Reciprocal Graphs - Sketching (AQA A-Level Mathematics): Revision Notes

Identify the Basic Form:

• The most basic reciprocal graph is $y = \frac{1}{x}$, which can be modified by changing the value of $\ k$ or by adding transformations. Determine Asymptotes:

• Vertical Asymptote: The graph has a vertical asymptote where the denominator is zero. For $\ y = \frac{k}{x}$ , this is at $\ x = 0$. For$\ y = \frac{k}{\ a \ x + b} ,$ set $\ axe + b = 0$ to find the vertical asymptote.

• Horizontal Asymptote: The graph has a horizontal asymptote as $\ x$ approaches infinity or negative infinity. For $\ y = \frac{k}{x}$, this is $\ y = 0$. Similarly, for $\ y = \frac{k}{ax + b}$, the horizontal asymptote is $\ y = 0$. Identify Key Points:

• Consider specific values of $\ x$ to determine key points on the graph. For example:

• $\ ( x = 1 ): \ y = \frac{k}{1} = k$

• $\ ( x = -1 ): \ y = \frac{k}{-1} = -k$

• These points help establish the general shape of the graph. Consider Symmetry:

• The graph of $\ y = \frac{k}{x}$is symmetric about the origin (rotational symmetry of $180$ degrees).

• If $\ k > 0$, the graph lies in the first and third quadrants. If $\ k < 0$, it lies in the second and fourth quadrants. Plot the Graph:

• Plot the vertical and horizontal asymptotes.

• Plot key points, such as where $\ x = 1, -1$ and others as necessary.

• Sketch the graph, ensuring it approaches the asymptotes but never touches them.

Example 1: Basic Reciprocal Graph

Graph $\ y = \frac{2}{x} :$

1. Asymptotes:

• Vertical asymptote at $\ x = 0$.
• Horizontal asymptote at $\ y = 0 .$

2. Key Points:

• At $\ ( x = 1 ), \ ( y = 2 )$.
• At $\ ( x = -1 ), \ ( y = -2 ).$

3. Symmetry:

• The graph is symmetric about the origin.

4. Sketch:

• The graph passes through$\ (1, 2)$ and $\ (-1, -2)$, and it approaches the asymptotes.

Example 2: Transformed Reciprocal Graph

Graph $\ y = \frac{3}{x - 2}$:

5. Asymptotes:

• Vertical asymptote where $\ ( x - 2 = 0 ), so \ ( x = 2 ).$
• Horizontal asymptote at$\ y = 0 .$

6. Key Points:

• At $\ ( x = 3 ), \ y = \frac{3}{1} = 3 .$
• At $\ ( x = 1 ), \ y = \frac{3}{-1} = -3 .$

7. Sketch:

• The graph passes through $\ (3, 3) \ and \ (1, -3) .$
• It approaches the vertical asymptote $\ x = 2$ and the horizontal asymptote $\ y = 0 .$

• Asymptotes: Identify where the graph will have vertical and horizontal asymptotes.
• Key Points: Calculate values for key points on the graph, such as where $\ x = 1, -1 .$
• Symmetry: Consider the symmetry of the graph to guide your sketching.
• Plot and Sketch: Combine the asymptotes, key points, and symmetry to sketch the graph.

By following these steps, you can accurately sketch reciprocal graphs, which are crucial for understanding inverse relationships and behaviours in mathematics.