4.3 Reciprocal Graphs
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Learning Objectives
- Understand the shape and properties of reciprocal functions
- Sketch basic reciprocal graphs like y = 1/x and y = 1/x²
- Identify vertical and horizontal asymptotes
- Recognise the effect of transformations on reciprocal graphs
- Solve equations and inequalities involving reciprocal functions
Key Concepts
What Is a Reciprocal Graph?
A **reciprocal function** has the variable in the denominator, such as _y = 1/x_ or _y = 1/x²_.
These functions are undefined when the denominator is 0, which creates a **vertical asymptote** at x = 0.
As x gets larger or smaller, y approaches 0, forming a **horizontal asymptote** at y = 0.
Graph of y = 1/x
This graph has two branches, one in the first quadrant and one in the third quadrant.
As x → 0⁺, y → +∞, and as x → 0⁻, y → −∞.
The graph has asymptotes at x = 0 and y = 0, and it never touches either axis.
Example 1
Sketch the graph of y = 1/x
Solution:
The graph has vertical asymptote at x = 0 and horizontal asymptote at y = 0. Two curved branches in quadrants I and III.
Graph of y = 1/x²
This graph has a U-like shape but never touches the x-axis or y-axis.
Both branches are in the first and second quadrants because the output is always positive.
As x → ±∞, y → 0, and as x → 0, y → +∞.
Example 1
What are the asymptotes of y = 1/x²?
Solution:
Vertical asymptote: x = 0. Horizontal asymptote: y = 0. The graph stays above the x-axis.
Transformations of Reciprocal Graphs
Transformations apply in the usual way:
- _y = 1/(x − a)_ shifts the graph **right** by _a_.
- _y = 1/x + b_ shifts the graph **up** by _b_.
- _y = −1/x_ reflects it in the **x-axis**.
- _y = 1/(−x)_ reflects it in the **y-axis**.
Example 1
Describe the transformation from y = 1/x to y = −1/(x − 2) + 3
Solution:
Reflect in x-axis, shift right 2 units, then up 3 units.
Solving Equations and Inequalities
To solve reciprocal equations, clear the fraction and rearrange the equation.
For inequalities like _1/x < 2_, consider where the graph of _y = 1/x_ lies below the line _y = 2_.
Remember to consider the sign of x, as reciprocal graphs change sign at x = 0.
Example 1
Solve the inequality 1/x < −1
Solution:
Graph y = 1/x and y = −1. The inequality is true when the 1/x graph lies below y = −1. Solution: x < −1.
Test Your Knowledge
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