GCSE · Algebra

Recognising Graphs

Every type of equation draws its own kind of shape. Straight lines, U-shaped parabolas, S-shaped cubics, the two-piece reciprocal graph, exponential growth, repeating waves and circles: once you know the families, you can sketch a graph from its equation, or name an equation from its graph, in seconds.

GCSE subject content: recognise, sketch and interpret graphs of linear, quadratic, simple cubic and reciprocal (y = 1/x) functions; identify roots, intercepts and turning points of quadratics graphically; Higher: exponential functions y = kˣ, the trigonometric functions y = sin x, cos x, tan x, the equation of a circle centred at the origin and the tangent to a circle at a given point

Recap. You should already be able to plot a graph from a table of values and to read y = mx + c (see Linear graphs and Quadratic graphs). A sketch is not an accurate plot: it shows the right shape and marks the important points (where it crosses the axes, turning points) with their coordinates.

1 · The families, and quadratics in detail

Here are the graph families for GCSE. Learn the shapes first; the details come after.

A quadratic graph, y = ax² + bx + c, is a parabola. Its key features: it is symmetrical about a vertical line; it has one turning point (a minimum if a > 0, a U shape; a maximum if a < 0, an ∩ shape); it crosses the y-axis at (0, c); and its roots are where it crosses the x-axis (0, 1 or 2 of them). The turning point is halfway between the roots.

a = b = c =

Roots

y-intercept

Turning point

2 · Cubic graphs

A cubic has x³ as its highest power. The basic shape, y = x³, comes up from the bottom left, flattens out through the origin and shoots up to the top right. Most cubics have a wiggle in the middle: two turning points, and up to three roots. Written in factorised form, y = (x − p)(x − q)(x − r), the roots are p, q and r. A negative x³ flips the shape: it comes down from the top left.

a = p = q = r =

3 · Reciprocal graphs: y = k/x

In y = 1/x, y is the reciprocal of x (1 divided by x). You can't divide by 0, so there is no point at x = 0. As x gets close to 0, y gets enormous; as x gets large, y gets close to 0 but never reaches it. The graph gets closer and closer to the axes without ever touching them. A line like that is called an asymptote. Here both axes, x = 0 and y = 0, are asymptotes. Drag the point along the curve.

k =

Spot it: two separate curves, one in each of two opposite quadrants. Positive k: top right and bottom left. Negative k: top left and bottom right. Never join the two pieces across x = 0, and never let the curve touch an axis.

4 · Exponential graphs: y = kˣ Higher

In an exponential function the variable is in the power: y = 2ˣ. Each step of 1 in x multiplies y by 2 (compare a linear function, where each step adds the same amount). Because k⁰ = 1, every graph y = kˣ passes through (0, 1). For k > 1 it is exponential growth; for 0 < k < 1, decay. The x-axis is an asymptote: kˣ is always positive.

k = compare with y = x²

5 · Trigonometric graphs Higher

Take a circle of radius 1 and a point moving round it, starting at angle 0°. The point's height is sin x and its distance across is cos x (the same SOHCAHTOA ratios, with hypotenuse 1). Plot them against the angle and you get waves that repeat every 360°. tan x = sin x ÷ cos x, which blows up wherever cos x = 0 (at 90° and 270°), giving vertical asymptotes. Drag the angle, then use the line y = k to solve equations.

y = angle x = solve f(x) =

Graph	Starts at	Max / min	Crosses x-axis (0°–360°)	Repeats every
y = sin x	(0°, 0)	1 at 90°, −1 at 270°	0°, 180°, 360°	360°
y = cos x	(0°, 1)	1 at 0° and 360°, −1 at 180°	90°, 270°	360°
y = tan x	(0°, 0)	none (asymptotes at 90°, 270°)	0°, 180°, 360°	180°

6 · Circles and tangents Higher

Every point (x, y) on a circle with centre (0, 0) and radius r makes a right-angled triangle with sides x and y and hypotenuse r. By Pythagoras, x² + y² = r²: that is the circle's equation. So x² + y² = 25 is a circle of radius 5 (not 25!).

