Modeling with tables, equations, and graphs

Math is all about relationships. For example, how can we describe the relationship between a person's height and weight? Or how can we describe the relationship between how much money you make and how many hours you work?

The three main ways to represent a relationship in math are using a table, a graph, or an equation. In this article, we'll represent the same relationship with a table, graph, and equation to see how this works.

Example relationship: A pizza company sells a small pizza for $\mathrm{\$}6$‍ . Each topping costs $\mathrm{\$}2$‍.

We know that the cost of a pizza with $0$‍ toppings is $\mathrm{\$}6$‍, the cost of a pizza with $1$‍ topping is $\mathrm{\$}2$‍ more which is $\mathrm{\$}8$‍, and so on. Here's a table showing this:

Of course, this table just shows the total cost for a few of the possible number of toppings. For example, there's no reason we couldn't have $7$‍ toppings on the pizza. (Other than that it'd be gross!)

Let's see how this table makes sense for a small pizza with $4$‍ toppings.

Here's the cost of just the pizza:

Here's the cost of the $4$‍ toppings:

This leads to the total cost of

Let's write an equation for the total cost $y$‍ of a pizza with $x$‍ toppings.

Here's the cost of just the pizza:

Here's the cost of $x$‍ toppings:

So here's the equation for the total cost $y$‍ of a small pizza:

Let's see how this makes sense for a small pizza with $3$‍ toppings:

We can create ordered pairs from the $x$‍ and $y$‍ values:

We can use these ordered pairs to create a graph:

Cool! Notice how the graph helps us easily see that the total cost of the small pizza increases as we add more toppings.

We represented the situation where a pizza company sells a small pizza for $\mathrm{\$}6$‍, and each topping costs $\mathrm{\$}2$‍ using a table, an equation, and a graph.

What's really cool is we used these three methods to represent the same relationship. The table allowed us to see exactly how much a pizza with different number of toppings costs, the equation gave us a way to find the cost of a pizza with any number of toppings, and the graph helped us visually see the relationship.

Now let's give you a chance to create a table, an equation, and a graph to represent a relationship.

An ice cream shop sells $2$‍ scoops of ice cream for $\mathrm{\$}3$‍. Each additional scoop costs $\mathrm{\$}1$‍.

We learned that the three main ways to represent a relationship is with a table, an equation, or a graph.