• Quadratic graphs (y = ax² + bx + c) are smooth U-shaped or n-shaped curves called parabolas, with a single turning point (vertex) and up to 2 roots.
• Cubic graphs (y = ax³ + bx² + cx + d) are S-shaped wiggly curves with up to 2 turning points and up to 3 roots.
• Reciprocal graphs (y = a/x) never touch either axis — they have two asymptotes and split into two separate curve branches.
• Exponential graphs (y = aˣ) always pass through (0, 1), have one horizontal asymptote at y = 0, and either grow or decay depending on the base.
• Across all four graph types, the same three questions unlock the sketch: where does it cross the y-axis? where does it cross the x-axis? what shape/direction does it take?

What is it, really?

A quadratic graph comes from an equation of the form y = ax² + bx + c, where a is never zero (if a were zero, the x² term would vanish and it wouldn't be quadratic anymore — it'd just be a straight line). Think of a, b, and c as three separate "dials": a controls the shape and steepness, b nudges the curve sideways, and c is simply where the curve crosses the y-axis.

Picture throwing a ball straight up in the air. It starts low, rises to a peak, then falls back down — that arc shape is exactly what a quadratic graph looks like when a is negative. Flip it upside down (a is positive) and you get a valley shape instead, like a skateboard ramp. This U or n shape is called a parabola, and every quadratic graph — no matter the numbers — has this same symmetrical, smooth curve. It's never jagged, never has sharp corners, and it always has exactly one line of symmetry running vertically through its turning point.

Every quadratic graph always crosses the y-axis exactly once (because for any curve, there's only one y-value when x = 0). But it can cross the x-axis 0, 1, or 2 times — imagine sliding the U-shape up and down: sometimes it dips below the x-axis and crosses it twice, sometimes it just grazes the axis at one point (touches but doesn't cross), and sometimes it floats entirely above or below the axis and never touches it at all. The points where it does cross are called the roots (also called x-intercepts or solutions).

Sketching a quadratic step by step

Here's the reliable 4-step method that works every time:

1. Find the y-intercept — substitute x = 0. For y = ax² + bx + c, this is always just (0, c) — you can read it straight off the equation.
2. Find the roots — set y = 0 and solve ax² + bx + c = 0, either by factorising or using the quadratic formula.
3. Check the sign of a — positive means U-shape, negative means n-shape.
4. Sketch a smooth curve through all the points you've found, respecting the symmetry.

Finding the vertex — the turning point

When a quadratic is written in completed square form, y = a(x − p)² + q, the vertex (minimum or maximum point) has coordinates (p, q) — but here's the classic trap: watch the sign flip.

The value of a doesn't move the vertex at all — it only changes how "wide" or "narrow" the curve looks, and whether it opens upward (a > 0, minimum) or downward (a < 0, maximum).

Working backwards: finding the equation from a graph

Sometimes you're given a sketch and need to reconstruct the equation. There are two routes in, depending on what information the graph shows you:

What is it, really?

A cubic function has the form ax³ + bx² + cx + d, where a, b, c, d are just constants (numbers) — the only rule is that a can never be zero (otherwise the x³ term disappears and it's no longer a cubic). Notice that b, c, and d are allowed to be zero — for example y = x³ is a perfectly valid, if simple, cubic.

The key skill here is different from quadratics: to actually sketch a cubic, you need it in factorised form — e.g. (2x − 1)(x + 2)(x − 3) — rather than expanded form. Why? Because factorised form instantly hands you the roots by inspection, the same way (x − 2)(x − 3) instantly told you the roots were 2 and 3 for a quadratic. You generally won't be asked to factorise a complicated cubic from scratch — but you should be comfortable factorising simpler ones, like x³ − 4x = x(x² − 4) = x(x − 2)(x + 2), and comfortable expanding three brackets together to go the other way.

What does the graph actually look like?

Where a quadratic only ever has one basic shape (a U or an n), a cubic has more personality — it's a smooth, continuous "wiggle." Think of it like a road that goes up a hill, dips into a valley, then climbs another hill (or the mirror image of that). There are four general shapes, split by the sign of a:

What decides the exact shape a particular cubic takes? Three things: the number and value of its roots (x-intercepts), its y-intercept, and the sign of a (which decides whether the curve "starts" bottom-left and "ends" top-right, or the reverse). It can also have up to two turning points — one local maximum and one local minimum — though the simplest cubics like y = x³ have no turning points at all and simply cross the x-axis once, at the origin.

