In 6 sentences :
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.
1) Function with direct variation:
In this function, y varies directly with x. This means that as x increases, y also increases at a constant rate. The graph of this function is a straight line passing through the origin (0,0). A real-world example of direct variation is 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 proportionately.
2) Function with inverse variation:
In this function, y varies inversely with x. This means that as x increases, y decreases and vice versa, while their product remains constant. The graph of this function is a hyperbola, with the asymptotes being the x and y-axes. A real-world example of inverse variation is the relationship between the speed of a vehicle and the time it takes to travel a fixed distance. As the speed increases, the time taken to cover the distance decreases, and vice versa.
3) Function with neither inverse nor direct variation:
In this function, there is no direct or inverse relationship between x and y. The graph of this function can take various shapes, depending on the specific equation. A real-world example of this type of relationship could be the correlation between the temperature outside and the number of ice cream cones sold. The number of ice cream cones sold may increase during hotter temperatures, but it may not necessarily decrease during cooler temperatures, as other factors like seasonal trends or personal preferences come into play.