Math Problem Statement
Solution
Let's break down the problem.
Function:
The score function is given by: where represents the game number, , and (whole numbers).
Questions:
a) After which game was Maya's score equal to zero?
For this, we set : This equation has three solutions:
Thus, Maya's score was zero after games 0, 4, and 6.
b) After which game was Maya's score -5?
We set : This is a cubic equation and would need to be solved numerically or by trial and error (since must be a whole number, ). By checking whole numbers from 0 to 10, there isn't any game number that satisfies this equation exactly, so it's possible that the answer involves estimation or the model isn't intended to give a score of -5.
c) After which game was Maya's score 16?
We set : Again, this is a cubic equation, but we can solve it numerically or by checking specific values of from 0 to 10.
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For : This isn't a match. We could keep trying other values until we find , which yields:
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For : This is closer to 16, but not exact, so there seems to be no exact game number yielding 16. The closest values can be tried.
d) Sketch and explanation:
The sketch of the graph of would be a cubic curve with roots at , , and . For , it would display typical cubic behavior, starting from negative infinity, crossing the x-axis at these points, and then heading toward positive infinity as increases.
This model may not be perfect for the game night because:
- It allows negative scores, which may not make sense for a game score.
- The cubic nature might not represent real-world behavior if scores were expected to be more linear or simpler.
Would you like further clarifications or a more detailed breakdown of any part? Let me know!
Here are 5 related questions to expand on this:
- How would you solve the cubic equation algebraically?
- Why might whole numbers be a better model for game scores than real numbers?
- What does the shape of a cubic function typically look like?
- How could the score function be adjusted to better represent game outcomes?
- What real-life factors could affect the accuracy of this mathematical model for scoring?
Tip: Always check the context of your problem to ensure that your mathematical model fits the scenario well. In this case, a non-linear function may not always represent a simple game score.
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Math Problem Analysis
Mathematical Concepts
Cubic Functions
Roots of Polynomials
Numerical Solutions
Formulas
S(x) = x(x - 4)(x - 6)
Theorems
Polynomial Roots Theorem
Cubic Function Behavior
Suitable Grade Level
Grades 9-11