Functions and Graphs

A taxi meter starts at 3.50 the second the doors close, then adds 1.20 for every kilometer the wheels turn. Ride 8 kilometers and the meter reads 3.50 + 1.20 times 8, which is 13.10. Ride twice as far and nobody recomputes anything by hand: the rule does the work, the same way, every time. That pair of numbers, a starting point and a rate, is one of the most useful patterns in all of applied mathematics, and it appears in taxi fares, phone bills, motor speeds, and the voltage across a resistor.

Mathematics is the discipline where rules like that get written down exactly, with no room for misreading. The vocabulary for writing them is the subject here: variables, functions, slope, and the graphs that turn all three into a single picture. Every later lesson in this track, from falling stones to charging capacitors, is a function with a different costume on.

Variables are names for quantities

A variable is a name that stands in for a quantity, the way a pronoun stands in for a person. Instead of saying "the number of kilometers traveled" fourteen times in one sentence, give it a short name: k. The taxi rule becomes fare = 3.50 + 1.20 times k, and that one line now covers every ride the cab will ever make. The letter carries no magic. t usually means time, d means distance, and any letter will do, but picking a name that hints at the quantity keeps the math readable.

A variable holds a number, and numbers can be added, multiplied, and compared exactly like the values they stand in for. Writing 3.50 + 1.20k is not shorthand for an idea; it is the idea, stated once and for all cases. The moment a rule is written with variables instead of one worked example, it can be reused, checked, and handed to a computer.

A function is a machine with one output per input

A function is a rule that takes an input and produces exactly one output. Feed it a number, it hands back a number, and the same input always produces the same output. A vending machine is the usual comparison: press B4, get the same bag of chips every time, because a machine that returned random items on different presses would be broken as a machine and useless as a model.

The standard notation writes the rule as f(x), read aloud as "f of x". The f names the rule, the x in parentheses is the input, and f(x) is the output that the rule produces from that input. The taxi rule written this way is f(k) = 3.50 + 1.20k. To ask what the meter shows after a 15 kilometer ride, evaluate the function at 15: f(15) = 3.50 + 1.20 times 15 = 21.50. The notation is compact to the point of terseness, but every part of it has a job: f is the machine, x is what goes in, f(x) is what comes out.

Two informal words describe what a function accepts and what it can produce. The domain is the set of inputs the rule makes sense for: kilometers from 0 up to, say, 200 for the taxi, since negative distances do not happen. The range is the set of outputs it can produce: 3.50 and up, in steps of 1.20. Neither word needs to be feared; domain means what can go in, range means what can come out.

Slope is a change divided by a change

Slope answers the question "per what?" It is the change in output divided by the change in input between two points, and it carries units: dollars per kilometer, meters per second, degrees per hour. Take two rides from the meter: 2 kilometers costs 5.90, and 6 kilometers costs 10.70. The slope between them is (10.70 - 5.90) / (6 - 2) = 4.80 / 4 = 1.20 dollars per kilometer. The rate hiding inside the rule falls out of two observations of the output.

On a graph, slope is steepness. A gentle line gains output slowly as the input grows; a steep line gains it fast. The nickname "rise over run" describes the picture: rise is the vertical change, run is the horizontal change, and their ratio is the slope. What the nickname leaves out is the units, and the units are the meaning. A slope of 1.20 is meaningless; a slope of 1.20 dollars per kilometer is a taxi rate. Whenever a slope shows up, ask per what before doing anything else.

Slope is also how a graph betrays its rate. On a distance-versus-time graph the slope is a speed, because meters divided by seconds is meters per second. On a cost-versus-quantity graph it is a price per unit. The slope always inherits the vertical axis units divided by the horizontal axis units, which is why the same picture can carry completely different meanings depending on what the axes say.

The line y = mx + b

Every straight line is a rate plus a starting point. The standard way to write it is y = mx + b, where m is the slope and b is the value of y when x is zero, the starting point. The taxi rule fits the pattern exactly: y = 1.20x + 3.50, so m is 1.20 dollars per kilometer and b is 3.50, the charge that appears before the wheels turn at all. The b is where the line crosses the vertical axis, which is why graphs of real systems so often start partway up instead of at zero.

Because the slope is the same between every pair of points on a straight line, the rule adds the same amount for every step. That is the whole reason the graph is straight. Predicting is then trivial arithmetic: a 15 kilometer ride costs 3.50 + 1.20 times 15 = 21.50, no matter which two known points on the line were used to discover m and b. Learn the pattern rather than the letters: any letter can play the role of m or b, but a rate plus a starting point is always what a straight line is saying.

Reading a graph like a sentence

A graph is a picture of a function: the horizontal axis carries the input, the vertical axis carries the output, and each point on the curve is one input and its output sitting together. Reading one fluently means asking three questions in order. What is rising and what is falling? Where is it flat? And where is it steepest? The steepness is the rate, so the steepest stretch of a curve is where the change happens fastest.

A distance-versus-time graph of a delivery scooter tells the whole trip at a glance. A flat segment means the scooter is parked, because time passes but distance does not grow. A steep segment means fast movement, and the steepest segment of the whole graph is the fastest leg of the ride. If the first half hour covers 10 kilometers and the second half hour covers 5, the slopes are 20 kilometers per hour and then 10 kilometers per hour, and the graph shows the slowdown as a visible flattening without a single number being read.

A temperature graph over a day behaves the same way with different units. The curve climbs through the morning, peaks in mid-afternoon, and slides overnight. The height of the curve says how hot it is; the steepness says how fast it is changing. The steepest stretch, often mid-morning, is not the hottest moment, it is the moment the temperature is rising fastest. Confusing height with steepness is the single most common mistake in reading graphs, and the two are different quantities with different units.

Why curves bend

A straight line has one slope everywhere, which is exactly why it stays straight. A curve has a different slope at every point, and the bending is what carries that information. Nothing in the natural world with a changing rate graphs as a straight line, so curves are not decorations; the bend is the data.

Watch it happen in numbers. Any object in free fall near Earth's surface gains 9.8 meters per second of downward speed each second, a fact stated here and used heavily in the physics lessons ahead. The distance it falls grows as the square of the time: d = 4.9 t^2, where the 4.9 is half of 9.8. Compare that with the taxi rule, one row of output at a time:

def fare(km):
    return 3.5 + 1.2 * km          # fare for a ride of km kilometers

def drop(t):
    return 4.9 * t ** 2            # meters fallen after t seconds

for t in range(5):
    print(t, fare(t), drop(t))

The fare column gains exactly 1.20 every row, so its graph is a straight line with slope 1.20. The drop column gains 4.9, then 14.7, then 24.5, then 34.3. Each gain is bigger than the last because the stone is moving faster with every second it has already spent falling. Plot those points and the curve bends upward, and the bend steepens row by row. A graph that bends downward tells the opposite story: a rate that is shrinking, like a cooling drink whose temperature drops quickly at first and then barely at all. Engineers learn to read the bend before they read any numbers off the graph, because the bend is where the physics is.

Every rule so far has grown by adding: the same 1.20 each kilometer, or a gain that at least grows steadily. But some relationships multiply instead, doubling or halving at every step, and they produce numbers that addition-based rules can never reach in a reasonable number of steps. Exponents and Logarithms takes up those relationships, along with the logarithm that runs multiplication backward to ask "how many doublings was that?"