the input value when the output will be zero, we could say that Elan will have no money left after 8.75 weeks.

Some real-world problems provide the $y\text{-}$ intercept, which is the constant or initial value. Once the $y\text{-}$ intercept is known, the $x\text{-}$ intercept can be calculated. Suppose, for example, that Hannah plans to pay off a no-interest loan from her parents. Her loan balance is $1,000. She plans to pay $250 per month until her balance is $0. The $y\text{-}$ intercept is the initial amount of her debt, or $1,000. The rate of change, or slope, is -$250 per month. We can then use the slope-intercept form and the given information to develop a linear model.

\[\begin{array}{l}f(x)=mx+b \\ =-250x+1000\end{array}\]

Now we can set the function equal to 0, and solve for $x$ to find the $x\text{-}$ intercept.

\[\begin{array}{l}\ 0=-250x+1000 \\ 1000=250x \\ 4=x \\ x=4\end{array}\]

The $x\text{-}$ intercept is the number of months it takes her to reach a balance of $0. The $x$-intercept is 4 months, so it will take Hannah four months to pay off her loan.

Condensed — the full section is in OpenStax Precalculus 2e.

Building Systems of Linear Models

Real-world situations including two or more linear functions may be modeled with a system of linear equations. Remember, when solving a system of linear equations, we are looking for points the two lines have in common. Typically, there are three types of answers possible, as shown in .

Example

Try it.

Jamal is choosing between two truck-rental companies. The first, Keep on Trucking, Inc., charges an up-front fee of $20, then 59 cents a mile. The second, Move It Your Way, charges an up-front fee of $16, then 63 cents a mileRates retrieved Aug 2, 2010 from http://www.budgettruck.com and http://www.uhaul.com/. When will Keep on Trucking, Inc. be the better choice for Jamal?

Solution

The two important quantities in this problem are the cost and the number of miles driven. Because we have two companies to consider, we will define two functions.

Input	$d,$ distance driven in miles
Outputs	$K(d):$ cost, in dollars, for renting from Keep on Trucking $M(d)$ cost, in dollars, for renting from Move It Your Way
Initial Value	Up-front fee: $K(0)=\text{2}0$ and $M(0)=\text{16}$
Rate of Change	$K(d)=\text{\$}0.\text{59}$ /mile and $P(d)=\text{\$}0.\text{63}$ /mile

A linear function is of the form $f(x)=mx+b.$ Using the rates of change and initial charges, we can write the equations

\[\begin{array}{l}K(d)=0.59d+20 \\ M(d)=0.63d+16\end{array}\]

Using these equations, we can determine when Keep on Trucking, Inc., will be the better choice. Because all we have to make that decision from is the costs, we are looking for when Move It Your Way, will cost less, or when \(K(d)

These graphs are sketched in , with $K(d)$ in blue.

To find the intersection, we set the equations equal and solve:

\[\begin{array}{l}\ K(d)=M(d) \\ 0.59d+20=0.63d+16 \\ 4=0.04d \\ 100=d \\ d=100\end{array}\]

This tells us that the cost from the two companies will be the same if 100 miles are driven. Either by looking at the graph, or noting that $K(d)$ is growing at a slower rate, we can conclude that Keep on Trucking, Inc. will be the cheaper price when more than 100 miles are driven, that is $d>100.$

Modeling with Linear Functions

We can use the same problem strategies that we would use for any type of function.
When modeling and solving a problem, identify the variables and look for key values, including the slope and y-intercept. See .
Draw a diagram, where appropriate. See and .
Check for reasonableness of the answer.
Linear models may be built by identifying or calculating the slope and using the y-intercept.
The x-intercept may be found by setting $y=0,$ which is setting the expression $mx+b$ equal to 0.
The point of intersection of a system of linear equations is the point where the x- and y-values are the same. See .
A graph of the system may be used to identify the points where one line falls below (or above) the other line.

Modeling with Linear Functions

Try it.

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

Solution

Determine the independent variable. This is the variable upon which the output depends.

Try it.

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

Try it.

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

Solution

To determine the initial value, find the output when the input is equal to zero.

Try it.

