At GCSE you need to recognise, sketch and interpret six types of function graph. Each has a characteristic shape you must learn.
Linear
$y = mx + c$
Straight line — constant gradient
Quadratic
$y = ax^2 + bx + c$
Parabola — U or ∩ shape
Cubic
$y = ax^3 + \ldots$
S-shaped curve
Reciprocal
$y = \dfrac{k}{x}$
Two-branch hyperbola
Exponential
$y = k^x$, $k > 0$
Rapid growth or decay
Trigonometric
$y = \sin x$, $\cos x$, $\tan x$
Wave / periodic
Three linear functions
Key features
- Positive $m$ → slopes upward (↗)
- Negative $m$ → slopes downward (↘)
- $m = 0$ → horizontal line $y = c$
- $|m|$ larger → steeper slope
- y-intercept always at $(0,\ c)$
- x-intercept (root): set $y=0$, solve for $x$
$y = x^2 - 2x - 3$ with key features labelled
Key features
- Roots: where curve meets x-axis ($y=0$)
- y-intercept: the constant $c$ at $(0,c)$
- Vertex: highest/lowest point; x-coord $= -\tfrac{b}{2a}$
- Axis of symmetry: $x = -\tfrac{b}{2a}$
- 0, 1 or 2 real roots possible
$y = x^3$ — the basic cubic S-curve
Key features of $y = x^3$
- Passes through the origin $(0, 0)$
- S-shaped: rises steeply for large $|x|$
- As $x \to +\infty$, $y \to +\infty$
- As $x \to -\infty$, $y \to -\infty$
- Rotational symmetry about origin
Variations
| Equation | Effect |
|---|---|
| $y = x^3 + c$ | Shift up/down by $c$ |
| $y = -x^3$ | Reflects in x-axis |
| $y = 2x^3$ | Steeper S-curve |
| $y = (x-a)^3$ | Shift right by $a$ |
| $x$ | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| $x^3$ | -27 | -8 | -1 | 0 | 1 | 8 | 27 |
| $-4x$ | 12 | 8 | 4 | 0 | -4 | -8 | -12 |
| $y$ | -15 | 0 | 3 | 0 | -3 | 0 | 15 |
$y = \frac{1}{x}$ — two branches, axes are asymptotes
Key features
- Two branches — one for $x>0$, one for $x<0$
- Asymptotes: $x = 0$ (y-axis) and $y = 0$ (x-axis)
- Never crosses either axis
- Rotational symmetry of order 2 about origin
- If $k>0$: branches in Q1 and Q3
- If $k<0$: branches in Q2 and Q4
$y=2^x$ (growth) and $y=\left(\tfrac{1}{2}\right)^x$ (decay)
Key features
- Always passes through $(0,\ 1)$ since $k^0 = 1$
- Horizontal asymptote: $y = 0$ (x-axis)
- Never negative (always $y > 0$)
- $k > 1$: exponential growth — rises steeply to the right
- $0 < k < 1$: exponential decay — falls toward zero
- Passes through $(1,\ k)$ always
| $x$ | $y=2^x$ | $y=3^x$ |
|---|---|---|
| $-2$ | $0.25$ | $0.11$ |
| $0$ | $1$ | $1$ |
| $2$ | $4$ | $9$ |
| $3$ | $8$ | $27$ |
Sine and Cosine — $y = \sin x$ and $y = \cos x$
$y = \sin x$ and $y = \cos x$ for $0° \leq x \leq 360°$
$y = \sin x$
• Starts at $(0°, 0)$
• Maximum $1$ at $90°$
• Crosses zero at $0°$, $180°$, $360°$
• Minimum $-1$ at $270°$
• Period: $360°$
$y = \cos x$
• Starts at $(0°, 1)$
• Maximum $1$ at $0°$ and $360°$
• Crosses zero at $90°$ and $270°$
• Minimum $-1$ at $180°$
• Period: $360°$
• Same shape as sin, shifted $90°$ left
Key values
| Angle | sin | cos |
|---|---|---|
| $0°$ | $0$ | $1$ |
| $30°$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ |
| $45°$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ |
| $60°$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ |
| $90°$ | $1$ | $0$ |
Tangent — $y = \tan x$
$y = \tan x$ for $0° \leq x \leq 360°$ — asymptotes at $90°$ and $270°$
$y = \tan x$ — key facts
• $\tan x = \dfrac{\sin x}{\cos x}$
• Period: $180°$
• Undefined at $90°$, $270°$, …
• Passes through $(0°,0)$, $(180°,0)$, $(360°,0)$
• $\tan 45° = 1$, $\tan 135° = -1$
Using graphs to find angles
If $\sin \theta = 0.5$, the graph of $y = \sin x$ shows this at $\theta = 30°$ and $\theta = 150°$ in $0°$–$360°$. Always check for a second solution using the graph's symmetry.
