## Model an Application With a Linear Function

### Learning Outcomes

• Write the equation for a linear function given an application

In the real world, problems are not always explicitly stated in terms of a function or represented with a graph. Fortunately, we can analyze the problem by first representing it as a linear function and then interpreting the components of the function. As long as we know or can figure out the initial value and the rate of change of a linear function, we can solve many different kinds of real-world problems.

### How To: Given a linear function $f$ and the initial value and rate of change, evaluate $f(c)$

1. Determine the initial value and the rate of change (slope).
2. Substitute the values into $f\left(x\right)=mx+b$.
3. Evaluate the function at $x=c$.

Initial value is a term that is typically used in applications of functions.  It can be represented as the starting point of the relationship we are describing with a function. In the case of linear functions, the initial value is typically the y-intercept. Here are some characteristics of the initial value:

• The point $(0,y)$ is often the initial value of a linear function
• The y-value of the initial value comes from b in slope-intercept form of a linear function, $f\left(x\right)=mx+b$
• The initial value can be found by solving for b or substituting $0$ in for x in a linear function.

In our first example, we are given a scenario where Marcus wants to increase the number of songs in his music collection by a fixed amount each month. This is a perfect candidate for a linear function because the increase in the number of songs stays the same each month. We will identify the initial value for the music collection and write an equation that represents the number of songs in the collection for any number of months t.

### Example

Marcus currently has $200$ songs in his music collection. Every month he adds $15$ new songs. Write a formula for the number of songs, N, in his collection as a function of time, t, the number of months. How many songs will he own in a year?

In the example we just completed, notice that N is an increasing linear function. As the input (the number of months) increases, the output (number of songs) increases as well.

In our next example, we will show that you can write the equation for a linear function given two data points. In this case, Ilya’s weekly income depends on the number of insurance policies he sells. We are given his income for two different weeks and the number of policies sold. We first find the rate of change and then solve for the initial value.

### Example

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(n) and interpret the meaning of the components of the equation.

In the following video example, we show how to identify the initial value, slope, and equation for a linear function.

We will show one more example of how to write a linear function that represents the monthly cost to run a company given monthly fixed costs and production costs per item.

### Example

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 where C(x) is the cost for x items produced in a given month.

The following video example shows how to write a linear function that represents how many miles you can travel in a rental car given a fixed amount of money.

In the next example, we will take data that is in tabular (table) form to write an equation that describes the rate of change of a rat population.

### Example

The table below relates the number of rats in a population to time (in weeks). Use the table to write a linear equation.

 w, number of weeks $0$ $2$ $4$ $6$ P(w), number of rats $1000$ $1080$ $1160$ $1240$