Linear Functions

Learning Objectives

• Define linear functions and describe their characteristics.
• Compare and contrast proportional and non-proportional linear functions.
• Explain the components of the linear function equation.

Every morning, Miguel makes a cup of coffee by dumping `2.5` tablespoons of ground coffee beans into his coffee maker. Sometimes after a late night, he needs `2` cups of coffee to wake up, so he puts twice as many grounds into the machine. If he has friends over for dinner and wants to serve coffee with dessert, he makes an even bigger pot of coffee. His recipe is simple—he puts in `2.5` tablespoons of grounds for every cup of coffee he wants to brew.

Miguel can easily adapt his recipe, because there is a relationship between the amount of grounds and the number of cups of coffee—as the amount of coffee increases, so does the amount of grounds needed. Let’s see what that relationThe relationship between variables that change together. looks like on a graph.

We can see that this relation is a functionA kind of relation in which one variable uniquely determines the value of another variable.—for each independent value (cups of coffee), there is a single dependent value (tablespoons of grounds needed). We can also see that the graph of paired inputsThe independent variable of a function—input determines output. and outputsThe dependent variable of a function—output is determined by input. is a straight line. That makes this a linear functionA function with a constant rate of change and a straight line graph.—a function is linear if its graph forms a straight line. The line is straight because the variables change at a constant rate. That is another characteristic of linear functions—they have a constant rate of changeThe constant in a proportional function equation; it describes the ratio or proportional relationship of the independent and dependent variables—also called the constant of variation or the constant of proportionality..

Let’s look at a couple of different kinds of linear functions to get a better idea of how they work.

While Miguel drinks his coffee, he decides to compare some text plans to see which one would work better for him. His cell phone company offers two plans. The basic plan charges `7` cents per text message sent or received. The advanced plan lets Miguel send text, pictures and video, and it only charges `2` cents per message. Score! Oh, but wait—the advanced plan has a `$10` monthly fee. So how is Miguel going to decide which plan is actually the better deal for him?

I know! Both plans are functions—the cost of texting depends on the number of texts he sends. If we graph each one, we can compare how the costs vary.

We’ll start with the basic plan. We can figure out the cost (`y`) of sending and receiving any number of text messages (`x`) with the simple equation `y = 0.07x`. (By using `0.07` instead of `7`, the cost (`y`) will be measured in dollars, not in cents.) With that equation, we can calculate the following points:

The graph of those points looks like this—a straight line rising from `(0,0)`. 

Now we’ll look at the advanced plan. To figure out the cost (`y`) of sending and receiving `x` number of messages per month with this plan, we can begin with an equation a lot like the one we used before: `y = 0.02x`, since every message costs `2` cents. But remember the monthly fee? We have to add on `$10` for that, so our equation is `y = 0.02x + 10`. With that equation, we can calculate the following values for the advanced plan:

Let’s plot those points now.

Now the difference between the plans is clear—if Miguel sends fewer than `200` texts a month, the basic plan is cheaper. If he sends more than `200` texts, he’d save money using the advanced plan. That’s good for Miguel to know, but we are more interested in the functions themselves. They are both straight lines with a steady rate of change, so they are both linear functions. But the basic plan function includes the origin, while the advanced plan does not.

There’s something special about linear functions that include the origin—there is a very simple multiplicative relationship between the independent variablesA value or variable that changes or can be manipulated by circumstances. and dependent variablesA value or variable that depends upon the independent value.. This means that the variables have a proportional relationship, and these are proportional functionsA function in which the input times a constant equals the output.. For proportional functions, if we multiply both variables by the same number, their relationship will be maintained. For example, let’s go back to the chart for the basic text plan:

The cost of one text `= $0.07`. If Miguel wanted to find out the cost of sending `10` texts under this plan, he could take the equation `1" text" = $0.07` and multiply both sides by `10`. He would arrive at a cost of `$0.70` for `10` texts. Multiply both quantities by `0` and you’ll see that the graph must begin at the origin. This makes sense, because `0` messages cost `$0`.

Now let’s look at the other function, for the advanced plan:

The cost of one text ` = $10.02`. Can we calculate the cost of `10` texts by multiplying both sides of that relationship by `10`? No. `10` texts do not cost `$100.20`! What went wrong? While the advanced plan is a linear function, it is not a proportional function—the independent and dependent variables are not related by just a constant and they do not have a proportional relationship. With non-proportional relationships, multiplying both input and output by the same number doesn’t work. Because of this, non-proportional linear functions do not include the origin—multiplying the input by `0` will not make the output `0`.

We’ve seen that all linear functions have certain things in common. They all involve independent and dependent variables (inputs and outputs) that show a constant rate of change. For proportional functions, these features are sufficient to write the equation of the function: output `=` constant rate of change `*` input. The standard algebraic way of writing this equation is `y=kx`.

For non-proportional linear functions, there is another factor, an adjustment that moves the graphed line of the function away from the origin. In our text plan example, this adjustment was the monthly charge, which is applied regardless of the value of the independent and dependent variables. We have to include this factor in the equation that defines these functions: output `=` (constant rate of change `*` input) `+` adjustment. The standard algebraic way of writing this equation is `y = mx + b`. 

Linear functions have a constant rate of change and describe a straight line on a graph. Linear functions can be defined by the equation `y = mx + b`. Proportional functions are linear functions that include the origin and can be defined by the equation `y = kx`. All proportional functions are linear, but not all linear functions are proportional.