E7

Direct and Inverse Variation

Introduction to Variation

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

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

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

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

Graphs of Rational Functions

Identifying Key Features

Consider the rational function $f(x) = \frac{x-3}{x^2-4}$. To analyze and graph this function, follow these steps:

• Domain: The set of all real numbers except where the denominator is zero.

• Holes: Occur where a factor cancels from numerator and denominator.

• Vertical Asymptotes: Set denominator equal to zero and solve for $x$ (excluding holes).

• Horizontal Asymptotes: Compare degrees of numerator and denominator.

• x-intercepts: Set numerator equal to zero and solve for $x$.

• y-intercept: Evaluate $f(0)$.

Example: For $f(x) = \frac{x-3}{x^2-4}$:

• Denominator: $x^2-4 = (x-2)(x+2)$, so $x \neq 2, -2$

• No common factors, so no holes.

• Vertical asymptotes at $x=2$ and $x=-2$.

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

• x-intercept at $x=3$.

• y-intercept at $f(0) = \frac{-3}{-4} = \frac{3}{4}$.

Proportions

Definition and Properties

A proportion is an equation stating that two ratios are equal:

$\frac{a}{b} = \frac{c}{d}$

• Means: $b$ and $c$

• Extremes: $a$ and $d$

• Cross Product: $a \cdot d = b \cdot c$

Example: $\frac{2}{3} = \frac{4}{6}$ since $2 \times 6 = 3 \times 4$

Direct Variation

Definition and Equation

Variable $y$ varies directly with $x$ if $y = kx$, where $k$ is the constant of variation.

• Graph: A straight line through the origin with slope $k$.

• Proportionality: $\frac{y}{x} = k$

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

Applications

• Tuition: $T = 400x + 300$ (not direct, since $+300$ is a fixed fee)

• Commission: $C = 0.15x$ (direct variation)

Inverse Variation

Definition and Equation

Variable $y$ varies inversely with $x$ if $y = \frac{k}{x}$, where $k$ is the constant of variation.

• Product: $y \cdot x = k$

• Graph: A rectangular hyperbola in quadrants I and III (for $k>0$)

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

Joint and Combined Variation

Definitions

• Joint Variation: $y$ varies jointly as $x$ and $z$ if $y = kxz$.

• Combined Variation: $y$ varies directly as $x$ and inversely as $z$: $y = \frac{kx}{z}$.

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

Solving Variation Problems

General Steps

1. Write the variation equation based on the problem statement.

2. Substitute known values to solve for the constant $k$.

3. Use the equation to solve for unknowns.

Example: The volume of blood $B$ in a human varies directly as body weight $W$. If a 160-pound person has 5 quarts of blood, $B = kW$, so $5 = k \times 160 \implies k = \frac{5}{160} = 0.03125$. For a 220-pound person: $B = 0.03125 \times 220 = 6.875$ quarts.

Applications and Examples

• Boyle's Law (Inverse Variation): $P = \frac{k}{V}$, where $P$ is pressure and $V$ is volume at constant temperature.

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

• Frequency of a vibrating string: $f = \frac{k}{l}$, where $l$ is the length of the string.

• BMI (Body Mass Index): $BMI = \frac{W}{H^2}$, where $W$ is weight in kg and $H$ is height in meters.

Practice Problems

Practice problems involve setting up and solving equations for direct, inverse, and joint variation. For example:

• If $y$ varies directly as $x$, $y=45$ when $x=9$. Find $y$ when $x=12$.

• If $y$ varies inversely as $x$, $y=8$ when $x=3$. Find $y$ when $x=12$.

• If $y$ varies jointly as $x$ and $z$, $y=25$ when $x=2$ and $z=5$. Find $y$ when $x=4$ and $z=3$.

Summary Table: Types of Variation

Type	Equation	Key Feature	Example
Direct	$y = kx$	Linear through origin	Commission: $C = 0.15x$
Inverse	$y = \frac{k}{x}$	Product $y \cdot x = k$	Boyle's Law: $P = \frac{k}{V}$
Joint	$y = kxz$	Direct with multiple variables	Volume: $V = k r^2 h$
Combined	$y = \frac{kx}{z}$	Direct and inverse together	Speed: $S = \frac{kd}{t}$

Additional info: The notes also include worked examples, graphical interpretations, and real-world applications such as tuition models, BMI, and physical laws (Boyle's Law, gravity, etc.), reinforcing the connection between algebraic variation and practical scenarios.