Module 1: Mathematical Foundations · Theory · assumes Foundation 1
Learning to read a curve — before learning to compute on one.
Mahavakya
A graph is not a picture of a formula. It is the whole behaviour of a system, laid out where the eye can take it in at once.
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Learning objectives
After this reading you should be able to:
- State what a function is, and why physical situations restrict its domain.
- Recognise the standard function families by shape and name where each shows up in mechanics.
- Predict how shifting, stretching or reflecting a formula moves its graph — and what that means physically.
- Use even and odd symmetry to check an answer before computing anything.
- Read velocity from the steepness of a position–time graph and displacement from the area under a velocity–time graph.
A note on what this module does not do. Everything here is read off a graph by eye. Nothing is differentiated or integrated — those arrive in Foundations 8 and 9. The order is deliberate: it is much easier to compute a thing once you already know what it looks like.
1. What a function is, and what physics demands of it
A function is a rule that takes each input and returns exactly one output. Feed it t and it gives back one value of x — never two, never none. The set of allowed inputs is the domain; the set of values that come out is the range.
That “exactly one” is not a technicality in mechanics. It is the mathematical form of a physical commitment: at a given instant, a particle is somewhere, and it is only in one place. A rule that returned two positions for one time would not be describing anything that could happen.
The vertical line test
Draw any vertical line across the curve. If it ever crosses more than once, the curve is not a function of that horizontal variable. A position–time graph must pass this test. A trajectory — the path a projectile traces through space — often fails it, and that is fine, because a path is not claiming to be a function of anything.
Physics also narrows domains in ways pure mathematics does not. Time usually starts at zero rather than running back forever. The gravitational force F(r) = GMm/r² is undefined at r = 0, and no amount of algebra rescues it — the formula is telling you that a point mass is an idealisation that breaks down at its own centre. A mass, a length, a speed: each carries constraints before any equation is written.
This is Foundation 1’s principle in a new setting. Every formula has a domain of validity, and the domain is part of the physics rather than a mathematical afterthought.
2. The families worth recognising on sight
Most of mechanics is written with a small number of function types. Knowing the shape before you know the numbers is worth a great deal in an exam.
- Linear — x(t) = v0t + x0. A straight line. Constant velocity. Steepness is the same everywhere, which is exactly what “constant” means graphically.
- Quadratic — x(t) = ½at² + v0t + x0. A parabola. Uniform acceleration, and also the shape of a projectile’s path through space. One turning point, and symmetry about it.
- Cubic and higher — more turning points. Potential energy curves with several dips and humps live here.
- Inverse square — F(r) = k/r². Steep near the origin, flattening toward zero far away, never quite reaching it. Gravity and electrostatics share this shape.
- Sine and cosine — x(t) = A cos(ωt + φ). Repeating. Oscillation of every kind: springs, pendulums, waves.
- Exponential — N(t) = N0e−λt. Decay that slows as it proceeds, approaching zero without arriving. Radioactive decay, and a falling body approaching terminal speed.
The oscillating family carries three quantities worth naming now, because they are read straight off the picture. The amplitude A is the height of the peaks — how far the system swings from equilibrium. The period T is the horizontal distance between one peak and the next, and the angular frequency ω = 2π/T counts how fast the cycle turns. The phase φ shifts the whole curve sideways, and its physical job is to record where the system started.
Which is why sine and cosine describe the same motion. A pendulum released from full swing follows a cosine; released as it passes through the centre, a sine. Same pendulum, same physics, different starting moment — and cos(ωt) is just sin(ωt + π/2).
3. Transformations
Change a formula in a standard way and the graph moves in a standard way. This is worth knowing because it lets you adapt a known curve to new conditions without starting again.
| Change | Effect on the graph | What it means physically |
|---|---|---|
| f(x) + d | Shifts up by d | A different zero — choosing where to measure potential energy from |
| f(x − c) | Shifts right by c | The same event, starting later |
| a · f(x) | Stretches vertically | A larger amplitude, or a stronger force |
| f(bx) | Squeezes horizontally | Faster oscillation — a higher frequency |
| −f(x) | Flips top to bottom | The same size, opposite direction |
| f(−x) | Flips left to right | Running the situation backwards |
The one that catches everyone is f(x − c) shifting the graph to the right, when the formula says minus. The reason is worth holding rather than memorising: to get the output that used to appear at x = 0, you now need x = c, because only then does the bracket contain zero again. Every value arrives later, so the whole picture slides forward.
4. Symmetry
Two symmetries appear constantly in mechanics, and spotting one can save an entire calculation.
Even: f(−x) = f(x)
A mirror image about the vertical axis. The energy stored in a spring, U = ½kx², is even: squeeze it by 3 cm or stretch it by 3 cm and you have stored exactly the same energy. The spring does not care which way you pulled.
Odd: f(−x) = −f(x)
Rotate the graph half a turn about the origin and it lands on itself. The spring’s restoring force, F = −kx, is odd: push it left and it pushes back right, with equal strength. The force does care which way you pulled — that is the whole point of a restoring force.
