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.
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
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