Lesson 6: 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

When graphed, the relationship between two proportional variables has two key characteristics:

1. The graph is a straight line.
2. The graph passes through the origin, which is the point $(0, 0)$.

These features occur because the rate of change is constant and because if one variable is zero, the other must also be zero.

Examples

• A graph shows the cost of bulk almonds. The point $(4, 24)$ is on the line, meaning 4 pounds cost 24 dollars. Since the graph is a line through the origin, the unit price is constant: $\frac{24}{4} = 6$ dollars per pound.
• The graph of a monthly bus pass cost is a horizontal line at $y=50$. This is not proportional to the number of rides because it does not pass through $(0,0)$ and the cost is constant regardless of the number of rides.
• A caterer's fee is shown on a graph that is a straight line through $(0,0)$ and $(10, 150)$. The relationship is proportional. The unit rate (cost per person) is $\frac{150}{10} = 15$ dollars per person. For 30 people, the cost would be $30 \times 15 = 450$ dollars.

Explanation

Think of a proportional graph as a perfectly straight ramp that starts right at the ground. It is straight because the steepness (the rate) never changes, and it starts at $(0,0)$ because zero input means zero output, like working 0 hours earns 0 dollars.