Notice the pair. Energy is even, force is odd, and they describe the same spring. That is not a coincidence, and Foundation 8 will show why: the relationship between them systematically turns one symmetry into the other.
5. Reading physics off a graph
Here is where a graph stops being an illustration and becomes an instrument. Two features carry almost everything.
Steepness and area — the two things a graph tells you before any algebra begins.
Steepness is a rate. Take two points on a curve and the straight line joining them has a slope, Δy/Δx — that is the average rate of change between them. Bring the two points closer together and the line pivots; keep going and it settles onto the line that just touches the curve at a single point. The slope of that line is the instantaneous rate. On a position–time graph it is the velocity; on a velocity–time graph, the acceleration.
You do not need to compute it to use it. Steep means fast. Flat means momentarily at rest. Rising means moving forward, falling means moving back, and the exact instant of turning is where the curve levels off. In the left-hand graph above, the particle moves forward, stops, comes back, stops again, and sets off forward once more — and every one of those statements is read off the shape.
Area is an accumulation. The region between a curve and the horizontal axis measures how much has piled up. Under a velocity–time graph it is displacement; under a force–position graph, work; under a force–time graph, impulse.
Where the graph is made of straight lines you can compute it now, with school geometry. The right-hand graph shows a body at 4 m/s for 3 seconds, then slowing steadily to rest over the next 2. Displacement is a rectangle plus a triangle: (4 × 3) + (½ × 2 × 4) = 12 + 4 = 16 m. No calculus required. Foundation 9 arrives to handle the curved cases, but the idea is the one you have just used.
Turning points and equilibrium. On a potential energy curve, the flat points are where a particle can sit at rest. A dip is stable — nudge the particle and it rolls back, like a marble in a bowl. A hump is unstable — nudge it and it rolls away, like a marble balanced on a dome. You can tell which by looking, which is exactly why the curve is worth drawing.
Contemplation
Throw a ball and draw two graphs. One plots its height against horizontal distance: the arc you would see if you stood back and watched. The other plots its height against time: how high it was at each moment. Under constant gravity both come out as parabolas, and students routinely treat them as the same picture. They are not. The first is a shape in space — you could walk along it, photograph it, trace it with your finger. The second is a history, and nothing about it exists anywhere; it is a record of a sequence of moments, and no part of the ball ever travelled along that curve. The axes are what distinguish them, and the axes are the part students skip. A graph is not a picture of the motion. It is a picture of a relationship, and you have to be told which relationship before you know what you are looking at.
Key insight
Steepness and area are not two facts about graphs that happen to be useful. They are the two questions calculus exists to answer — how fast is this changing, and how much has accumulated. You have been answering both by eye in this module. Foundations 8 and 9 supply the machinery for the cases where the eye is not enough.
Common misconceptions
1. Zero velocity means zero position
A flat point on a position–time graph means the particle is momentarily still. It says nothing about where it is — it can be still at 6 m just as easily as at 0. Height on the graph is position; steepness is velocity. Two different features, read differently.
2. A downward slope means slowing down
On a position–time graph it means moving backwards, which is a direction, not a speed. A steep downward section is fast backward motion. Speeding up or slowing down is about whether the steepness is increasing or decreasing, not about its sign.
3. Any curve can be a position–time graph
It has to pass the vertical line test, because a particle cannot be in two places at one instant. A circle drawn on x–t axes is not a slow error of drawing; it is a claim that something impossible happened.
4. Sine motion and cosine motion are different
They are the same oscillation started at different moments, since cos(ωt) = sin(ωt + π/2). The choice records the initial condition, not the physics. Nothing about the spring changes when you decide to start your stopwatch.
5. Read the curve, ignore the axes
The most expensive habit in this topic. A parabola on y–x axes is a path through space; the same parabola on y–t axes is a history in time. Identical shapes, entirely different claims. Always read the axes first.
Reflection
- Two pendulums are released at the same instant, one from full swing and one from the centre. Their graphs have identical shape and different phase. What in the physical setup does the phase record?
- The area under a velocity–time graph is a displacement. What has to be true about the units on the two axes for an area to come out in metres at all?
- You are shown a smooth curve with no axis labels. What can you still say about the physics, and what becomes impossible to say?
Key takeaways
- A function returns exactly one output per input — which is what makes it able to describe a particle’s position.
- Physical situations restrict domains before any algebra does.
- Linear, quadratic, inverse-square, sinusoidal and exponential shapes cover most of mechanics.
- Shifts, stretches and reflections of a formula correspond to delays, amplitude changes and reversals.
- Even symmetry belongs to energies; odd symmetry belongs to restoring forces.
- Steepness is a rate; area is an accumulation. Both can be read before any calculus.
- Read the axes before the curve. The same shape means different things on different axes.
Prerequisite: Foundation 1 · Reading: 15 min · Practice: 20 min · Difficulty: Beginner