E7

Direct and Inverse Variation

Introduction to Variation

Variation describes how one quantity changes in relation to another. In College Algebra, direct and inverse variation are foundational concepts used to model relationships between variables in equations and real-world applications.

• Direct Variation: One variable increases or decreases in direct proportion to another.

• Inverse Variation: One variable increases as another decreases, such that their product remains constant.

Graphs and Asymptotes of Rational Functions

Analyzing Rational Functions

Rational functions are quotients of polynomials. Their graphs often feature asymptotes and holes, which are important for understanding their behavior.

• Definition: A rational function is any function of the form $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.

• Holes: Occur at values of $x$ that make both numerator and denominator zero (common factors).

• Vertical Asymptotes: Occur at values of $x$ that make the denominator zero but not the numerator.

• Horizontal Asymptotes: Determined by the degrees of the numerator and denominator polynomials.

Example: For $f(x) = \frac{x-3}{x^2-9}$:

• Factor denominator: $x^2-9 = (x-3)(x+3)$

• Hole at $x=3$ (since $x-3$ cancels)

• Vertical asymptote at $x=-3$

• Horizontal asymptote at $y=0$ (degree of numerator < denominator)

Proportions

Understanding Proportions

A proportion is an equation stating that two ratios are equal. Proportions are used to solve for unknowns in direct and inverse variation problems.

• General form: $\frac{a}{b} = \frac{c}{d}$

• Cross-multiplication: $ad = bc$

• Used to solve for missing values in variation problems

Example: If $\frac{4}{x} = \frac{8}{6}$, then $4 \times 6 = 8 \times x \implies x = 3$

Direct Variation

Definition and Properties

In direct variation, $y$ varies directly with $x$ if there is a constant $k$ such that $y = kx$.

• Constant of Variation: $k$ is the constant ratio $\frac{y}{x}$.

• Graph: The graph of $y = kx$ is a straight line through the origin.

Example: If $y$ varies directly with $x$ and $y=15$ when $x=3$, then $k=5$ and $y=5x$.

Applications of Direct Variation

• Distance traveled at constant speed: $d = rt$

• Circumference of a circle: $C = 2\pi r$

• Area of a circle: $A = \pi r^2$ (directly with the square of the radius)

Inverse Variation

Definition and Properties

In inverse variation, $y$ varies inversely with $x$ if there is a constant $k$ such that $y = \frac{k}{x}$.

• Constant of Variation: $k = yx$

• Graph: The graph of $y = \frac{k}{x}$ is a rectangular hyperbola with asymptotes at $x=0$ and $y=0$.

Example: If $y$ varies inversely with $x$ and $y=4$ when $x=5$, then $k=20$ and $y=\frac{20}{x}$.

Applications of Inverse Variation

• Boyle's Law (gas pressure and volume): $PV = k$

• Intensity of light inversely with the square of the distance: $I = \frac{k}{d^2}$

Joint and Combined Variation

Definitions

• Joint Variation: A variable varies directly as the product of two or more other variables. $y = kxz$

• Combined Variation: Involves both direct and inverse variation. $y = \frac{kx}{z}$

Example: The volume of a cylinder varies jointly with the square of the radius and the height: $V = k r^2 h$

Solving Variation Problems

General Steps

1. Identify the type of variation (direct, inverse, joint, or combined).

2. Write the appropriate equation using a constant $k$.

3. Use given values to solve for $k$.

4. Substitute $k$ and other known values to solve for the unknown.

Example Problems

• Direct Variation: If $y$ varies directly with $x$ and $y=12$ when $x=4$, find $y$ when $x=10$.

  • Equation: $y = kx$

  • $k = \frac{12}{4} = 3$

  • When $x=10$, $y = 3 \times 10 = 30$

• Inverse Variation: If $y$ varies inversely with $x$ and $y=8$ when $x=2$, find $y$ when $x=4$.

  • Equation: $y = \frac{k}{x}$

  • $k = 8 \times 2 = 16$

  • When $x=4$, $y = \frac{16}{4} = 4$

Sample Table: Types of Variation

Type of Variation	Equation	Graph Shape	Example
Direct	$y = kx$	Straight line through origin	Distance = Rate × Time
Inverse	$y = \frac{k}{x}$	Rectangular hyperbola	Boyle's Law
Joint	$y = kxz$	Varies (depends on variables)	Volume = $k r^2 h$
Combined	$y = \frac{kx}{z}$	Varies	Electrical resistance

Practice Problems

• Write variation equations for given scenarios.

• Solve for unknowns using direct, inverse, joint, or combined variation.

• Interpret real-world applications such as BMI, sound intensity, and physical laws.

Summary

• Direct and inverse variation are essential for modeling proportional relationships.

• Recognizing the type of variation allows for correct equation setup and solution.

• Applications span geometry, physics, economics, and more.