Explain how to determine the slope in a word problem that uses a linear function.

For the following exercises, use the graph in , which shows the profit, $\text{y,}$ in thousands of dollars, of a company in a given year, $\text{t,}$ where $t$ represents the number of years since 1980.

Try it.

Find the linear function $y,$ where $y$ depends on $t,$ the number of years since 1980.

Solution

$y=-2t\text{+180}$

Try it.

Find and interpret the y-intercept.

Try it.

Find and interpret the x-intercept.

Solution

In 2070, the company’s profit will be zero.

Try it.

Find and interpret the slope.

For the following exercises, use the graph in , which shows the profit, $y,$ in thousands of dollars, of a company in a given year, $t,$ where $t$ represents the number of years since 1980.

Try it.

Find the linear function $y,$ where $y$ depends on $t,$ the number of years since 1980.

Solution

$y=30t-300$

Try it.

Find and interpret the y-intercept.

Try it.

Find and interpret the x-intercept.

Solution

(10, 0) In 1990, the profit earned zero profit.

Try it.

Find and interpret the slope.

Condensed — the full section is in OpenStax Precalculus 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

A town’s population has been growing linearly. In 2004 the population was 6,200. By 2009 the population had grown to 8,100. Assume this trend continues.
1. ⓐ Predict the population in 2013.
2. ⓑ Identify the year in which the population will reach 15,000.
Жауап беріңіз
The two changing quantities are the population size and time. While we could use the actual year value as the input quantity, doing so tends to lead to very cumbersome equations because the $y\text{-}$ intercept would correspond to the year 0, more than 2000 years ago!
To make computation a little nicer, we will define our input as the number of years since 2004:
- Input: $t,$ years since 2004
- Output: $P(t),$ the town’s population
To predict the population in 2013 $(t=9),$ we would first need an equation for the population. Likewise, to find when the population would reach 15,000, we would need to solve for the input that would provide an output of 15,000. To write an equation, we need the initial value and the rate of change, or slope.
To determine the rate of change, we will use the change in output per change in input.
\[m=\frac{\text{change in output}}{\text{change in input}}\]
The problem gives us two input-output pairs. Converting them to match our defined variables, the year 2004 would correspond to $t=0,$ giving the point $(0,\text{6200}).$ Notice that through our clever choice of variable definition, we have “given” ourselves the y-intercept of the function. The year 2009 would correspond to $t=\text{5,}$ giving the point $(5,\text{8100}).$
The two coordinate pairs are $(0,\text{6200})$ and $(5,\text{8100}).$ Recall that we encountered examples in which we were provided two points earlier in the chapter. We can use these values to calculate the slope.
\[\begin{array}{l}\begin{array}{l} \\ m=\frac{8100-6200}{5-0}\end{array} \\ =\frac{1900}{5} \\ =380\text{ people per year}\end{array}\]
We already know the y-intercept of the line, so we can immediately write the equation:
\[P(t)=380t+6200\]
To predict the population in 2013, we evaluate our function at $t=9.$
\[\begin{array}{l}P(9)=380(9)+6,200 \\ =9,620\end{array}\]
If the trend continues, our model predicts a population of 9,620 in 2013.
To find when the population will reach 15,000, we can set $P(t)=15000$ and solve for $t.$
\[\begin{array}{l}15000=380t+6200 \\ 8800=380t \\ t\approx 23.158\end{array}\]
Our model predicts the population will reach 15,000 in a little more than 23 years after 2004, or somewhere around the year 2027.
A company sells doughnuts. They incur a fixed cost of $25,000 for rent, insurance, and other expenses. It costs $0.25 to produce each doughnut.
ⓐ Write a linear model to represent the cost $C$ of the company as a function of $x,$ the number of doughnuts produced.
ⓑ Find and interpret the y-intercept.
Жауап беріңіз
1. ⓐ $C(x)=0.25x+25,000$