Symmetry rules
$\sin(180° - x) = \sin x$
$\cos(360° - x) = \cos x$
$\tan(180° + x) = \tan x$
These help you find all solutions in $0°$–$360°$.
| Function | Equation | Shape | Key features |
|---|---|---|---|
| Linear | $y = mx + c$ | Straight line | Gradient $m$; y-intercept $c$ |
| Quadratic | $y = ax^2 + bx + c$ | U or ∩ (parabola) | Roots, vertex, axis of symmetry |
| Cubic | $y = ax^3 + \ldots$ | S-curve | Up to 3 roots; $a>0$ rises right |
| Reciprocal | $y = k/x$ | Two-branch hyperbola | Asymptotes $x=0$, $y=0$; never crosses axes |
| Exponential | $y = k^x$, $k>0$ | Steep curve | Passes through $(0,1)$; asymptote $y=0$ |
| Sine | $y = \sin x$ | Wave | $(0,0)$; max $1$ at $90°$; period $360°$ |
| Cosine | $y = \cos x$ | Wave (sin shifted) | $(0,1)$; max $1$ at $0°$, $360°$; period $360°$ |
| Tangent | $y = \tan x$ | Repeating branches | Asymptotes at $90°$, $270°$; period $180°$ |
State the type of function for each of the following:
(a) $y = 5x - 3$ (b) $y = \dfrac{4}{x}$ (c) $y = x^3 - 2x$ (d) $y = 3^x$ (e) $y = x^2 - 4x + 1$
▶ Show solution
(a) $y = 5x - 3$ — Linear (highest power of $x$ is 1, in the form $y=mx+c$)
(b) $y = \frac{4}{x}$ — Reciprocal (form $y = k/x$)
(c) $y = x^3 - 2x$ — Cubic (highest power is 3)
(d) $y = 3^x$ — Exponential (variable is in the exponent, $k=3>0$)
(e) $y = x^2 - 4x + 1$ — Quadratic (highest power is 2)
For the line $y = -4x + 6$:
(a) State the gradient and y-intercept. (b) Find the x-intercept. (c) Sketch the line.
▶ Show solution
(a) Gradient $m = -4$; y-intercept $c = 6$, so the line crosses the y-axis at $(0, 6)$.
(b) Set $y = 0$: $-4x + 6 = 0 \Rightarrow x = \frac{6}{4} = 1.5$. x-intercept: $(1.5,\ 0)$.
(c) Sketch: plot $(0,6)$ and $(1.5,0)$; draw a straight line with negative gradient (slopes downward to the right).
The graph of $y = x^2 - 4x + 3$ is a parabola.
(a) Find the roots. (b) Find the turning point. (c) Write the y-intercept. (d) State the axis of symmetry.
▶ Show solution
(a) $x^2 - 4x + 3 = 0 \Rightarrow (x-1)(x-3) = 0 \Rightarrow x = 1$ or $x = 3$. Roots at $(1,0)$ and $(3,0)$.
(b) Axis: $x = -\frac{-4}{2} = 2$. Sub: $y = 4-8+3 = -1$. Minimum at $(2, -1)$.
(c) y-intercept: $(0, 3)$ (the constant term).
(d) Axis of symmetry: $x = 2$.
Complete the table for $y = x^3 + x^2 - 2x$ and state the roots of the equation $x^3 + x^2 - 2x = 0$.
| $x$ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $2$ |
|---|---|---|---|---|---|---|
| $y$ | ? | ? | ? | ? | ? | ? |
▶ Show solution
Substitute each $x$ into $y = x^3 + x^2 - 2x$:
| $x$ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $2$ |
|---|---|---|---|---|---|---|
| $y$ | $-12$ | $0$ | $2$ | $0$ | $0$ | $8$ |
Roots (where $y=0$): $x = -2,\ x = 0,\ x = 1$. (We can verify by factorising: $x(x^2+x-2) = x(x+2)(x-1) = 0$.)
(a) For $y = \dfrac{6}{x}$, calculate the values of $y$ when $x = -3,\ -1,\ 1,\ 2,\ 6$.
