Question

Rational functions discussion: Write three functions. In the first function, Y should vary directly with X. in the second function, Y should vary inversely with X. In the third function, the relationship between X and Y should be neither inverse variation nor direct variation. Describe the graph of each function and give a real world example for each.

Answer 1

1. Proportional relationship is also called a direct variation of y with x. The equation that describes such variation is:

y = kx

Where k is a constant of proportionality.

One common example of a direct variation is:

The pound of candies cost $0.45. Write the equation for the cost of x pounds of candies.

The equation for the problem is y = 0.45x

The graph of this function is a line that passes through the origin.

2. Inverse variation. The equation for this function is:

yx = k

Where k is the constant of inverse proportionality.

An example could be: One worker builds a house (alone) in 60 days. Write the equation of the time needed to build the house for x workers at the same pace.

The equation is:

yx = 60

The graph of this equation is one branch of a hyperbola. See the figure below:

3. Neither inverse nor direct variation. Any other form of the equation cannot be classified as inverse or direct variation. We could use, for example:

y = mx + b

A real situation can be stated as follows: A gym charges $40 as a fixed fee and $25 per month. The equation for the cost of the gym as a function of the number of months (x) is:

y = 25x + 40

The graph of this equation is a line that does not pass through the origin, so it's not a direct variation.