2. ⓑ The y-intercept is $(0,25,000).$ If the company does not produce a single doughnut, they still incur a cost of $25,000.
A city’s population has been growing linearly. In 2008, the population was 28,200. By 2012, the population was 36,800. Assume this trend continues.
1. ⓐ Predict the population in 2014.
2. ⓑ Identify the year in which the population will reach 54,000.
Жауап беріңіз
1. ⓐ 41,100
2. ⓑ 2020
Anna and Emanuel start at the same intersection. Anna walks east at 4 miles per hour while Emanuel walks south at 3 miles per hour. They are communicating with a two-way radio that has a range of 2 miles. How long after they start walking will they fall out of radio contact?
Жауап беріңіз
In essence, we can partially answer this question by saying they will fall out of radio contact when they are 2 miles apart, which leads us to ask a new question:
“How long will it take them to be 2 miles apart?”
In this problem, our changing quantities are time and position, but ultimately we need to know how long will it take for them to be 2 miles apart. We can see that time will be our input variable, so we’ll define our input and output variables.
- Input: $t,$ time in hours.
- Output: $A(t),$ distance in miles, and $E(t),$ distance in miles
Because it is not obvious how to define our output variable, we’ll start by drawing a picture such as .
Initial Value: They both start at the same intersection so when $t=0,$ the distance traveled by each person should also be 0. Thus the initial value for each is 0.
Rate of Change: Anna is walking 4 miles per hour and Emanuel is walking 3 miles per hour, which are both rates of change. The slope for $A$ is 4 and the slope for $E$ is 3.
Using those values, we can write formulas for the distance each person has walked.
\[\begin{array}{l}A(t)=4t \\ E(t)=3t\end{array}\]
For this problem, the distances from the starting point are important. To notate these, we can define a coordinate system, identifying the “starting point” at the intersection where they both started. Then we can use the variable, $A,$ which we introduced above, to represent Anna’s position, and define it to be a measurement from the starting point in the eastward direction. Likewise, can use the variable, $E,$ to represent Emanuel’s position, measured from the starting point in the southward direction. Note that in defining the coordinate system, we specified both the starting point of the measurement and the direction of measure.
We can then define a third variable, $D,$ to be the measurement of the distance between Anna and Emanuel. Showing the variables on the diagram is often helpful, as we can see from .
Recall that we need to know how long it takes for $D,$ the distance between them, to equal 2 miles. Notice that for any given input $t,$ the outputs $A(t),E(t),$ and $D(t)$ represent distances.
shows us that we can use the Pythagorean Theorem because we have drawn a right angle.
Using the Pythagorean Theorem, we get:
\[\begin{array}{ll}D{(t)}^{2}=A{(t)}^{2}+E{(t)}^{2} & \\ ={(4t)}^{2}+{(3t)}^{2} & \\ =16{t}^{2}+9{t}^{2} & \\ =25{t}^{2} & \\ D(t)=\pm \sqrt{25{t}^{2}} & \text{Solve for }D(t)\text{ using the square root} \\ =\pm 5|t| & \end{array}\]
In this scenario we are considering only positive values of $t,$ so our distance $D(t)$ will always be positive. We can simplify this answer to $D(t)=5t.$ This means that the distance between Anna and Emanuel is also a linear function. Because $D$ is a linear function, we can now answer the question of when the distance between them will reach 2 miles. We will set the output $D(t)=2$ and solve for $t.$
\[\begin{array}{l}D(t)=2 \\ 5t=2 \\ t=\frac{2}{5}=0.4\end{array}\]
They will fall out of radio contact in 0.4 hours, or 24 minutes.
There is a straight road leading from the town of Westborough to Agritown 30 miles east and 10 miles north. Partway down this road, it junctions with a second road, perpendicular to the first, leading to the town of Eastborough. If the town of Eastborough is located 20 miles directly east of the town of Westborough, how far is the road junction from Westborough?

