At GCSE you meet six families of graph — linear, quadratic, cubic, reciprocal, exponential, and trigonometric — each with a distinctive shape. Recognising the family from its equation, or matching an equation to a sketch, is a common examination skill at both Foundation and Higher tier.
The six families at a glance
| Family | General form | Key visual features |
|---|---|---|
| Linear | y = mx + c | Straight line; gradient m, y-intercept c |
| Quadratic | y = ax² + bx + c | Parabola; U-shape (a > 0) or ∩-shape (a < 0) |
| Cubic | y = ax³ + bx² + cx + d | S-shaped curve; one inflection point |
| Reciprocal | y = k/x (or y = k/x²) | Two branches in opposite (or same) quadrants; asymptotes at x = 0 and y = 0 |
| Exponential | y = aˣ or y = kaˣ | Smooth growth or decay curve; never touches x-axis; passes through (0, k) |
| Trigonometric | y = sin x, cos x, tan x | Wave or repeating asymptote pattern; period 360° for sin/cos |
Linear graphs
A linear equation contains x (and possibly a constant) but no higher powers of x.
Key pointers:
- If the equation can be written as y = mx + c, it is a straight line.
- m is the gradient: positive → slopes up to the right; negative → slopes down to the right; zero → horizontal.
- c is where the line crosses the y-axis.
Examples: y = 3x − 2, y = −x + 5, y = 4, x = −2 (vertical line).
Exam trap: x = k is a vertical line, gradient is undefined (not zero).
Quadratic graphs
A quadratic contains x² as the highest power, with no x³ term.
Key pointers:
- Positive leading coefficient (a > 0): U-shaped parabola opening upward.
- Negative leading coefficient (a < 0): ∩-shaped parabola opening downward.
- The vertex (turning point) is the minimum or maximum.
- The parabola is symmetric about the vertical line through its vertex.
Examples: y = x² − 4x + 3 (U-shape), y = −2x² + 3 (∩-shape, vertex at (0, 3)).
How to spot it from a sketch: one turning point, symmetric shape, opens up or down.
Cubic graphs
A cubic contains x³ as the highest power.
Key pointers:
- The curve has an S-shape with one point of inflection.
- A cubic always has an odd number of x-intercepts (1, 2, or 3), depending on the equation.
- Positive leading coefficient: rises from bottom-left to top-right.
- Negative leading coefficient: falls from top-left to bottom-right.
Examples: y = x³, y = x³ − 3x, y = −x³ + 2x².
How to spot it from a sketch: the curve enters from one direction and exits from the opposite — it changes direction twice (or appears to have a flat section at the inflection).
Reciprocal graphs
A reciprocal graph arises from y = k/x or y = k/x².
Key pointers for y = k/x:
- Two separate branches, one in each of two opposite quadrants.
- If k > 0: branches in quadrants 1 and 3 (top-right and bottom-left).
- If k < 0: branches in quadrants 2 and 4.
- Asymptotes: x = 0 (y-axis) and y = 0 (x-axis) — the curve approaches but never reaches either.
Key pointers for y = k/x²:
- Both branches are in the same two quadrants (both above or both below the x-axis).
Examples: y = 3/x (positive k, quadrants 1 and 3), y = −2/x (negative k, quadrants 2 and 4).
How to spot it from a sketch: two distinct branches with asymptotes; the curve approaches axes without touching them.
Exponential graphs
An exponential has x in the exponent (power), not as a base raised to a constant.
Key pointers:
- y = aˣ where a > 1: growth curve, rising steeply to the right, levelling out toward y = 0 to the left.
- y = aˣ where 0 < a < 1: decay curve, falling to the right, levelling out toward y = 0.
- For y = k × aˣ: the curve passes through (0, k) because a⁰ = 1.
- The x-axis is a horizontal asymptote — the curve approaches y = 0 but never reaches it.
Examples: y = 2ˣ (growth), y = (½)ˣ (decay), y = 3 × 2ˣ (growth, y-intercept at 3).
How to spot it from a sketch: smooth curve approaching an asymptote on one side, rising or falling rapidly on the other.
Trigonometric graphs
y = sin x: wave starting at (0, 0), reaching maximum 1 at x = 90°, returning to 0 at x = 180°, minimum −1 at x = 270°, back to 0 at x = 360°. Repeats every 360°.
y = cos x: same wave shape as sin x but shifted left by 90° — it starts at (0, 1), not (0, 0). Reaches 0 at x = 90°, minimum −1 at x = 180°, returns to 1 at x = 360°. Period 360°.
y = tan x: not a wave. Rises steeply, has vertical asymptotes at x = 90°, 270°, 450°, etc. (every 180°). Period 180°.
How to spot them:
- Regular wave with amplitude 1 and period 360° → sin or cos.
- If the wave passes through (0, 0): sin x. If through (0, 1): cos x.
- Repeating asymptote pattern, period 180°: tan x.
Quick identification checklist
When you see a graph or equation in an exam, ask:
- Is it a straight line? → Linear.
- Is the highest power x²? → Quadratic (parabola).
- Is the highest power x³? → Cubic (S-curve).
- Is x in the denominator? → Reciprocal.
- Is x in the exponent (power)? → Exponential.
- Does it involve sin, cos, or tan? → Trigonometric.
Frequently asked questions
Can one equation belong to more than one family?
No — each equation belongs to exactly one family based on its structure. However, be careful: y = x³ − x² + x − 1 is a cubic, not a quadratic, even though it contains x² as well. Always identify the highest power of x to determine the family.
How do I tell sin and cos apart on a sketch?
Both are smooth waves with the same shape, period (360°), and amplitude (1). The difference is where they cross the y-axis: y = sin x passes through (0, 0); y = cos x passes through (0, 1). Check the y-intercept first.
What does a negative coefficient do to a quadratic or cubic?
For a quadratic, a negative leading coefficient flips the parabola: instead of opening upward (U), it opens downward (∩). For a cubic, a negative leading coefficient reverses the S-direction: instead of rising from bottom-left to top-right, it falls from top-left to bottom-right.
Do I need to draw accurate graphs for a "recognising" question?
Usually not — "match the equation to the sketch" or "which graph shows y = 2ˣ?" only require you to identify the correct shape from a set of options. You need to know the key features (shape, asymptotes, intercepts) rather than plotting precise coordinate values.
Explore all six graph families with guided questions from Professor Pi at aitutors.me.