Think of two real-life quantities that are related to each other and construct a function that models the relationship between the two quantities. The function should be a piecewise function where not all components are linear. Describe the scenario you are modeling, the definitions of the variables involved in the function, and a graph or equation of the function.
Post a detailed response to the Solving Problems with Functions Discussion prompt. Then respond to two classmates' posts. In your response to each post, do the following:
State whether or not you agree or disagree with your classmates' models representing the scenario. Use evidence to support your answer.
Evaluate the usefulness of the model. What would make the model more useful?
I have chosen to model the relationship between the number of hours a person studies for an exam and their exam score. The function will be piecewise, as the relationship between the two quantities may not be linear.
Scenario: A student is preparing for a math exam and wants to determine how their exam score is affected by the number of hours they study.
Variables:
- x: Number of hours studied
- y: Exam score
Piecewise Function:
Let's assume that for the first 10 hours studied, the linear relationship between hours studied and exam score is given by:
y = 3x + 50
After studying for 10 hours, the relationship between hours studied and exam score may change, and we will use a quadratic model to represent this relationship:
y = -0.5x^2 + 15x + 70
Graph:
The graph of the piecewise function will have a linear portion for the first 10 hours studied and a quadratic portion after that.
Equation:
y = { 3x + 50 if 0 ≤ x ≤ 10
-0.5x^2 + 15x + 70 if x > 10 }
Evaluation:
I agree with this model as it accurately captures the non-linear relationship between the number of hours studied and the exam score. The piecewise function allows for a more realistic representation of how studying more hours can have diminishing returns on exam performance.
To make the model more useful, we could gather more data points and refine the quadratic portion of the function to better fit the actual relationship between hours studied and exam score. Additionally, incorporating other factors such as the student's prior knowledge or study habits could make the model even more accurate and useful.