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.
First function:
In this case, we can say that y varies directly with x. The equation for this function would be y = kx, where k is a constant. As x increases, y also increases by the same factor. Consequently, the graph of this function would be a straight line passing through the origin, with a positive slope.
Example:
Consider the relationship between the number of hours worked and the amount of money earned. As the number of hours worked increases, the amount of money earned also increases proportionally.
Second function:
In this function, y should vary inversely with x. The equation for this function would be y = k/x, where k is a constant. As x increases, y decreases, and conversely, as x decreases, y increases. The graph of this function would be a curve that approaches zero as x approaches positive or negative infinity.
Example:
Let's consider the relationship between the speed of a car and the time it takes to reach a certain destination. As the speed of the car increases, the time it takes to reach the destination decreases inversely. For instance, if a car is traveling at a high speed, it will take less time to reach the destination compared to a car traveling at a lower speed.
Third function:
In this case, the relationship between x and y should be neither inverse variation nor direct variation. The equation for this function can take any form, as long as it is not directly proportional or inversely proportional. The graph of this function can be any curve that does not follow a straight line or a hyperbolic shape.
Example:
Consider the relationship between the number of hours a student spends studying and their test score. While studying for more hours may initially result in higher test scores, there could be a point where studying beyond a certain threshold does not yield significant score improvements. The connection between the number of hours studied and test scores is not directly proportional or inversely proportional because there are additional factors at play, such as the effectiveness of study methods and the student's aptitude for the subject.