Further Graphs & Tangents

Revise Further Graphs & Tangents for Mathematics 0580 (O Level) — revision notes and instant AI marking.

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Types of Graphs
Drawing Graphs from Tables
Solving Equations from Graphs
Finding Gradients of Tangents
Summary & Review
Concepts Checklist
What to Memorize
Exam Tips

Types of Graphs

The Five Main Graph Types You Must Know

For IGCSE Extended Maths, you need to recognize, sketch, and interpret five fundamental graph types. Each has a distinctive shape and specific properties.

1. Linear Graphs

y = mx + c

Shape: Straight line
Key Features: Constant gradient m, y-intercept at c
Examples: y = 2x + 3, y = −x + 5
What to Look For: Positive m slopes uphill, negative m slopes downhill

2. Quadratic Graphs

y = ax² + bx + c

Shape: Parabola (U-shaped or ∩-shaped)
Key Features: Has a vertex (turning point), vertical line of symmetry
Positive a: Opens upward (U-shape, minimum)
Negative a: Opens downward (∩-shape, maximum)
Examples: y = x², y = −2x² + 4x − 1
Symmetry: Mirror image on either side of the vertical line x = −b/2a

3. Cubic Graphs

y = ax³ + bx² + cx + d

Shape: S-shaped curve with up to 2 turning points
Positive a: Starts bottom-left, ends top-right
Negative a: Starts top-left, ends bottom-right
Examples: y = x³, y = −x³ + 2x
Key Feature: Can have local maximum and minimum before the general trend

4. Reciprocal Graphs

y = a/x + b

Shape: Two separate branches (hyperbola)
Asymptotes: Vertical at x = 0, horizontal at y = b
Key Rule: NEVER crosses either asymptote
No y-intercept: Can't substitute x = 0 (division by zero)
Positive a: Branches in quadrants 1 and 3
Negative a: Branches in quadrants 2 and 4
Special Note: y = 1/x² is similar but always positive

5. Exponential Graphs

y = ak^x + b

Growth (k > 1): Starts slow, then curves upward rapidly
Decay (0 < k < 1): Starts high, then curves downward, leveling off
Horizontal Asymptote: y = b (the graph approaches but never reaches this line)
y-intercept: (0, a + b)
Examples: y = 2^x, y = 400 × (1/2)^x + 100
Real-World Use: Population growth, radioactive decay, compound interest

Understanding Asymptotes

An asymptote is a line that a graph gets closer and closer to, but never actually touches or crosses.

Vertical asymptote: A value of x that makes the function undefined. For reciprocal graphs like y = a/x, the vertical asymptote is at x = 0.

Horizontal asymptote: The value that y approaches as x becomes very large or very negative. For y = a/x, it's y = 0. For y = ak^x + b, it's y = b.

Try This

Which graph type has equation y = 3^x − 2? What is its horizontal asymptote?

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Exam Tips & Common Mistakes
Recognizing Graphs from Shapes
The Step-by-Step Process
Worked Example: Creating and Using a Table
Special Cases to Watch For
Recognizing Points That Don't Fit
Finding Intersection Points
Solving Simultaneous Equations Graphically

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Further Graphs & Tangents