Mathematics 4024 ยท Syllabus tracker
Graphs Of Functions
Tables of values and graphs of linear, quadratic, cubic, reciprocal and exponential functions, their shapes and key points.
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Quick revision
Graphs Of Functions on one sheet
For the night before. The full notes teach the topic; this is just what you must remember.
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 โ 4 | โช if x2 term positive, โฉ if negative; symmetrical |
| Cubic | y = x3 โ 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 โ x2, and where is its turning point?
- Quadratic with a negative x2 term: โฉ-shaped.
- At x = 0, y = 5.
โฉ-shaped parabola, maximum at (0, 5)
2Tables of values and plotting
- Use brackets for negatives: x = โ2 in x2 โ 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 โ 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 = โ2.
- x = 4: 3 + 4 = 7.
- x = โ2: โ6 + (โ2) = โ8.
7 and โ8
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 โ p)2 + q โ (p, q).
- Cubic in factorised form y = (x โ a)(x โ b)(x โ c) crosses the x-axis at a, b and c.
Worked example
For y = x2 โ 4x โ 5, find the intercepts and the turning point.
- x = 0: y = โ5.
- x2 โ 4x โ 5 = (x โ 5)(x + 1) = 0 โ x = 5 or โ1.
- Halfway: x = 2, y = 4 โ 8 โ 5 = โ9.
(0, โ5); (5, 0) and (โ1, 0); minimum (2, โ9)
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 โ 3x crosses the x-axis three times. Where?
- x(x2 โ 3) = 0.
- x = 0 or x = ยฑโ3 โ ยฑ1.73.
x = โ1.7, 0 and 1.7
โCommon mistakes
| Donโt write | Write instead |
|---|---|
| (โ2)2 entered as โ22 = โ4 | Use brackets: (โ2)2 = 4 |
| Joining points with straight lines | Draw one smooth curve |
| Joining the two branches of y = k/x | They never meet โ break the curve at x = 0 |
| Putting the turning point of (x โ 3)2 + 4 at (โ3, 4) | It is at (3, 4) |
โ 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.