What is Direct Variation?
Direct Proportionality, or Direct Variation, without a power occurs when the X value and the Y value of a line increase at the same, constant rate. When seeing a line on a graph, it is easy to tell if it is Directly Proportional. If it is straight and passes through the point of origin (0,0).
Example:
Here is the time taken for a runner running different distances
Note:
That while time increases at a constant rate, so too does the distance. Over 20 minutes, the distance in meters increased by 3000. Therefore, the constant “k” that describes the distance travelled in a certain amount of time, more commonly referred to as speed, can be applied here.
The equation for time is written as T = kD, where T represents time, D represents distance, and k represents the constant of speed. In this case, the formula would be T = 150D. This is because when looking at the speed of 3000 meters in 20 minutes, one can divide 3000 evenly by 20, resulting in a quotient of 150. This means that in 1 minute, the runner travels 150 meters.
What is the formula for Direct Proportionality?
The general formula for Direct Proportionality, when considering concepts other than distance and time, is Y = kX.
The value of the constant k, and by extension any point on a line, can be found through observing one point on that line.
Example:
The price of apples is directly proportional to their weight. The price of an apple that weighs 100 grams is 80 cents.
What is the price of an apple that weighs 150 grams?
What is the weight of an apple that costs 1.5 Euros?
The first thing to do is to find the constant (k). This can be easily calculated by using the above formula, Y = kX.
Let the price of apples, the dependent variable, be Y and the weight of apples, the independent variable, be X. One can observe that an apple worth 80 cents weighs 100 grams, providing a data point. This point’s Y and X values can be used to find k:
80 = k(100)
80/100 = k
0.8 = k
Once k is obtained, the formula of Direct Proportionality can be applied to any point on the line. The formula now reads:
Y = (0.8)X
Returning to the earlier example:
Find the cost of an apple that weighs 150 grams. In this instance, the price of the apple, Y, is the unknown value:
Y = (0.8)X
Y = (0.8)(150)
Y = 1. 20 Euros
Find the weight of an apple that costs 1.5 euros. In this instance, the weight of the apple, X, is the unknown value:
Y = (0.8)X
1.5 = (0.8)X
1.5/(0.8) = X
188 grams = X
What is Inverse Variation?
Inverse Variation, or Inverse Proportionality is expressed by the formula Y = k/X. Even though it may not seem like it at first glance, its graph looks very different to that of Direct Variation. Here are some examples of k values ranging from 1 to 4:
Note:
The lines do not pass through the point of origin (a vertical asymptote), and are not straight at any point on the graph.
Instead, the lines:
Continue decreasing infinitely, with the rate of descent decreasing exponentially
Move mainly along the two axes, but
Never touch the axes
These lines, the x- and y-axis, are called asymptotes.
What is Power Variation?
Power Variation is similar to Direct Variation and Inverse Variation. However, as implied in the name, Power Variation is put to a power. It is expressed with the equation Y = kXn.
Power Variation works in the same way as Direct and Inverse Variation, except its models are graphed differently. Instead of a straight line going through the point of origin, or a constantly decreasing line, Power Variation is represented by an exponentially curved line going through the point of origin. Here are some power models where k = 1:
That they all intersect at the value of the constant (1,1).
Note:
Inverse Power Variation models are also asymptotes and therefore are similar to Inverse Variation models.