Function 1: y = kx
This function shows direct variation, with y increasing at a constant rate as x increases. The graph of this function would be a straight line passing through the origin.
Real-world example: The amount of money earned by an employee working hourly. As the number of hours worked (x) increases, the amount earned (y) also increases at a constant rate, assuming a fixed hourly rate.
Function 2: y = k/x
This function shows inverse variation, with y decreasing as x increases. The graph of this function would be a hyperbola, with the x and y axes as asymptotes.
Real-world example: The time it takes to complete a task is inversely proportional to the number of people working on it. As the number of people (x) working on a task increases, the time taken to complete it (y) decreases.
Function 3: y = x^2 + 1
This function does not show direct or inverse variation, as y does not increase or decrease linearly with x. The graph of this function would be a curve, specifically a parabola opening upwards.
Real-world example: The height of a ball thrown into the air over time. As time (x) increases, the height of the ball (y) will increase initially, peak at a certain point, and then start decreasing as the ball falls back to the ground. This relationship is not direct or inverse, as the height does not increase or decrease proportionally with time.
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...
3 answers
use simpler terms
Function 1: y = 2x
This function shows direct variation, with y increasing twice as fast as x increases. The graph of this function would be a straight line that passes through the origin (0,0).
Real-world example: The distance traveled by a car is directly proportional to the time it has been driving at a constant speed.
Function 2: y = 4/x
This function shows inverse variation, with y decreasing as x increases. The graph of this function would be a curve that approaches the x-axis but never touches it.
Real-world example: The time it takes to finish a job is inversely proportional to the number of workers assigned to it.
Function 3: y = x^2 + 3
This function does not show a direct or inverse relationship between x and y. The graph of this function would be a curve that opens upwards, forming a U-shape.
Real-world example: The height of a helicopter above the ground as time passes, taking into account factors such as wind speed and altitude adjustments.
This function shows direct variation, with y increasing twice as fast as x increases. The graph of this function would be a straight line that passes through the origin (0,0).
Real-world example: The distance traveled by a car is directly proportional to the time it has been driving at a constant speed.
Function 2: y = 4/x
This function shows inverse variation, with y decreasing as x increases. The graph of this function would be a curve that approaches the x-axis but never touches it.
Real-world example: The time it takes to finish a job is inversely proportional to the number of workers assigned to it.
Function 3: y = x^2 + 3
This function does not show a direct or inverse relationship between x and y. The graph of this function would be a curve that opens upwards, forming a U-shape.
Real-world example: The height of a helicopter above the ground as time passes, taking into account factors such as wind speed and altitude adjustments.
1 answer