Problem 34 - Algebra 2

Question

BASKETBALL For Exercises \(33-36,\) use the following information. Zonta plays basketball for Centerville High School. So far this season, she has made 6 out of 10 free throws. She is determined to improve her free-throw percentage. If she can make \(x\) consecutive free throws, her free-throw percentage can be determined using \(P(x)=\frac{6+x}{10+x}\) What part of the graph is meaningful in the context of the problem?

Step-by-Step Solution

Answer

The meaningful part of the graph is where \( x \geq 0 \).

Step 1: Understand the Context

Zonta has currently made 6 out of 10 free throws. The function \( P(x) = \frac{6+x}{10+x} \) represents her free-throw percentage after making \( x \) more consecutive shots.

Step 2: Define the Graph

The graph of \( P(x) = \frac{6+x}{10+x} \) shows the relationship between the number of additional made free throws (\( x \)) and her resulting free-throw percentage.

Step 3: Evaluate the Domain of \( P(x) \)

In this context, \( x \) must be a non-negative integer because it represents the number of free throws Zonta can make, which cannot be negative. Thus, the domain of \( x \) is all non-negative integers \( \{0, 1, 2, 3, ...\} \).

Step 4: Assess the Meaningful Part of the Graph

The meaningful part of the graph in this context is where \( x \) is 0 or greater, corresponding to the cases where Zonta makes 0 or more consecutive free throws. Negative values of \( x \) do not make sense in a real-world scenario as Zonta cannot make a negative number of free throws.

Key Concepts

Graph Analysis

Understanding the graph of a function like \( P(x) = \frac{6+x}{10+x} \) involves analyzing how Zonta's free-throw percentage changes as she makes more consecutive free throws. The graph of this function is a curve that shows the free-throw success rate visually, helping us identify trends and make predictions.

In this context, the horizontal axis represents the number of free throws Zonta makes (\( x \)), while the vertical axis shows her resulting free-throw percentage. As \( x \) increases, so does the percentage \( P(x) \), suggesting that making more free throws will improve her ratio significantly but will also exhibit diminishing returns as it approaches the upper limit close to 100%.

Graph analysis allows us to better understand patterns in the data. By examining the curve, one can determine how quickly her free-throw success percentage rises with each successful shot. Graphical plots are not just visual tools, but they provide insights that pure numbers alone may not convey. The graph thus becomes an essential part of understanding Zonta's progress in her sport.

Domain of a Function

The domain of a function refers to the set of all possible inputs, or values \( x \) can take, for which the function \( P(x) \) is defined. In the case of Zonta's free-throw function \( P(x) = \frac{6+x}{10+x} \), the domain represents the number of additional free throws she can potentially attempt.

For Zonta, since she cannot make a negative number of free throws, \( x \) must be a non-negative integer. Therefore, the domain is \( \{0, 1, 2, 3, \ldots \} \), encompassing all non-negative whole numbers. This means she can attempt any whole number of free throws starting from zero upwards.

It's important to understand domains in practical situations like this one. The domain ensures that the function is valid within the context of real-world actions Zonta can take. If other values were used, such as negative integers or fractions, the situation would not reflect the physical reality Zonta faces.

Real-World Applications

Functions such as \( P(x) = \frac{6+x}{10+x} \) have a wide range of real-world applications beyond basketball. They are used in fields like economics, biology, and engineering to model phenomena where a ratio or percentage needs to be expressed as a function of variable changes.

For instance:

• In economics, similar rational functions might model the price elasticity of supply or demand.
• In biology, they can describe populations growth rates depending on resource availability.

In Zonta's example, this function helps her project and target improvements in her game performance.

The concept of adjusting a percentage through a variable number of attempts or trials is beneficial. It offers a strategy for improvement and measurement. Just as Zonta aims to improve her skills over time by analyzing her free-throw success, other fields use similar models to optimize outcomes and drive strategic decisions.

Rational Functions

Rational functions, such as \( P(x) = \frac{6+x}{10+x} \), are ratios of two polynomials. They are critical in various mathematical applications as they allow for describing relationships where quantities change at different rates.

These functions often have key characteristics such as asymptotes, which are lines that the graph approaches but never actually reaches. In Zonta's case, there is an asymptote at \( y=1 \) (or 100%) since her free-throw percentage cannot exceed 100%. Another characteristic might be vertical asymptotes, which can occur in more complex rational functions when the denominator equals zero, but for our function, it happens only theoretically at \( x=-10 \), which is beyond her domain.

Rational functions are widely utilized in resources allocations, equipment efficiency, and strategy optimization. They provide vital insights into how dependent variables respond when independent variables change. Their importance is reflected in the accurate predictions and strategic enhancements they offer across sectors and disciplines.