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: y varies directly with x
In this case, the function can be represented as y = kx, where k is a constant. As x increases or decreases, y will also increase or decrease proportionally. The graph of this function is a straight line that passes through the origin (0,0) with a positive slope.
Example: The relationship between distance and time travelled at a constant speed. As the time increases, the distance travelled also increases proportionally.
Second function: y varies inversely with x
In this case, the function can be represented as y = k/x, where k is a constant. As x increases, y will decrease and vice versa. The graph of this function is a hyperbola, with the curve decreasing as x increases and vice versa.
Example: The relationship between the number of workers and the time it takes to complete a task. As the number of workers increases, the time it takes to complete the task decreases.
Third function: The relationship between x and y is neither inverse variation nor direct variation
In this case, the function can be represented by a polynomial or any function that does not follow a direct or inverse relationship. The graph can be a curve, a combination of straight lines, or any other shape that does not obey direct or inverse variation.
Example: The relationship between the volume of water in a container and the temperature of the surroundings. The volume of water can be influenced by various factors such as temperature, air pressure, and container shape, resulting in a complex relationship that does not follow direct or inverse variation.