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Question 1167884: Construct a situation that relates to the concept of a rational function, the scenario must specifically be about a basketball game. What is the importance of rational function to the scenario? Set a table of values, equation, solution and answers to support the answers please.
Answer by CPhill(2138)   (Show Source):
You can put this solution on YOUR website! Sure, let's construct a scenario involving a basketball game and a rational function.
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## Scenario: Player's Free Throw Percentage during a Game
Imagine a basketball player, **Alex**, who is a decent but not perfect free-throw shooter. We want to track his free-throw percentage *during* a game as he attempts more free throws. A rational function can effectively model this.
**The Situation:**
Alex starts the game having made 0 out of 0 free throws. As the game progresses, he attempts and makes (or misses) free throws. We want to represent his current free-throw percentage based on the total free throws attempted and made.
**Why a Rational Function is Important to this Scenario:**
A rational function is crucial here because:
1. **Ratio/Proportion:** A percentage is inherently a ratio (successful attempts / total attempts), which is the definition of a rational expression.
2. **Asymptotic Behavior:** As the number of attempts gets very large, the player's percentage will stabilize and approach his true long-term free-throw percentage (if he were to continue shooting indefinitely under similar conditions). This "leveling off" is represented by a horizontal asymptote in the rational function.
3. **Initial Undefined State:** At the very beginning of the game, before any free throws are attempted (0 made / 0 attempted), the percentage is undefined. A rational function naturally handles this as a division by zero.
4. **Impact of Early Attempts:** The function will clearly show how a single successful or missed free throw early in the game can drastically change the percentage, while the impact of a single shot diminishes as the total number of attempts increases.
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**Mathematical Representation:**
Let:
* $x$ = Total number of free throws Alex has *attempted* so far in the game.
* $y$ = Total number of free throws Alex has *made* so far in the game.
Alex's free-throw percentage, $P(x)$, can be represented as a rational function:
$P(x) = \frac{\text{Free Throws Made}}{\text{Free Throws Attempted}}$
To make it a function of a single variable $x$ (attempts), we need a rule for how many he makes. Let's assume Alex makes free throws at a consistent rate *once he starts shooting*.
**Let's refine the scenario with a specific sequence of shots:**
Suppose Alex starts with 0 made and 0 attempted.
* He attempts his first free throw and makes it.
* He attempts his second free throw and misses it.
* He attempts his third free throw and makes it.
* ...and so on.
This sequence is too complex for a simple rational function of $x$. Instead, let's define the function based on his *average success rate* over attempts, or a simpler scenario.
**Simpler Scenario: Alex's Free Throw Percentage after a certain number of attempts where he is expected to make a certain fraction.**
Let's say Alex makes 2 free throws for every 3 attempts.
If he has made 2 shots for every 3 attempts, his long-term percentage is $2/3 \approx 66.7\%$.
Let:
* $x$ = total free throws *attempted* by Alex during the game (from the start of the game).
* Assume Alex's performance can be approximated by saying he makes about $2/3$ of his free throws. However, this isn't his *current* percentage if we are tracking shot-by-shot.
**Better Scenario: Analyzing the impact of a specific set of future shots on his current percentage.**
Let's say Alex has already attempted $A$ free throws and made $M$ of them. His current percentage is $M/A$.
Now, he's about to attempt $x$ more free throws, and we assume he will make $k$ of those $x$ attempts (e.g., $k = 0.8x$ if he's shooting 80%).
Let:
* Alex's current free throw record: 10 attempts, 7 made. So, his current percentage is $7/10 = 70\%$.
* Alex is about to attempt $x$ more free throws.
* For these $x$ future attempts, he expects to make them at his long-term average, which is 60%. So, he expects to make $0.6x$ free throws out of these $x$ attempts.
His new total percentage, $P(x)$, after these $x$ additional attempts will be:
**Equation:**
$P(x) = \frac{\text{Current Made} + \text{Expected Made in future}}{\text{Current Attempted} + \text{Future Attempted}}$
$P(x) = \frac{7 + 0.6x}{10 + x}$
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**Table of Values:**
Let's see how his percentage changes as he attempts more free throws ($x$).
| $x$ (Additional Attempts) | Expected Made ($0.6x$) | Total Made ($7 + 0.6x$) | Total Attempted ($10 + x$) | $P(x)$ (Percentage) |
| :---------------------- | :--------------------- | :----------------------- | :------------------------- | :------------------ |
| 0 | 0 | 7 | 10 | $7/10 = 0.70$ (70%) |
| 1 | 0.6 | 7.6 | 11 | $7.6/11 \approx 0.691$ (69.1%) |
| 5 | 3 | 10 | 15 | $10/15 \approx 0.667$ (66.7%) |
| 10 | 6 | 13 | 20 | $13/20 = 0.65$ (65%) |
| 20 | 12 | 19 | 30 | $19/30 \approx 0.633$ (63.3%) |
| 50 | 30 | 37 | 60 | $37/60 \approx 0.617$ (61.7%) |
| 100 | 60 | 67 | 110 | $67/110 \approx 0.609$ (60.9%) |
| 1000 | 600 | 607 | 1010 | $607/1010 \approx 0.601$ (60.1%) |
*(Note: While you can't make 0.6 free throws, this model assumes a large number of future attempts where the fraction represents the expected average. If we were tracking discrete shots, the function would jump. This continuous model helps illustrate the long-term trend.)*
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**Solution and Answers (Analysis):**
* **Initial State:** When $x=0$, Alex's percentage is $70\%$. This is his current performance.
* **Impact of Future Shots:** As Alex attempts more free throws, his overall percentage $P(x)$ gradually moves from his current $70\%$ towards his expected long-term average of $60\%$.
* **Horizontal Asymptote:** For the rational function $P(x) = \frac{7 + 0.6x}{10 + x}$, as $x$ approaches infinity, the term $7$ and $10$ become negligible compared to $0.6x$ and $x$.
$\lim_{x \to \infty} \frac{7 + 0.6x}{10 + x} = \lim_{x \to \infty} \frac{x(0.6 + 7/x)}{x(1 + 10/x)} = \frac{0.6}{1} = 0.6$
This means the horizontal asymptote is $y=0.6$, or $60\%$. This represents Alex's hypothesized long-term free-throw percentage, which his game percentage will approach as he takes more and more shots.
* **No Vertical Asymptote in Context:** The vertical asymptote would occur when the denominator is zero ($10+x=0 \implies x=-10$). In the context of "additional attempts," $x$ must be non-negative, so this asymptote is not relevant to the practical scenario of future shots in the game.
* **Practical Use:** This rational function allows the coach, analysts, or fans to understand how a player's initial hot (or cold) streak will be "averaged out" by their long-term performance as the sample size of their attempts grows during a game or season. It highlights that early performance swings are more dramatic, and as more data points are added, the percentage stabilizes closer to the player's true ability.
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