EDEXCEL GCSE HIGHER MATHS - What are Graphs (Linear, Quadratic, and Cubic)

Edexcel GCSE Higher Maths: Graphs (Linear, Quadratic, and Cubic)

1. Introduction to Graphs

Graphs are powerful visual tools used in mathematics to represent relationships between variables. In this tutorial, we'll explore three fundamental types of graphs: linear, quadratic, and cubic.

2. Linear Graphs

a) Definition:

A linear graph represents a relationship where the change in one variable is directly proportional to the change in the other. This means the graph is a straight line.

b) Equation:

Linear equations are written in the form y = mx + c, where:

• y represents the dependent variable (usually plotted on the vertical axis)
• x represents the independent variable (usually plotted on the horizontal axis)
• m is the slope of the line (representing the rate of change)
• c is the y-intercept (the point where the line crosses the y-axis)

c) Examples:

• y = 2x + 1: This line has a slope of 2 and a y-intercept of 1.
• y = -3x + 5: This line has a slope of -3 and a y-intercept of 5.

d) Key Features:

• Constant Slope: The line has the same steepness throughout.
• Straight Line: The graph is a straight line.
• One y-intercept: The line crosses the y-axis at one point.

3. Quadratic Graphs

a) Definition:

A quadratic graph represents a relationship where one variable is related to the square of the other. This results in a curved shape called a parabola.

b) Equation:

Quadratic equations are written in the form y = ax² + bx + c, where:

• a, b, and c are constants.

c) Examples:

• y = x² + 2x + 1: This parabola opens upwards and has a minimum point.
• y = -x² + 4x - 3: This parabola opens downwards and has a maximum point.

d) Key Features:

• Parabola Shape: The graph forms a U-shaped curve (either opening upwards or downwards).
• Turning Point: The parabola has a minimum or maximum point called the turning point.
• Symmetry: The parabola is symmetrical about a vertical line passing through the turning point.

4. Cubic Graphs

a) Definition:

A cubic graph represents a relationship where one variable is related to the cube of the other. These graphs are more complex than linear or quadratic graphs.

b) Equation:

Cubic equations are written in the form y = ax³ + bx² + cx + d, where:

• a, b, c, and d are constants.

c) Examples:

• y = x³ + 2x² - x - 2: This cubic graph has a local maximum and minimum point.
• y = -x³ + 3x² - 2x + 1: This cubic graph has a local maximum point and a point of inflection.

d) Key Features:

• S-Shape: The graph typically has an 'S' shape with one or more turning points.
• Points of Inflection: The graph can have points of inflection, where the curvature changes.
• Asymptotes: Some cubic graphs have asymptotes, lines that the graph approaches but never touches.

5. Summary

Understanding the fundamental characteristics of linear, quadratic, and cubic graphs is essential for success in GCSE Higher Maths. Remember to pay attention to their shapes, key features, and how their equations relate to their graphical representation.