Question

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. in three ways

Answer 1

[AI-generated answer — Bot GPT 3.5]

Function 1:

y = kx
This function represents a direct variation, where y increases or decreases in proportion to x. The graph of this function is a straight line passing through the origin (0,0). As x increases, y also increases at a constant rate.
Real-world example: The relationship between the number of hours worked and the amount earned. As someone works more hours, their earnings increase proportionally.

Function 2:
y = k/x
This function represents an inverse variation, where y decreases as x increases, and vice versa. The graph of this function is a hyperbola, with y decreasing as x increases and approaching zero but never reaching it.
Real-world example: The relationship between the distance traveled and time taken to travel that distance at a constant speed. As the speed increases, the time taken to travel a certain distance decreases inversely.

Function 3:
y = x^2 + 3
This function does not represent direct or inverse variation, as the relationship between x and y is not proportional or inversely proportional. The graph of this function is a parabola that opens upwards, with the vertex at (0,3).
Real-world example: The relationship between the number of years since a tree was planted and its height. The growth of a tree is not directly proportional to the number of years since it was planted, as it also depends on other factors such as soil quality and weather conditions.