EDEXCEL GCSE HIGHER MATHS - What are Reciprocal, Exponential, and Logarithmic Graphs

Edexcel GCSE Higher Maths: Reciprocal, Exponential, and Logarithmic Graphs

1. Reciprocal Graphs

Definition: A reciprocal graph is a graph of the function y = 1/x.

Key Features:

• Asymptotes: The graph has two asymptotes:
  • Vertical Asymptote: x = 0 (the y-axis)
  • Horizontal Asymptote: y = 0 (the x-axis)
• Shape: The graph has two separate branches:
  • One in the first quadrant (where both x and y are positive)
  • Another in the third quadrant (where both x and y are negative)
• Symmetry: The graph is symmetrical about the line y = x.

Example:

y = 1/x 

How to sketch:

1. Plot the asymptotes.
2. Plot a few points on either side of the vertical asymptote.
3. Connect the points smoothly, ensuring the graph approaches the asymptotes.

2. Exponential Graphs

Definition: An exponential graph is a graph of the function y = a^x, where a is a positive constant (base) and x is the variable.

Key Features:

• Growth/Decay: The graph shows exponential growth if a > 1 and exponential decay if 0 < a < 1.
• Asymptote: There is a horizontal asymptote at y = 0 (the x-axis)
• y-intercept: The graph always intersects the y-axis at (0, 1)

Example:

y = 2^x 

How to sketch:

1. Plot the y-intercept (0, 1)
2. Plot a few more points by substituting values for x.
3. Connect the points smoothly, ensuring the graph approaches the asymptote.

3. Logarithmic Graphs

Definition: A logarithmic graph is the graph of the function y = log_a(x), where a is a positive constant (base) and x is the variable.

Key Features:

• Domain: The domain of the function is x > 0, as the logarithm is only defined for positive values.
• Asymptote: There is a vertical asymptote at x = 0 (the y-axis)
• x-intercept: The graph always intersects the x-axis at (1, 0)
• Relationship with Exponential Graphs: The logarithmic graph is the reflection of the corresponding exponential graph across the line y = x.

Example:

y = log_2(x)

How to sketch:

1. Plot the x-intercept (1, 0)
2. Plot a few more points by substituting values for x.
3. Connect the points smoothly, ensuring the graph approaches the asymptote.

Important Notes:

• The base a of both exponential and logarithmic functions affects the steepness of the graph. A larger base results in a steeper graph.
• You can use transformations (translations, stretches, reflections) to modify the basic graphs of reciprocal, exponential, and logarithmic functions.
• Understanding the relationship between these different types of graphs is crucial for solving equations and inequalities involving them.