EDEXCEL GCSE FOUNDATION MATHS - What are Reciprocal Graphs

Edexcel GCSE Foundation Maths: What are Reciprocal Graphs?

Introduction

A reciprocal graph is a graph that represents the relationship between a variable and its reciprocal. The reciprocal of a number is simply 1 divided by that number. For example, the reciprocal of 2 is 1/2, and the reciprocal of 5 is 1/5.

Reciprocal graphs are typically shaped like a hyperbola, with two branches that extend infinitely in opposite directions.

The Equation of a Reciprocal Graph

The general equation for a reciprocal graph is:

y = k/x

Where:

• y is the dependent variable (typically plotted on the vertical axis)
• x is the independent variable (typically plotted on the horizontal axis)
• k is a constant that determines the shape and position of the graph

Key Features of Reciprocal Graphs

• Asymptotes: Reciprocal graphs have two asymptotes:
  • Vertical Asymptote: The graph approaches a vertical line as the value of x approaches zero. The equation of this asymptote is x = 0.
  • Horizontal Asymptote: The graph approaches a horizontal line as the value of x approaches positive or negative infinity. The equation of this asymptote is y = 0.
• Symmetry: The graph is symmetrical about the origin (0, 0). This means that if you reflect the graph across the x-axis and then again across the y-axis, it will return to its original position.
• Shape: The graph has two branches that extend infinitely in opposite directions, approaching the asymptotes but never actually touching them.

Examples of Reciprocal Graphs

• y = 1/x: This is the simplest reciprocal graph. It has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. The graph is symmetrical about the origin.

• y = 2/x: This graph has the same shape as y = 1/x, but it is stretched vertically by a factor of 2. The asymptotes remain the same.

• y = -1/x: This graph is reflected across the x-axis compared to y = 1/x. It has the same asymptotes but is flipped.

Drawing Reciprocal Graphs

To draw a reciprocal graph, follow these steps:

1. Identify the equation: Write down the equation of the reciprocal graph you want to draw.
2. Find the asymptotes: Determine the vertical and horizontal asymptotes of the graph.
3. Plot some points: Choose a few values of x and calculate the corresponding values of y.
4. Draw the graph: Connect the points you plotted, making sure the graph approaches the asymptotes.

Applications of Reciprocal Graphs

Reciprocal graphs have applications in various fields, including:

• Physics: Describing the relationship between the force of gravity and the distance between two objects.
• Economics: Representing supply and demand curves.
• Chemistry: Illustrating the relationship between the concentration of a solution and its volume.

Practice Problems

1. Draw the graph of y = 3/x.
2. What are the asymptotes of the graph y = -2/x?
3. Explain how the graph of y = 1/x changes when the constant k in the equation y = k/x is changed.

Conclusion

Understanding reciprocal graphs is essential for success in Edexcel GCSE Foundation Maths. By mastering the concepts of asymptotes, symmetry, and the general equation, you will be able to draw and interpret reciprocal graphs effectively.