Lesson 4-4: Direct Variation

Property

$y$ varies directly with $x$ if

where $k$ is a positive constant called the constant of variation. If $y$ varies directly with $x$, we may also say that $y$ is directly proportional to $x$. This relationship defines a linear function whose graph is a straight line passing through the origin.

Examples

• The total cost, $C$, of concert tickets varies directly with the number of tickets, $n$, purchased. If each ticket is 50 dollars, the relationship is $C = 50n$.
• The distance, $d$, you travel at a constant speed varies directly with time, $t$. If you are driving at 60 miles per hour, the formula is $d = 60t$.
• The amount of interest, $I$, earned in one year is directly proportional to the principal, $P$, invested. For a 4% interest rate, the formula is $I = 0.04P$.

Explanation

Think of this as a perfect partnership. When one variable changes, the other changes by the exact same multiplier. If you buy twice as many items, you pay twice the price. The ratio between the two quantities always stays constant.

Property

A relationship shows direct variation if its graph is a straight line that passes through the origin (0,0). This means when one variable is zero, the other must also be zero.

Examples

The graph for $P=4s$ is a line through the origin, showing direct variation.
The graph for $A=s^2$ is a curve, so it is not a direct variation.
The graph for $D=2m+3$ is a line but misses the origin, so it's not direct variation.

Explanation

Spotting this on a graph is a two-part test. Is it a perfectly straight line? Does it start at the (0,0) point, the origin? A graph must pass both tests to be considered a direct variation. If it fails either one, it's out!

Property

The constant of proportionality, often called the unit rate ($r$), is the constant ratio in a proportional relationship.
If quantities $x$ and $y$ are proportional, their relationship can be described by the equation $y = rx$.
The constant $r$ can be found by calculating the ratio $r = \frac{y}{x}$ for any corresponding pair $(x, y)$ where $x \neq 0$. On a graph, the unit rate is represented by the point $(1, r)$.

Examples

• A car travels 180 miles in 3 hours. The constant of proportionality is $\frac{180}{3} = 60$ miles per hour. The equation is $d = 60h$.
• A graph of a proportional relationship between cost and pounds of bananas passes through the point $(1, 0.55)$. The constant of proportionality is 0.55 dollars per pound.
• A table shows that 4 movie tickets cost 52 dollars. The unit rate (constant of proportionality) is $\frac{52}{4} = 13$ dollars per ticket.

Explanation

The constant of proportionality is the 'secret multiplier' that connects the two quantities. It tells you how much of the second quantity you get for exactly one unit of the first quantity, like price per item or miles per hour.