For the following table, find the algebraic equation that shows the proportional relationship between 𝑥 and 𝑦.
What is that mean — a proportional relationship? The variables in a function will change at a constant rate in a proportional relationship, and we use the formula 𝑦 equals 𝑘𝑥, where 𝑘 is the constant rate that is changing the variables. And we can use the formula 𝑦 equals 𝑘𝑥 to help us figure out what the constant rate of change in this table is.
We’re asking here, what is happening to 𝑥 to give us 𝑦? What we’re multiplying our 𝑥 by to give 𝑦? And our first instance: 𝑥 equals six and 𝑦 equals 12. How did we go from six to 12? 12 equals 𝑘 times six. We recognize that our 𝑘, our constant rate of change, is two. Two times six equals 12; our constant rate of change equals two.
Let’s check the other two points in our table to make sure that’s true for them as well. Does seven times two equal 14? Yes, it does. Does eight times two equal 16? Yes, so all three of our points are following a constant rate of change of two. But our work here isn’t finished. Our question is asking, what is the algebraic equation that shows this? We plug in two to our proportional relationship formula. And then our algebraic equation becomes 𝑦 equals two times 𝑥.