Graphs of Functions (Copy)
Graphs of Functions Cheat Sheet (IGCSE Mathematics 0580 – CORE)
Topic: Linear, Quadratic, Reciprocal Graphs & Graphical Solutions
1. Graph Types and Their Forms
| Graph Type | General Form | Description |
|---|---|---|
| Linear | y = ax + b | Straight line, constant gradient |
| Quadratic | y = ±x² + ax + b | Parabola, curved (U or ∩ shape) |
| Reciprocal | y = x⁻¹ or y = a/x | Curved graph, never touches axes |
2. Constructing Tables of Values
- Choose values for x (usually –3 to 3)
- Substitute into the equation to find y
- Use values to plot points (x, y) on graph
Example – Linear:
Equation: y = 2x + 1
x: –2, –1, 0, 1, 2
y: –3, –1, 1, 3, 5
Example – Quadratic:
Equation: y = x² – 2
x: –2, –1, 0, 1, 2
y: 2, –1, –2, –1, 2
3. Recognising Graph Shapes
- y = ax + b → Straight line
- Gradient = a, y-intercept = b
- Positive gradient: goes up
- Negative gradient: goes down
- y = x² + ax + b → Parabola
- If coefficient of x² is positive → U shape
- If negative → ∩ shape
- Symmetrical about the vertex (turning point)
- y = 1/x → Reciprocal
- Two branches in opposite quadrants
- Never touches x-axis or y-axis (asymptotes)
4. Solving Equations Graphically
- Find x where the graph crosses the x-axis
→ These are the roots (solutions to y = 0) - To solve equations graphically, plot both functions and find points of intersection.
Example:
To solve: x² – 3x + 2 = x + 1
- Plot y = x² – 3x + 2
- Plot y = x + 1
- Find x-values where the graphs intersect → these are the solutions
5. Interpreting Graphs
- x-intercepts: where graph crosses x-axis → roots
- y-intercept: where graph crosses y-axis
- Turning point (for quadratics): minimum or maximum
- Gradient of straight line: Rate of change
- Asymptotes (for reciprocal): lines graph gets close to but never touches
6. Important Forms and Graph Behaviour
| Equation | Type | Graph Shape / Notes |
|---|---|---|
| y = 2x – 4 | Linear | Straight line, gradient = 2, crosses y at –4 |
| y = x² – 4 | Quadratic | U-shape, crosses x-axis at ±2 |
| y = –x² + 1 | Quadratic | ∩-shape, vertex at (0,1) |
| y = 1/x | Reciprocal | Two branches, undefined at x = 0 |
| y = –1/x | Reciprocal | Reflection in x-axis |
7. Common Questions in Exams
- Complete a table of values for given x-values
- Draw a graph using plotted points
- Identify roots from graph (where y = 0)
- Estimate value of x or y from a point on the curve
- Find intersection between two graphs (solve equations)
Tip: In the exam, if given a graph and asked to “solve”, look for x where y = 0 (roots), or where two graphs intersect. Always label your axes and plot neatly using a ruler for straight lines.