Quick revision notes · Mathematics 0580 · Chapter 24 of 37
Graphs Of Functions
Linear, quadratic, cubic, reciprocal and exponential graphs: shapes, tables, plotting and key features.
4 sectionsWorked examples
Common mistakes
At a glance
Linear, quadratic, cubic, reciprocal and exponential graphs: shapes, tables, plotting and key features.
1Shapes of graphs
- Recognise each family from its equation and sketch its shape.
| Type | Example | Shape and features |
|---|---|---|
| Linear | y = 2x + 1 | straight line |
| Quadratic | y = x2 − 4 | ∪ if x2 term positive, ∩ if negative; symmetrical |
| Cubic | y = x3 − 3x | S-shaped; up to two turning points |
| Reciprocal | y = 2/x | two branches; asymptotes x = 0 and y = 0 |
| Exponential | y = 2x | through (0, 1); asymptote y = 0; grows fast |
Worked example
Which type is y = 5 − x2, and where is its turning point?
- Quadratic with a negative x2 term: ∩-shaped.
- At x = 0, y = 5.
∩-shaped parabola, maximum at (0, 5)
2Tables of values and plotting
- Use brackets for negatives: x = −2 in x2 − 3x + 1 gives 4 + 6 + 1 = 11.
- Use the calculator's TABLE mode to check every value.
- Plot with small crosses, then draw one smooth curve through all points — no straight segments, no joining across an asymptote.
How to do it
- Complete the table (check with TABLE mode).
- Plot each point accurately.
- Draw a smooth curve; extend only as far as the table.
Worked example
For y = 12/x + x, find y when x = 4 and x = −2.
- x = 4: 3 + 4 = 7.
- x = −2: −6 + (−2) = −8.
7 and −8
3Key features
- y-intercept: put x = 0. Roots (x-intercepts): solve y = 0.
- Turning point of a quadratic: halfway between the roots, or from completed-square form (x − p)2 + q → (p, q).
- Cubic in factorised form y = (x − a)(x − b)(x − c) crosses the x-axis at a, b and c.
Worked example
For y = x2 − 4x − 5, find the intercepts and the turning point.
- x = 0: y = −5.
- x2 − 4x − 5 = (x − 5)(x + 1) = 0 → x = 5 or −1.
- Halfway: x = 2, y = 4 − 8 − 5 = −9.
(0, −5); (5, 0) and (−1, 0); minimum (2, −9)
4Reading from a graph
- Read solutions of 'curve = 0' where the curve crosses the x-axis.
- Read a y value: go up from x to the curve, then across.
- Give values to the accuracy of the grid (often to the nearest half small square).
Worked example
The graph of y = x3 − 3x crosses the x-axis three times. Where?
- x(x2 − 3) = 0.
- x = 0 or x = ±√3 ≈ ±1.73.
x = −1.7, 0 and 1.7
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| (−2)2 entered as −22 = −4 | Use brackets: (−2)2 = 4 |
| Joining points with straight lines | Draw one smooth curve |
| Joining the two branches of y = k/x | They never meet — break the curve at x = 0 |
| Putting the turning point of (x − 3)2 + 4 at (−3, 4) | It is at (3, 4) |
★In the exam
- Draw curves in pencil with one confident stroke.
- Use the symmetry of a quadratic to check your table.
- Label the curve with its equation when there is more than one on the grid.