Graphs of Functions
2.10 Graphs of Functions – Cheat Sheet
1. Forms of Functions
| Form | Example | Notes |
|---|---|---|
| Polynomial axⁿ | y = x³ + x − 4 | n can be −2, −1, −½, 0, ½, 1, 2, 3 |
| Rational a/(xⁿ) | y = 2x + 3/x² | Vertical asymptotes where denominator = 0 |
| Exponential abˣ + c | y = (1/4) × 2ˣ | Growth if b > 1, decay if 0 < b < 1 |
2. Constructing Tables of Values
- Choose x-values (including negatives if needed).
- Substitute into equation carefully (brackets for negatives).
- Plot (x, y) points on graph.
Example: y = x² − 3x + 2
| x | y |
|---|---|
| −1 | 6 |
| 0 | 2 |
| 1 | 0 |
| 2 | 0 |
| 3 | 2 |
3. Recognising Graph Shapes
| Function | Shape |
|---|---|
| y = x² | U-shaped parabola |
| y = x³ | Cubic S-shape |
| y = 1/x | Two separate curves, asymptotes at x = 0, y = 0 |
| y = √x | Starts at (0, 0), rises slowly |
| y = 2ˣ | Exponential growth |
| y = (1/2)ˣ | Exponential decay |
4. Solving Equations Graphically
- Roots = where graph crosses x-axis (y = 0).
- Intersection of two graphs = solutions to simultaneous equations.
Example: Solve x² − 3x + 2 = 0 by graph → Roots at x = 1 and x = 2.
5. Exponential Growth and Decay Graphs
- Growth: curve rises faster as x increases (b > 1).
- Decay: curve falls towards zero as x increases (0 < b < 1).
- Common in population growth, radioactive decay, and compound interest.
Example: y = 100 × (1.05)ˣ → 5% growth per step.
6. Estimating Gradient of a Curve
- Draw tangent at the required point.
- Gradient = vertical change ÷ horizontal change.
Example: Tangent rises 4 units over 2 units run → Gradient = 2.
7. Examples Table for Quick Reference
| Problem | Process | Answer |
|---|---|---|
| Table for y = x² | Sub x = −2 to 2 | y-values: 4, 1, 0, 1, 4 |
| Roots of y = x² − 3x + 2 | y = 0 points | x = 1, x = 2 |
| Shape of y = 1/x² | Reciprocal squared | Two positive U-shaped branches |
| Growth: y = 2ˣ | Exponential | Doubles each step |
| Decay: y = (1/3)ˣ | Exponential | Divides by 3 each step |