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In this section you will:
  • Build linear models from verbal descriptions.
  • Model a set of data with a linear function.
A skyline view of Seattle with a focus on the Space Needle.
(credit: EEK Photography/Flickr)

Emily is a college student who plans to spend a summer in Seattle. She has saved $3,500 for her trip and anticipates spending $400 each week on rent, food, and activities. How can we write a linear model to represent her situation? What would be the x -intercept, and what can she 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 her situation, there are two changing quantities: time and money. The amount of money she has remaining while on vacation depends on how long she stays. 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: She saved $3,500, so $3,500 is the initial value for M .              Rate of Change: She anticipates 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 she is spending money each week.

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

Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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