# Linear Functions

## Modeling with Linear Functions


### Learning Objectives

1. Graph and interpret applications of slope–intercept form of a linear function. (IA 3.2.5)

### Objective 1: Graph and interpret applications of slope–intercept form of a linear function. (IA 3.2.5)


### Graph and Interpret Applications of Slope–Intercept form of linear equations.

Many real-world applications are modeled by linear functions. We will take a look at a few applications here so you can see how equations written in slope–intercept form describe real world situations.
Usually, when a linear function uses real-world data, different letters are used to represent the variables, instead of using only  and  . The variables used remind us of what quantities are being measured.
Also, we often will need to adjust the axes in our rectangular coordinate system to different scales to accommodate the data in the application. Since many applications have both independent and dependent variables that are positive our graphs will lie primarily in Quadrant I. 

### Linear Functions

A linear function is a function whose graph is a line. Linear functions can be written in the slope-intercept form of a line

  
where b is the initial or starting value of the function (when input, x=0), and m is the constant rate of change, or slope of the function. The y-intercept is at (0,b),

  .
When interpreting slope, it will be important to consider the units of measurement. Make sure to always attach these units to both the numerator and denominator when they are provided to you.


### Practice Makes Perfect


Graph and interpret applications of slope–intercept form of a linear function.

Elan is a college student who plans to spend a summer in Seattle. Elan has saved $3,500 for their trip and anticipates spending $400 each week on rent, food, and activities. How can we write a linear model to represent this situation? What would be the x-intercept, and what can Elan learn from it? To answer these and related questions, we can create a model using a linear function. Models such as this one can be extremely useful for analyzing relationships and making predictions based on those relationships. In this section, we will explore examples of linear function models.

### Building Linear Models from Verbal Descriptions 

When building linear models to solve 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:
1. Identify changing quantities, and then define descriptive variables to represent those quantities. When appropriate, sketch a picture or define a coordinate system.
2. 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.
3. Carefully read the problem to determine what we are trying to find, identify, solve, or interpret.
4. 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.
5. When needed, write a formula for the function.
6. Solve or evaluate the function using the formula.
7. Reflect on whether your answer is reasonable for the given situation and whether it makes sense mathematically.
8. Clearly convey your result using appropriate units, and answer in full sentences when necessary.
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.
So, the amount of money remaining depends on the number of weeks: 
  .
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 she is spending money each week.
The rate of change is constant, so we can start with the linear model 
  Then we can substitute the intercept and slope provided.
To find the t-intercept (horizontal axis intercept), we set the output to zero, and solve for the input.
The t-intercept (horizontal axis 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.
When modeling any real-life scenario with functions, there is typically a limited domain over which that model will be valid—almost no trend continues indefinitely. Here the domain refers to the number of weeks. In this case, it doesn’t make sense to talk about input values less than zero. A negative input value could refer to a number of weeks before Elan saved $3,500, but the scenario discussed poses the question once they saved $3,500 because this is when the trip and subsequent spending starts. It is also likely that this model is not valid after the t-intercept (horizontal axis intercept), unless Elan uses a credit card and goes into debt. The domain represents the set of input values, so the reasonable domain for this function is 
 

In this example, we were given a written description of the situation. We followed the steps of modeling a problem to analyze the information. However, the information provided may not always be the same. Sometimes we might be provided with an intercept. Other times we might be provided with an output value. We must be careful to analyze the information we are given, and use it appropriately to build a linear model.

### Using a Given Intercept to Build a Model

Some real-world problems provide the vertical axis intercept, which is the constant or initial value. Once the vertical axis intercept is known, the t-intercept (horizontal axis 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-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.
Now we can set the function equal to 0, and solve for 
  to find the x-intercept.
The x-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.

###  Using a Given Input and Output to Build a Model

Many real-world applications are not as direct as the ones we just considered. Instead they require us to identify some aspect of a linear function. We might sometimes instead be asked to evaluate the linear model at a given input or set the equation of the linear model equal to a specified output.

### Using a Diagram to Build a Model 

It is useful for many real-world applications to draw a picture to gain a sense of how the variables representing the input and output may be used to answer a question. To draw the picture, first consider what the problem is asking for. Then, determine the input and the output. The diagram should relate the variables. Often, geometrical shapes or figures are drawn. Distances are often traced out. If a right triangle is sketched, the Pythagorean Theorem relates the sides. If a rectangle is sketched, labeling width and height is helpful.

### Modeling a Set of Data with Linear Functions

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 .

### Key Concepts

1. We can use the same problem strategies that we would use for any type of function.
2. When modeling and solving a problem, identify the variables and look for key values, including the slope and y-intercept. See .
3. Draw a diagram, where appropriate. See  and .
4. Check for reasonableness of the answer.
5. Linear models may be built by identifying or calculating the slope and using the y-intercept.




### Section Exercises


### Verbal


### Algebraic

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. By 2012 the population had dropped to 4,700. Assume this trend continues.
For the following exercises, consider this scenario: A town’s population has been increased at a constant rate. In 2010 the population was 46,020. By 2012 the population had increased to 52,070. Assume this trend continues.
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.
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.
For the following exercises, consider this scenario: The number of people afflicted with the common cold in the winter months steadily decreased by 205 each year from 2005 until 2010. In 2005, 12,025 people were inflicted.

### Graphical

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

### Numeric

For the following exercises, use the median home values in Mississippi and Hawaii (adjusted for inflation) shown in . Assume that the house values are changing linearly.
For the following exercises, use the median home values in Indiana and Alabama (adjusted for inflation) shown in . Assume that the house values are changing linearly.

### Real-World Applications
