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Recognizing Linear Functions

A function describes a relationship in which each xx-value corresponds with exactly one yy-value. There are different ways to group functions based on certain characteristics or qualities they have. Here, how a function’s rate of change can make it unique, will be explored.
Rule

Rate of Change

The rate of change, ROC, is a ratio that describes the average change between two related points. It can represent the average speed of a car over a certain period of time, or the average growth rate for bacteria in an experiment. It's determined by dividing the change in the vertical direction (yy) by the change in the horizontal direction (xx).

rate of change=change in ychange in x\text{rate of change}=\dfrac{\text{change in } y}{\text{change in }x}

The Greek letter Δ\Delta (Delta) is commonly used to describe a difference, which leads to an alternative notation of the formula.

rate of change=ΔyΔx\text{rate of change} = \dfrac{\Delta y}{\Delta x}

Exercise

Some of the xx- and yy-values of a function are given in the table.

Determine the rate of change.

Solution

To find the ROC, we must determine the change between xx-coordinates and the change between yy-coordinates. Let's start by looking at the left column. The difference between each row is constant, 1.1.

In the right column, the difference between each row is also constant, 2.2.

Since the rate of change is determined by the ratio of the change in yy and the change in xx, we get ROC=ΔyΔx=21=2. \text{ROC}=\dfrac{\Delta y}{\Delta x}=\dfrac{2}{1}=2. The rate of change is 2.2.

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Rule

Constant Rate of Change

If the rate of change of a function is constant, meaning that for all points on the function, the ROC is always the same, the function is said to be linear. By determining whether or not the rate of change is constant, it's possible to separate linear functions from non-linear functions.
Exercise

In the tables, the xx- and yy-values for two different functions are given.

Determine which, if either, function has a constant rate of change.

Solution

To find the rate of change for each function, we must analyze the change between coordinates. We can start by noting that the left columns for both functions are the same. Between rows, they each increase by 2.2.

Now, let's look at the left table. The first yy-value is 11 and the second is 4.4. The difference between 11 and 44 is 3.3. If we add 33 to 4,4, we get 77 which is the next yy-value. In fact, the difference between each yy-value is 3.3.

Thus, the ROC of the first function is constant and equals 32.\frac{3}{2}. Using the same method, we can note the difference in yy-coordinates for the second function.

Notice that the difference between yy-coordinates changes. Thus, the ROC of the second function is not constant.

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Concept

Linear Function

If a function has a constant rate of change, it is linear. Graphically, a linear function is a straight line.

Using that line, it's possible to determine the rate of change by finding the horizontal change (Δx)(\Delta x) and the vertical change (Δy)(\Delta y) between any two given points on the line. Any function whose graph is not a straight line cannot be linear.
Exercise

A small box contains three golf balls.

Golfballs.svg

The number of golf balls, g,g, that fit inside a golf cart depends on how many small boxes, b,b, fit in the cart. Determine if gg is a linear function.

Solution

If gg has a constant rate of change, then it is linear. To begin, we can make a table of values that shows how many boxes fit in the cart. First, if only one box fits, there will be 33 balls in the cart.

If the golf cart can fit two boxes, then it will contain 23=62\cdot3=6 balls.

If the cart contains 33 boxes, there will be 33=93\cdot3=9 golf balls, and so on. Let's extend the table to five boxes.

Now that we have a table that represents g,g, we can determine the rate of change. The left column increases by 11 each step and the right column increases by 3.3.

Thus, ROC=31ROC=\frac{3}{1} and remains constant. This is because each time 11 additional box is added to the cart, the number of balls increases by 3.3. Thus, gg is linear, and has a discrete domain.

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