Study Notes: Inverse Variation and the Reciprocal Function

Section 4.1 – Inverse Variation and the Reciprocal Function

Introduction

• Date: October 27, 2025
• Focus: Exploring inverse variation and the properties of the reciprocal function related to graph translations.

Model & Discuss

Problem Setup

• Two rectangles both have an area of 144 square units. Examples of rectangle pairs include:
  • Length: 12, Width: 12
  • Length: 8, Width: 18
  • Length: 9, Width: 16
  • Length: 6, Width: 24
  • Length: 4, Width: 36
  • Length: 3, Width: 48
  • Length: 2, Width: 72
  • Length: 1, Width: 144

Part A

• Task: Sketch and organize data for additional rectangles maintaining the same area of 144 square units.

Part B

• Consideration: As the length of the rectangle increases, the width decreases.

Essential Question

Inquiry

• How are inverse variations related to the reciprocal function?

Identify Inverse Variation

Determination Process

• To determine if a relationship represents an inverse variation:

  
  
  
  
  • A table of values is provided:
    

x	1	2	3	4	6	12
y	12	6	4	3	2	1
Explanation:
An inverse variation is a relationship where as one variable increases, the other decreases proportionally.
If the product of the pairs of values remains constant (K), it confirms inverse variation.
Example 1: Identify Inverse Variation

• Given a set of values:
  • x: 1, 2, 3, 4, 5, 6
  • y: 12, 6, 4, 3, 2, 1
• Since all products (xy) equal 12, the relationship depicts inverse variation.

Study Tip

• Always check the products of each pair of values before concluding about the relationship.

Examples

Example 2: Inverse Variation Assessment

• Evaluate the tables:

  
  
  
  

  • Table A:
    
    

    x	1	2	3	5	15
    y	25.5	12.75	8.5	5.1	1.7

Conclusion: Yes, it represents an inverse variation.

All products equal 25.5.

Table B:

x35791113-------------------------y111111

Conclusion: No, it does not represent an inverse variation.

Concept of Inverse Variation

• If a relation between x and y is an inverse variation, we express it as:
  • $xy = k$ where $k <br />\neq 0$.
• Here, $k$ is the constant of variation.
  • Example pairs:
    | x | 1 | 2 | 4 | 8 |
    |---|----|----|---|----|
    | y | 24 | 12 | 6 | 3 |
  • Observations: As $x$ doubles, $y$ halves.

Example 3: Use Inverse Variation

• Scenario: If $x = 10$ when $y = 3$, find the equation.
  • Step 1: Write $y = k/x$. Solve for k:
  • 3 = k/10
    ightarrow k = 30.
  • Step 2: Substitute to find $y$ when $x = -6$:
  • y = 30 / (-6)
    ightarrow y = -5.

Common Error Highlight

• In direct variation, $y = kx$, both $x$ and $y$ increase or decrease simultaneously. In contrast, in inverse variation, $xy = k$, one variable's increase corresponds with the other's decrease.

Reciprocal Function Understanding

Graph Features

• The graph of the reciprocal function $f(x) = \frac{1}{x}$ features:
  • Horizontal Asymptote: $y = 0$.
  • Vertical Asymptote: $x = 0$.
  • Domain of f(x): $x <br />\neq 0$.
  • Range of f(x): $y <br />\neq 0$.

End Behavior

• As $x$ approaches 0 from the positive side, f(x) o + ext{∞}.
• As $x$ approaches 0 from the negative side, f(x) o - ext{∞}.

Example 8: Graph Translations of Reciprocal Function

Definition Updates

• Translate $g(x) = \frac{1}{x - 3} + 2$:
  • Vertical Asymptote: Shifted 3 units to the right (from $x = 0$).
  • Horizontal Asymptote: Shifted 2 units up (from $y = 0$).
• Domain: $x <br />\neq 3$; Range: $y <br />\neq 2$.

Generalization of Translations

• Horizontally translate the graph and adjust vertically using formats:
  • $g(x) = f(x - h) + k$, where $h$ is horizontal offset and $k$ is vertical offset.

Example 9: Graph Further Translations

• Task: Graph $g(x) = -\frac{1}{x + 2} - 4$. Identify:
  • Vertical Asymptote: $x = -2$.
  • Horizontal Asymptote: $y = -4$.