Functions and Graphs: An Eye for Change

1.Functions: One Output for Each Input

Press button 1 on a vending machine and you always get the same drink. A relationship like this, where choosing one input x settles exactly one output y, is called a function. "Distance traveled by taxi → fare," "hours studied → predicted test score," and "temperature → predicted ice cream sales" are all relationships that can be expressed as functions.

On the other hand, if one input has several outputs, it is not a function. "Person aged 30 → that person's height" is not a function, because there are many 30-year-olds and their heights all differ. To decide whether something is a function, ask: "Does the same input always give the same output?"

Broadly speaking, an AI model is also a giant function. You input a sentence or a photo, and it outputs an answer or a classification. We'll come back to this view in Lesson 10.

2.The Linear Function y = ax + b

The simplest and most commonly used function is the linear function. It has the form y = ax + b, and its graph is a straight line. a is called the slope, and b the y-intercept.

The y-intercept b is the value of y when x = 0, that is, the height at which the graph meets the y-axis. For a phone plan, it is the base fee you pay even if you use nothing. The slope a tells you how much y changes when x goes up by 1. For a phone plan, it is the extra charge per 1GB. If a is positive, the line rises to the right; if a is negative, it falls to the right; and the larger its absolute value, the steeper the line.

Making a table for y = 2x + 1 shows the meaning of slope at a glance. Every time x goes up by 1, y always goes up by 2.

3.Finding the Equation from Two Points

Two points are enough to pin down a single straight line. First find the slope between the two points, then plug one point into the equation to find b. When you "find the speed of change from values at two points in time" in real data, this is exactly the calculation you are doing.

4.When the Slope Is Negative or a Fraction

Even when the slope is not a whole number, the method is the same. But mistakes with signs and fractions are common, so subtract in the same order for both coordinates, and always check with the other point at the end.

A slope of −2/3 means "when x goes up by 3, y goes down by 2." Reading a fractional slope this way, as "how many squares you move up or down for so many squares across," makes the graph easy to draw. With a slope of 1/2, you go up 1 square for every 2 squares to the right; with a slope of −3, you go down 3 squares for every 1 square to the right.

5.Intersections: Where Two Options Are Equal

The point where two lines meet is where the two equations give the same value. Seen on a graph, solving the systems of equations from Lesson 3 is exactly the job of finding this intersection. In everyday life, it tells you "from what point the other option becomes the better deal."

6.Reading Graphs: Slope Is Speed

When you look at a graph, first check what the two axes are and what their units are. Then read the slope. If the horizontal axis is time and the vertical axis is distance, the slope is speed. If the horizontal axis is usage and the vertical axis is cost, the slope is the unit price. The slope's unit is "vertical-axis unit ÷ horizontal-axis unit," as in km/h or $/GB.

Real-world graphs are usually not perfectly straight. Even so, if you zoom in on a short stretch, it looks like a straight line, and the slope of that stretch is the rate of change at that moment. If the slope keeps getting bigger, the change is speeding up; if the slope gets close to 0, the change is coming to a stop. This idea of "the slope at a single point" leads into gradient descent in Lesson 10.

• Read the axis names and units first
• Put the y-intercept (starting value) and the slope (rate of change) into words, one at a time
• If there is a point where the slope changes, find out what changed
• Don't casually extend the line beyond the range of the graph; there's no guarantee the relationship stays a straight line