Functions and Graphs: Lines, Parabolas and Sine Waves

A point is two numbers

Imagine a grid like a city map. To name a spot you need two numbers: how far across and how far up. These are the coordinates (x, y). The point (3, 2) is three steps right and two steps up from the origin, the corner marked (0, 0). In 1637 the French thinker René Descartes published La Géométrie, tying geometry to algebra in this way, which is why we speak of Cartesian coordinates. Pierre de Fermat had a similar idea independently. One tale says Descartes thought of it while watching a fly on his ceiling, but it is probably a legend. By Pythagoras, the point (3, 2) lies √13 ≈ 3.61 from the origin.

The straight line

Take a rule: double x and add 1, written y = 2x + 1. Feed in numbers and you get points: x = 0 gives y = 1, x = 1 gives y = 3, x = 3 gives y = 7. Join the dots and they fall on a perfectly straight line. Every rule of the form y = kx + b makes a line. The number k is the slope: how much y rises when x goes up by 1. The number b is where the line crosses the vertical axis. So k tilts the line and b lifts it. The word 'function' for a rule that gives one output for each input comes from Leibniz in the late 1600s.

Reading the slope

To find k, pick two points and divide the rise by the run. A line passes through (0, 1) and (4, 9). The rise is 9 − 1 = 8, the run is 4 − 0 = 4, so k = 8 / 4 = 2 and b = 1: it is y = 2x + 1 once more. A negative k makes the line fall, and k = 0 makes it flat. Does this ring a bell? In the physics book, a distance–time graph is a line whose slope is the speed. In the sandbox choose the line and move k and b: the moving point shows its coordinates as it slides along.

The parabola

Now square the input: y = x². The values for x = −3, −2, −1, 0, 1, 2, 3 are 9, 4, 1, 0, 1, 4, 9. Plot them and you get a U-shaped curve called a parabola. It is symmetric, because a negative number squared is positive, and its lowest point sits at (0, 0). The curve is steep far away and flat near the bottom, because each step outward adds more than the last: from 1 to 2 adds 3, and from 2 to 3 adds 5. Doubling x quadruples y, which should remind you of falling bodies, where distance grows as the square of time.

A ball's path

Throw a ball straight up at 10 m/s on Earth. Its height after t seconds is h = 10t − 4.905 t², where 4.905 is g / 2 and g = 9.81. This is an upside-down parabola. The ball peaks when t = 10 / 9.81 ≈ 1.02 s, at a height of 10² / (2 × 9.81) ≈ 5.10 m, and it lands at t = 2 × 10 / 9.81 ≈ 2.04 s, when h is zero again. The same shape appears in the flight of a ball thrown at an angle, in the main cables of a suspension bridge and in the dish of a satellite antenna.

Reading a graph

A graph is a picture you can read. Where it crosses the horizontal axis, y is zero: these places are the zeros of the function. Take y = 2x − 6. Setting y = 0 gives 2x = 6, so x = 3. For y = x² − 4, the zeros are where x² = 4, at x = −2 and x = 2, and the lowest point is (0, −4). Highest and lowest points are called maximum and minimum. Where a curve climbs, y is increasing; where it falls, y is decreasing. In the sandbox, slide the point along the parabola and watch the readout drop to its minimum and rise again.

One rule, a wave

Rules can make shapes that repeat forever. Look at y = sin x, with x measured in radians. As x grows from 0, y rises to 1 at x = π/2 ≈ 1.571, returns to 0 at x = π ≈ 3.142, dips to −1 at x = 3π/2 ≈ 4.712 and is back at 0 when x = 2π ≈ 6.283. Then the pattern starts again. The wave never goes above 1 or below −1. Multiply by 3 and y = 3 sin x swings between −3 and 3. Sound, light and a swinging pendulum are all described by curves of this kind. Switch the sandbox to the sine and drag the point along the wave.

Functions and graphs