What is a graph of a function? (We explain on our fingers)
Imagine that a mathematical function is a magical gum dispenser. You throw a coin into it (this is the number x), the machine inside clicks something according to its own rule, and gives you some chewing gum (this is the number y). They threw two coins and got two pieces of gum. They threw five and got five.
So, function graph is just a photograph of how this machine works. This is a picture that shows us all the possible options: how many chewing gums we will get with any number of coins. Instead of holding long lists of numbers in our heads, we draw one clear line.
Where are graphs drawn? (Getting to know the axes)
To draw a graph, we need a special “canvas”. In mathematics it is called thecoordinate plane. Imagine the intersection of two roads:
- ➡️ X-axis (horizontal): It goes from left to right. It is also called the buzzword “abscissa axis”. Here we mark numbers that we set ourselves (our coins).
- ⬆️ Y-axis (vertical): It goes from bottom to top. This is the "y-axis". Here we note the result produced by the function (our chewing gum).
- 🎯 Origin: The place where these two roads intersect. This is point Zero (0; 0).
How to build your first chart? (Step by step)
Let's draw a graph for the simplest rule: “the result is always twice the starting number”. In the language of mathematics, this is written briefly: y = 2 × x.
To build a line, we need to find at least a couple of points. Let's make a small table:
| Take any number X | Count Y (multiply by 2) | Getting a point (Coordinates) |
|---|---|---|
| 1 | 2 × 1 = 2 | Point A (1; 2) |
| 2 | 2 × 2 = 4 | Point B (2; 4) |
| 3 | 2 × 3 = 6 | Point B (3; 6) |
Now we simply find these points at our “intersection” (we move back to the right along the X axis and go up along the Y axis), put bold points there with a pencil and connect them using a ruler. Congratulations! You've just graphed a function!
🔥 Interesting fact: What does the fly have to do with ceiling?
The coordinate plane was invented by the great scientist Rene Descartes. Legend has it that one day he was lying in bed and looking at a fly crawling on the ceiling. In order to accurately describe to his neighbor exactly where the fly was sitting, he came up with the idea of measuring the distance from two adjacent walls of the room. This is how the modern X and Y axes appeared!
What shapes do graphs have?
The graph doesn't always look like a boring straight line. Depending on the rule (function), the line can bend in any way:
- 📏 Straight line: The simplest shape. Occurs when we simply multiply or add ordinary numbers (for example, y = 3x + 1).
- 🪃 Parabola: Looks like a beautiful bowl or horseshoe. It appears when our number X is multiplied by itself, that is, raised to square (y = x2).
- 🌊 Sine wave: A line that jumps up and down, like waves on the sea or a heart cardiogram. Shows processes that are constantly repeated.
Why do we need graphs in real life?
Scientists and engineers don't draw graphs for fun. Our brain is designed in such a way that it is difficult for it to analyze tables of thousands of dry numbers. But taking one look at the picture (graph), we immediately understand the essence of what is happening:
- 🌡️ Weather forecast: The temperature graph shows when the heat will peak during the day (the line went up), and when it will get colder at night (the line went down).
- 💰 Money and business: Looking at the sales chart, the store director immediately sees whether his income is falling or growing.
- 🏃♂️ Sports: The runner’s speed graph will show at what minute he got tired and slowed down, and where he made the final push.