Жауап беріңіз

It might help here to draw a picture of the situation. See . It would then be helpful to introduce a coordinate system. While we could place the origin anywhere, placing it at Westborough seems convenient. This puts Agritown at coordinates $(\text{3}0,\text{ 1}0),$ and Eastborough at $(\text{2}0,0).$
Using this point along with the origin, we can find the slope of the line from Westborough to Agritown:
\[m=\frac{10-0}{30-0}=\frac{1}{3}\]
The equation of the road from Westborough to Agritown would be
\[W(x)=\frac{1}{3}x\]
From this, we can determine the perpendicular road to Eastborough will have slope $m=-3.$ Because the town of Eastborough is at the point (20, 0), we can find the equation:
\[\begin{array}{ll}E(x)=-3x+b & \\ 0=-3(20)+b & \text{Substitute in (20, 0)} \\ b=60 & \\ E(x)=-3x+60 & \end{array}\]
We can now find the coordinates of the junction of the roads by finding the intersection of these lines. Setting them equal,
\[\begin{array}{ll}\ \frac{1}{3}x=-3x+60 & \\ \frac{10}{3}x=60 & \\ 10x=180 & \\ x=18 & \text{Substituting this back into }W(x) \\ y=W(18) & \\ =\frac{1}{3}(18) & \\ =6 & \end{array}\]
The roads intersect at the point (18, 6). Using the distance formula, we can now find the distance from Westborough to the junction.
\[\begin{array}{l}\text{distance}=\sqrt{{({x}_{2}-{x}_{1})}^{2}+{({y}_{2}-{y}_{1})}^{2}} \\ =\sqrt{{(18-0)}^{2}+{(6-0)}^{2}} \\ \approx 18.974\text{ miles}\end{array}\]
There is a straight road leading from the town of Timpson to Ashburn 60 miles east and 12 miles north. Partway down the road, it junctions with a second road, perpendicular to the first, leading to the town of Garrison. If the town of Garrison is located 22 miles directly east of the town of Timpson, how far is the road junction from Timpson?

Жауап беріңіз

21.57 miles

Жауап беріңіз

The two important quantities in this problem are the cost and the number of miles driven. Because we have two companies to consider, we will define two functions.

Input	$d,$ distance driven in miles
Outputs	$K(d):$ cost, in dollars, for renting from Keep on Trucking $M(d)$ cost, in dollars, for renting from Move It Your Way
Initial Value	Up-front fee: $K(0)=\text{2}0$ and $M(0)=\text{16}$
Rate of Change	$K(d)=\text{\$}0.\text{59}$ /mile and $P(d)=\text{\$}0.\text{63}$ /mile

A linear function is of the form $f(x)=mx+b.$ Using the rates of change and initial charges, we can write the equations

\[\begin{array}{l}K(d)=0.59d+20 \\ M(d)=0.63d+16\end{array}\]

These graphs are sketched in , with $K(d)$ in blue.

To find the intersection, we set the equations equal and solve:

\[\begin{array}{l}\ K(d)=M(d) \\ 0.59d+20=0.63d+16 \\ 4=0.04d \\ 100=d \\ d=100\end{array}\]

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

Жауап беріңіз

Determine the independent variable. This is the variable upon which the output depends.
Explain how to find the output variable in a word problem that uses a linear function.
Explain how to interpret the initial value in a word problem that uses a linear function.

Жауап беріңіз

To determine the initial value, find the output when the input is equal to zero.
Explain how to determine the slope in a word problem that uses a linear function.
Find the area of a parallelogram bounded by the y-axis, the line $x=3,$ the line $f(x)=1+2x,$ and the line parallel to $f(x)$ passing through $(\text{2},\text{ 7}).$

Жауап беріңіз

6 square units
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.
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.

Жауап беріңіз

20.012 square units
Find the area of a parallelogram bounded by the x-axis, the line $g(x)=2,$ the line $f(x)=3x,$ and the line parallel to $f(x)$ passing through $(6,1).$
Predict the population in 2016.

Жауап беріңіз

2,300
Identify the year in which the population will reach 0.
Predict the population in 2016.

Жауап беріңіз

64,170
Identify the year in which the population will reach 75,000.
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.

Жауап беріңіз

$P(t)=75,000+2,500t$
Find a reasonable domain and range for the function $P.$
If the function $P$ is graphed, find and interpret the x- and y-intercepts.

Жауап беріңіз

(–30, 0) Thirty years before the start of this model, the town had no citizens. (0, 75,000) Initially, the town had a population of 75,000.
If the function $P$ is graphed, find and interpret the slope of the function.
When will the output reached 100,000?

