When we start learning about functions in math, especially in a Grade 9 Pre-Calculus class, it’s super important to tell the difference between functions and non-functions when we look at graphs.

This understanding helps us grasp a lot of ideas that come later!

What is a Function?

Let’s go over what a function is.

A function is a special type of relationship where each input (or "x-value") has exactly one output (or "y-value").

You can think of it like a vending machine.

When you put in a coin (input), it only gives you one specific item (output) for that coin.

For example, if you put in a quarter, you might get a soda, but you won’t get chips at the same time.

That’s the way we want to see functions on graphs!

The Vertical Line Test

So, how do we know if a graph shows a function?

One popular way is called the vertical line test. Here’s how it works:

1. Draw a vertical line: Picture or actually draw a straight up-and-down line on your graph.
2. Check for intersections: If this line crosses the graph at more than one point, it means there’s at least one input (an x-value) that has multiple outputs (y-values).
3. Function or not?: If the vertical line only touches the graph at one point (or not at all), then you have a function!

Example:

• Think about a circle. If you draw your vertical line anywhere, it will hit the circle at two points. So, a circle is not a function.
• Now, if you look at a straight line graph, the vertical line will only touch it at one point, which means it is a function.

Other Things to Notice

There are a few other tips to help recognize functions in graphs:

• Slope: If the line is sloped, like a linear function, it shows a steady output for each input. But remember, curves can still be functions as long as they pass the vertical line test!

• Specific shapes:

  • Parabolas (for example, the graph of $y = x^2$) are functions. They look like a U and will pass the vertical line test.
  • Circles (like the equation $x^2 + y^2 = r^2$) are not functions because for many x-values, there are two matching y-values.

Why This Matters

Knowing the difference between functions and non-functions is really important.

It helps us understand math relationships and gets us ready for more complex ideas like domain, range, and even calculus!

In short, when checking graphs, always remember the vertical line test.

If it crosses the graph more than once, then it’s not a function.

Practice these ideas regularly!

The more you get used to spotting functions, the more confident you will be in upcoming math challenges.

Make it fun and visual, and you'll understand it all in no time!