constant of variation
Determine the constant of variation k in direct and inverse variation: k equals y/x for direct and k equals xy for inverse, characterizing the proportional relationship throughout.
Key Concepts
The constant $k$ in the equation $A = kB$ is called the constant of variation.
1. A scooter travels 10 km in 30 minutes ($D=kT$). To find k, set up $10 = k(30)$, so the constant is $k = \frac{1}{3}$. 2. If 120 seconds have passed in 2 minutes ($S=kM$), then $120 = k(2)$, so the constant of variation is $k = 60$.
Meet 'k', the secret code of the relationship! This number is the specific multiplier that never changes for a given situation. Find 'k' first, and you can solve any problem related to that variation.
Common Questions
What is the constant of variation and how do you find it?
The constant of variation k is the fixed ratio that defines a variation relationship. For direct variation y=kx, compute k=y/x from any known data point. For inverse variation y=k/x, compute k=xy from any known data point.
How does the constant of variation change the graph?
In direct variation y=kx, a larger absolute value of k makes the line steeper. A positive k slopes upward; a negative k slopes downward. In inverse variation y=k/x, k determines how quickly the hyperbola approaches the axes.
How do you use the constant of variation to predict unknown values?
Once k is determined from one data point, substitute any new x or y value into the variation equation to predict the missing variable. For k=6 in direct variation, when x=4, y=kx=6(4)=24.