Жауап беріңіз

Ten years after the model began.
What is the output in the year 12 years from the onset of the model?
Find the linear function that models the baby’s weight $W$ as a function of the age of the baby, in months, $t.$

Жауап беріңіз

$W(t)=0.\text{5}t+\text{7}.\text{5}$
Find a reasonable domain and range for the function $W$.
If the function $W$ is graphed, find and interpret the x- and y-intercepts.

Жауап беріңіз

$(-15,0)$: The x-intercept is not a plausible set of data for this model because it means the baby weighed 0 pounds 15 months prior to birth. $(0,\text{ 7}.\text{5})$: The baby weighed 7.5 pounds at birth.
If the function W is graphed, find and interpret the slope of the function.
When did the baby weigh 10.4 pounds?

Жауап беріңіз

At age 5.8 months.
What is the output when the input is 6.2? Interpret your answer.
Find the linear function that models the number of people inflicted with the common cold $C$ as a function of the year, $t.$

Жауап беріңіз

$C(t)=12,025-205t$
Find a reasonable domain and range for the function $C.$
If the function $C$ is graphed, find and interpret the x- and y-intercepts.

Жауап беріңіз

$\text{(58.7, 0)}$: In roughly 59 years, the number of people inflicted with the common cold would be 0. $\text{(0,12,025)}$: Initially there were 12,025 people afflicted by the common cold.
If the function $C$ is graphed, find and interpret the slope of the function.
When will the number of people afflicted with the common cold reach 0?

Жауап беріңіз

2064
In what year will the number of people afflicted with the common cold be 9,700?
Find the linear function $y,$ where $y$ depends on $t,$ the number of years since 1980.

Жауап беріңіз

$y=-2t\text{+180}$
Find and interpret the y-intercept.
Find and interpret the x-intercept.

Жауап беріңіз

In 2070, the company’s profit will be zero.

Symbols used here

P(A),\ P(A \mid B)

probability, conditional probability
Chance of A; chance of A given that B happened.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

\binom{n}{k}

binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).

\pi

pi
Ratio of a circle's circumference to its diameter, 3.14159…

Euler's number
2.71828…, the base whose exponential is its own derivative.

imaginary unit
i² = −1.

\theta

theta
The usual name for an angle.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbf{v},\ \vec{v}

vector
A quantity with magnitude and direction; a column of numbers.

How to: Modeling with Linear Functions

Identify steps for modeling and solving.
Build linear models from verbal descriptions.
Build systems of linear models.
Identify changing quantities, and then define descriptive variables to represent those quantities. When appropriate, sketch a picture or define a coordinate system.
Carefully read the problem to identify important information. Look for information that provides values for the variables or values for parts of the functional model, such as slope and initial value.
Carefully read the problem to determine what we are trying to find, identify, solve, or interpret.
Identify a solution pathway from the provided information to what we are trying to find. Often this will involve checking and tracking units, building a table, or even finding a formula for the function being used to model the problem.
When needed, write a formula for the function.

Questions people ask

What is a function, really?

A rule that assigns exactly one output to each input. The vertical-line test on a graph is the same idea: no input may have two outputs.

Why do we need complex numbers?

Because x² + 1 = 0 has no real solution, and allowing one new number i with i² = −1 makes every polynomial equation solvable. They then turn out to describe rotation, waves and alternating current more naturally than real numbers do.

Өзіңіздіңіңізді сынап көріңіз

Parts of this page are adapted from OpenStax Precalculus 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Келесіде Precalculus

Complex numbers Polynomial functions Rational functions Sequences and series The binomial theorem Conic sections Vectors Exponential and logarithmic functions Polynomial division and the remainder theorem Parametric equations and polar coordinates

Input	\(d,\) distance driven in miles
Outputs	\(K(d):\) cost, in dollars, for renting from Keep on Trucking \(M(d)\) cost, in dollars, for renting from Move It Your Way
Initial Value	Up-front fee: \(K(0)=\text{2}0\) and \(M(0)=\text{16}\)
Rate of Change	\(K(d)=\text{\$}0.\text{59}\) /mile and \(P(d)=\text{\$}0.\text{63}\) /mile