Lesson 4: Direct Variation

New Concept

Direct variation describes a proportional relationship where one variable is a constant multiple of another ($y = kx$) or a power of another ($y=kx^n$). This lesson teaches you to identify and apply this concept using tables, graphs, and formulas.

What’s next

Next, you'll apply this concept in worked examples and practice cards to master finding the constant of variation and solving problems.

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

If we know any one pair of corresponding values for the variables in a direct variation, we can find the constant of variation, $k$. To do this, substitute the known values of $x$ and $y$ into the equation $y = kx$ and solve for $k$. Once $k$ is found, you can write the complete formula.

Examples

• The speed of a falling object, $v$, varies directly with time, $t$. If its speed is 49 meters per second after 5 seconds, we find $k$ from $49 = k(5)$, so $k = 9.8$. The formula is $v = 9.8t$.
• The property tax, $T$, on a home varies directly with its assessed value, $V$. A home valued at 200,000 dollars has a tax of 3,000 dollars. We have $3000 = k(200000)$, so $k = 0.015$. The formula is $T = 0.015V$.
• The weight of a bag of apples, $w$, is proportional to the number of apples, $n$. A bag with 10 apples weighs 3 pounds. We find $k$ from $3 = k(10)$, so $k = 0.3$. The formula is $w = 0.3n$.

Explanation

To find the specific 'rule' connecting two variables, you only need one matched pair of data. By plugging this pair into the general formula $y = kx$ or $y=kx^n$, you can solve for the constant, $k$, and unlock the full equation.

Property

$y$ varies directly with a power of $x$ if

where $k$ and $n$ are positive constants. To test if a dataset exhibits this relationship, check if the ratio $\frac{y}{x^n}$ remains constant for all data pairs.

Examples

• The area, $A$, of a circle varies directly with the square of its radius, $r$. The formula is $A = \pi r^2$, where $k = \pi$ and $n=2$.
• The volume, $V$, of a sphere varies directly with the cube of its radius, $r$. The formula is $V = \frac{4}{3}\pi r^3$, where $k = \frac{4}{3}\pi$ and $n=3$.
• The kinetic energy, $E$, of an object varies directly with the square of its velocity, $v$. The relationship is given by $E = \frac{1}{2}mv^2$, where the constant of variation is $k = \frac{1}{2}m$.

Explanation

This is a super-powered version of direct variation. Instead of $y$ being proportional to just $x$, it's proportional to $x$ raised to a power, like $x^2$ or $x^3$. As $x$ grows, $y$ grows much faster, creating a curve instead of a line.

Property

If $y$ varies directly with a power of $x$, such that $y = kx^n$, then multiplying the input $x$ by a factor $c$ causes the output $y$ to be multiplied by a factor of $c^n$. The new value of $y$ can be found with the formula:

The exponent $n$ is often called the scaling exponent.

Examples

• If the cost of a square carpet is proportional to its area ($A=s^2$), doubling the side length $s$ will multiply the cost by a factor of $2^2=4$.
• The weight of a solid sphere is proportional to its volume ($V \propto r^3$). If you increase its radius by a factor of 1.5, its weight will increase by a factor of $(1.5)^3 = 3.375$.
• If the power $P$ from a windmill varies with the cube of wind speed $w$ ($P=kw^3$), and the wind speed is halved, the power generated becomes $(\frac{1}{2})^3 = \frac{1}{8}$ of the original amount.

Explanation

Scaling reveals how much one variable changes when the other is multiplied. If $y$ is proportional to $x^n$, and you triple $x$, then $y$ will increase by a factor of $3^n$. This is a powerful shortcut for predicting outcomes without re-calculating everything.