Sketching a cubic step by step

1. Find the y-intercept — substitute x = 0.
2. Find the roots — set y = 0. In factorised form, each bracket set to zero gives you a root directly.
3. Decide the shape — is a positive or negative? Where does the curve "start" and "end"?
4. Place the turning points — logically, between/around the roots.
5. Draw a smooth curve through everything, labelling all the intercepts.

What is it, really?

A reciprocal graph has the form y = a/x. The reason it behaves so differently from quadratics and cubics comes down to one simple fact: you can never divide by zero. That single restriction is what shapes the entire graph.

Think about it logically: what happens to y = 1/x as x gets closer and closer to 0 from the positive side (like x = 0.1, then 0.01, then 0.001)? The value of y explodes upward toward infinity. And as x gets larger and larger (like x = 100, then 1000), y shrinks closer and closer to 0, but never quite touches it. This creates a curve made of two separate branches that never meet, and — critically — the curve never crosses either axis. There's no y-intercept (you can't substitute x = 0, it's undefined) and there are no roots (y can never actually equal 0, since a/x can only equal zero if a itself were zero).

Asymptotes — lines the curve chases but never reaches

An asymptote is a line that a curve gets infinitely close to, but mathematically never touches. Think of it like the horizon — you can walk toward it forever and it always stays just out of reach. A reciprocal graph has exactly two asymptotes: one horizontal, one vertical.

The shifted version: y = a/x + b

Add a constant b onto the reciprocal function and something neat happens — the entire shape of the graph stays exactly the same, it just slides vertically. The graph of y = a/x + b is identical in shape to y = a/x, but shifted upward by b units (or downward, if b is negative).

Since the whole curve slides up or down, the horizontal asymptote slides with it — but the vertical asymptote doesn't move, because it's still along the y-axis (x is still forbidden from equalling zero, regardless of b).

What is it, really?

An exponential graph has the form y = aˣ. What makes this fundamentally different from everything else in this chapter is where the x sits: in every graph so far, x has been the base (x², x³) — here, x has jumped up into the power itself. This tiny shift changes everything about how the graph behaves.

Think of exponential growth like compound interest in a bank account, or bacteria doubling every hour — the bigger the quantity gets, the faster it grows, which creates that unmistakable "hockey stick" curve that stays low for a while and then rockets upward. Exponential decay is the mirror image: think of a hot cup of coffee cooling down, or a radioactive substance losing half its mass at a time — it drops fast at first, then the rate of drop slows down as it approaches (but never quite reaches) zero.

Regardless of whether it's growth or decay, every exponential graph of this form shares three fixed features that never change:

Comparing two exponential graphs

When you sketch two exponential growth curves on the same axes (say y = 3ˣ and y = 4ˣ), here's the intuitive way to remember which sits where: for large positive x, the graph with the bigger base is climbing faster, so it ends up higher. But flip to negative x-values, and it's the opposite — the bigger base actually produces the smaller value (because you're now dividing by a bigger number over and over). Both curves still meet at exactly the same point though: (0, 1), since anything to the power 0 is 1, regardless of the base.

The same logic applies to decay curves with 0 < a < 1, just in reverse for which side is higher.

• I can identify a quadratic graph from its equation and state whether it's U-shaped or n-shaped
• I can find the y-intercept and roots of a quadratic, and sketch it with those points labelled
• I can find the vertex of a quadratic from completed square form, remembering the sign flip
• I can find the equation of a quadratic given its vertex or roots plus one other point
• I understand why cubics must be factorised before sketching, and can find their roots by inspection
• I can tell the difference between a root the graph crosses through and a repeated root it just touches
• I can sketch a cubic showing the correct "start/end" direction based on the sign of a
• I understand why reciprocal graphs have no y-intercept and no roots
• I can state the equations of both asymptotes for y = a/x and y = a/x + b
• I can distinguish exponential growth from decay by checking whether the base is above or below 1
• I know every exponential graph y = aˣ passes through (0, 1) and has asymptote y = 0
• I can compare two exponential graphs and say which is higher for positive and negative x