A function is linear if its rate of change is constant. Two ways that we can determine if a function has a constant rate of change — if it is linear or nonlinear — is by using a table or a graph. We are asked to explain how this works. Let's start with a table.

Using a Table

Here is a table showing the cost of a wedding reception $y$ (in dollars) for different numbers of guests $x .$ We want to know if this relationship is linear.

We need to see if the rate of change of the function is constant. If it is, then the function is linear. Let's then calculate the differences between consecutive $x -$ and $y -$ values.

Each time we invite $20$ more guests, the cost of the reception always increases by $$ 6000 .$ This tells us that the rate of change is constant. As a result, the function is linear.

Using a Graph

Here is a graph showing the cost of a wedding cake. The cost is a function of the number of guests invited to the reception.

Once again, we want to see if the function is linear. To do so, we need to check if the points all fall on the same line. If they do, then the function is linear. Let's then draw a line passing through as many points as possible.

Not all of the points fall on the line. This means that the function is nonlinear.

Exercise

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