Function Rules — I can interpret and write function rules in words and in equation form at my level.

People have been noticing patterns in numbers for thousands of years. Ancient farmers tracked how many seeds they planted and how much food they harvested. Merchants figured out the cost of buying many items at the same price. These everyday problems all follow a hidden pattern — a function rule that connects an input to an output.

Over time, mathematicians created a shorthand way to write these patterns. Instead of describing every step in words, they used letters and symbols. This made it faster to communicate rules and solve problems.

So here's the big question: how do you take a pattern you notice — like "every hour I earn $8 more" — and write it as a neat equation? That's exactly what this lesson is about.

A function is a rule that takes each input and gives exactly one output. Think of it like a machine: you put a number in, the machine does something to it, and one number comes out. A function rule is the set of instructions that tells the machine what to do.

The diagram below shows a function machine. You drop an input number in at the top. The machine applies the rule. The output pops out at the bottom. Notice how the rule is written in both words and as an equation inside the machine.

Notice that the same rule is written two ways inside the machine. The word version says "multiply by 3, then add 2." The equation version says y = 3x + 2. They mean the exact same thing. The equation is just quicker to write.

When you write a function rule as an equation, you use letters to stand for numbers. The letter x represents the input and y represents the output. The equation tells you what to do to x to get y.

Let's break down how to read this. The "2x" means "2 times x" (the multiplication sign is hidden — that's normal in algebra). The "+ 5" means you add 5 after multiplying. So if x = 4, you calculate 2 × 4 + 5 = 13. That means y = 13.

Sometimes you're given a function table (a list of input-output pairs) and you need to figure out the rule. The trick is to look at what happens between each input and its output. Ask yourself: "What do I do to x to get y?"

Notice the output goes up by 3 each time the input goes up by 1. That tells us the rule involves multiplying by 3. But 3 × 1 = 3, not 5. We need to add 2 to get 5. So the rule is y = 3x + 2.

Let's walk through a full example together. Suppose you babysit and earn money based on the number of hours you work. Here is your pay table:

Both word rules and equation rules describe the same thing. So when should you use each one? Let's compare.

The function rules you are learning now are the building blocks for algebra. In Algebra 1, you will see these same rules written as f(x) instead of y. For example, y = 3x + 2 becomes f(x) = 3x + 2. It means exactly the same thing.

You'll also learn about rules that aren't straight-line patterns, like y = x² (a parabola). But every function — no matter how fancy — still follows the same idea: one input goes in, one output comes out.