(b) State the equations of the asymptotes. (c) Describe the shape of the graph.
▶ Show solution
(a)
| $x$ | $-3$ | $-1$ | $1$ | $2$ | $6$ |
|---|---|---|---|---|---|
| $y = 6/x$ | $-2$ | $-6$ | $6$ | $3$ | $1$ |
(b) Asymptotes: $x = 0$ (y-axis) and $y = 0$ (x-axis).
(c) Two-branch hyperbola: one branch in the first quadrant ($x>0, y>0$) and one in the third quadrant ($x<0, y<0$). The curve approaches but never reaches either axis.
The population of bacteria follows $P = 2^t$ where $t$ is time in hours.
(a) What is the population at $t = 0$? (b) How long does it take to reach a population of 32? (c) Sketch the graph for $0 \leq t \leq 5$.
▶ Show solution
(a) $P = 2^0 = 1$. Population starts at 1.
(b) $2^t = 32 = 2^5$, so $t = 5$ hours.
(c) Plot points: $(0,1), (1,2), (2,4), (3,8), (4,16), (5,32)$. Draw a smooth increasing curve. The curve passes through $(0,1)$ and rises steeply. There is a horizontal asymptote at $P=0$.
(a) Write down the value of $\sin 270°$. (b) Solve $\sin \theta = \dfrac{\sqrt{3}}{2}$ for $0° \leq \theta \leq 360°$. (c) State the period of $y = \sin x$.
▶ Show solution
(a) From the sine graph, the minimum is $-1$ at $270°$: $\sin 270° = -1$
(b) $\sin^{-1}\!\left(\frac{\sqrt{3}}{2}\right) = 60°$ (first quadrant solution). Since $\sin$ is positive in Q1 and Q2: second solution $= 180° - 60° = 120°$.
$\boldsymbol{\theta = 60°}$ and $\boldsymbol{\theta = 120°}$
(c) Period of $y = \sin x$ is $\mathbf{360°}$.
Solve $\cos \theta = -\dfrac{\sqrt{2}}{2}$ for $0° \leq \theta \leq 360°$.
▶ Show solution
$\cos^{-1}\!\left(\frac{\sqrt{2}}{2}\right) = 45°$ — this is the reference angle.
Cosine is negative in the second and third quadrants:
Second quadrant: $\theta = 180° - 45° = 135°$
Third quadrant: $\theta = 180° + 45° = 225°$
$\theta = 135°$ and $\theta = 225°$
Match each description to one of the equations below:
Equations: A: $y = \sin x$ B: $y = x^3$ C: $y = \frac{1}{x}$ D: $y = 2^x$ E: $y = x^2 - 4$
(i) A U-shaped curve with roots at $x = \pm 2$ and minimum at $(0, -4)$.
(ii) A curve passing through $(0,1)$ with a horizontal asymptote at $y=0$ that increases steeply for $x>0$.
(iii) A curve with rotational symmetry about the origin, two branches that approach both axes but never cross them.
(iv) A curve that passes through the origin, with $y \to +\infty$ as $x \to +\infty$ and $y \to -\infty$ as $x \to -\infty$, S-shaped.
(v) A wave with period $360°$, value $0$ at $0°$ and maximum $1$ at $90°$.
▶ Show solution
(i) → E: $y = x^2 - 4$ — parabola, roots at $\pm 2$, vertex at $(0,-4)$
(ii) → D: $y = 2^x$ — exponential, passes through $(0,1)$, asymptote $y=0$
(iii) → C: $y = \frac{1}{x}$ — reciprocal, two-branch hyperbola
(iv) → B: $y = x^3$ — cubic S-curve through origin
(v) → A: $y = \sin x$ — sine wave, period $360°$, starts at $(0,0)$
A graph shows the curve $y = 4 \times 2^x$.
(a) What value does $y$ take when $x = 0$? (b) Write down the equation of the asymptote. (c) Is the curve increasing or decreasing? (d) Find $x$ when $y = 32$.
▶ Show solution
(a) $y = 4 \times 2^0 = 4 \times 1 = 4$. The y-intercept is $(0, 4)$.
(b) As $x \to -\infty$, $2^x \to 0$, so $y \to 0$. Asymptote: $\mathbf{y = 0}$.
(c) Since the base $2 > 1$, the function is increasing (exponential growth).
(d) $4 \times 2^x = 32 \Rightarrow 2^x = 8 = 2^3 \Rightarrow \mathbf{x = 3}$.