A tangent touches the circle at one point and is perpendicular (at right angles) to the radius there. Perpendicular gradients multiply to −1, so if the radius has gradient m, the tangent has gradient −1/m. Drag the point round the circle; it snaps to points with whole-number coordinates.

x² + y² =

7 · Match the equation to the graph

A graph appears; pick its equation. Look for the tell-tale feature first: straight? U or ∩? S-shaped? two separate pieces? through (0, 1)? a wave? If you get one wrong, the feedback says what your choice would look like instead.

include Higher graphs

Worked examples

Sketch y = x² − 4x − 5, marking the roots, y-intercept and turning point

Shape: positive x², so a U shape.

y-intercept: put x = 0: y = −5, so (0, −5).

Roots: x² − 4x − 5 = 0 → (x − 5)(x + 1) = 0 → x = 5 or x = −1.

Turning point: halfway between the roots, x = (−1 + 5) ÷ 2 = 2. y = 4 − 8 − 5 = −9. So (2, −9), a minimum.

Sketch y = (x + 2)(x − 1)(x − 3)

Roots: each bracket = 0: x = −2, 1, 3.

y-intercept: x = 0: (2)(−1)(−3) = 6, so (0, 6).

Shape: expanding would give +x³, so it comes up from the bottom left, crosses at −2, turns, comes down through (0, 6) and 1, turns again and goes up through 3.

Draw y = 6/x for −6 ≤ x ≤ 6

Table: x = 1, 2, 3, 6 give y = 6, 3, 2, 1. x = −1, −2, −3, −6 give y = −6, −3, −2, −1. x = 0 is impossible.

Plot and join each branch with a smooth curve: one in the top-right quadrant, one in the bottom-left.

Asymptotes: x = 0 and y = 0. Do not join the branches.

A colony of 500 bacteria doubles every hour. Write a formula and find the number after 5 hours. Higher

Each hour multiplies by 2: after t hours, N = 500 × 2ᵗ. That's an exponential graph through (0, 500).

t = 5: 2⁵ = 32, so N = 500 × 32 = 16 000.

Solve sin x = 0.5 for 0° ≤ x ≤ 360° Higher

Calculator: sin⁻¹(0.5) = 30°.

Use the graph: the sine curve is symmetrical about x = 90°, so the line y = 0.5 also crosses at 180° − 30° = 150°.

Answer: x = 30° or x = 150°. (Between 180° and 360° sin x is negative, so no more.)

Find the equation of the tangent to x² + y² = 25 at the point (3, 4) Higher

Gradient of the radius from (0, 0) to (3, 4): 4/3.

Tangent is perpendicular: gradient = −1 ÷ (4/3) = −3/4.

Through (3, 4): y − 4 = −¾(x − 3), so y = −¾x + 9/4 + 4 = −¾x + 25/4.

Tidy: multiply by 4: 4y = −3x + 25, or 3x + 4y = 25. Check (3, 4): 9 + 16 = 25 ✓

Quick check

What you've learned

Linear y = mx + c: straight line. Quadratic: a symmetrical U (or ∩ if x² is negative) with one turning point; roots where it crosses the x-axis, y-intercept at (0, c).
Cubic: S-shaped, up to three roots and two turning points; rises left to right if x³ is positive.
Reciprocal y = k/x: two separate branches with the axes as asymptotes; 1st and 3rd quadrants if k > 0, 2nd and 4th if k < 0.
Higher Exponential y = kˣ: always through (0, 1), always positive, the x-axis is an asymptote; growth if k > 1, decay if 0 < k < 1.
Higher sin and cos are waves between −1 and 1 repeating every 360°; tan repeats every 180° with asymptotes at 90° and 270°. Use symmetry to find second solutions.
Higher x² + y² = r² is a circle, centre (0, 0), radius r. The tangent at a point is perpendicular to the radius: gradient −1/m.