Problem 33 - 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 \(P(x)=\frac{6+x}{10+x}\) Graph the function.

Step-by-Step Solution

Answer

Graph the function \(P(x) = \frac{6+x}{10+x}\) using key points and observe it approaches 1.

Step 1: Understanding the Function

The function given is \(P(x) = \frac{6+x}{10+x}\), where \(x\) represents the number of free throws Zonta makes consecutively. The percentage \(P(x)\) is the ratio of successful free throws to the total number of attempts, expressed as a fraction.

Step 2: Analyzing the Behavior of the Function

To better predict the shape of the graph, let's identify key points:- When \(x = 0\), we have \(P(0) = \frac{6}{10} = 0.6\).- If \(x\) increases, the numerator increases more slowly than the denominator, possibly affecting the growth rate of \(P(x)\).- The horizontal asymptote as \(x\) increases will approach \(1\), because the highest degree terms \(x/x\) in the numerator and denominator dominate.

Step 3: Table of Values Creation

Create a table of values for different \(x\):- \(x = 0\), \(P(0) = \frac{6}{10} = 0.6\)- \(x = 1\), \(P(1) = \frac{7}{11} \approx 0.636\)- \(x = 2\), \(P(2) = \frac{8}{12} = 0.666\)- \(x = 3\), \(P(3) = \frac{9}{13} \approx 0.692\)This shows how the free throw percentage steadily increases with more successful throws.

Step 4: Sketching the Graph

Plot the points from the table on a coordinate plane. Mark the y-axis with values from \(0\) to \(1\) to represent percentages, while the x-axis represents the number of consecutive made free throws \(x\). Connect the points to form a curve that approaches \(y = 1\) asymptotically, showing that her percentage increases but never exceeds \(100\%\). This confirms the horizontal asymptote.

Key Concepts

Rational Functions

In mathematics, a rational function is a function that can be expressed as the ratio of two polynomial functions. The general form of a rational function is given by \( f(x) = \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials, and \( q(x) \) is not the zero polynomial. A rational function can provide valuable insight into a variety of real-world scenarios. For example, in this exercise, the function \( P(x) = \frac{6+x}{10+x} \) represents Zonta's free-throw percentage as a function of additional throws made.

Rational functions often have interesting characteristics, such as vertical and horizontal asymptotes, that can help us understand how the function behaves as its variables change. In the case of Zonta's percentage, the numerator \( 6 + x \) increases with each successful throw, while the denominator \( 10 + x \) increases as well due to the additional attempts.

The nature of rational functions means they can't have zero denominators, as this would make the function undefined, leading to what we call vertical asymptotes. In the context of free throw percentages, the denominator reflects an increasing total of attempted throws, keeping the function meaningful and well-defined for all \( x \geq 0 \).

Graphing Functions

Graphing functions is a powerful way to visualize mathematical relationships between variables. It allows us to clearly see how changes in one variable affect another. To graph the function \( P(x) = \frac{6+x}{10+x} \), we start by creating a table of values, which helps us identify some points on the graph.

• When \( x = 0 \), \( P(x) = 0.6 \) - this is the starting point reflecting Zonta's current percentage.
• Values like \( x = 1, \ 2, \ 3 \) show how the percentage improves with more consecutive successful throws.

By plotting these points, and connecting them smoothly, a curve begins to form. This curve shows that with more successful throws, Zonta's percentage increases steadily but slowly as it approaches the horizontal line \( y = 1 \).

Visualizing this helps in understanding that while improvement can be significant, it increasingly slows as she approaches the ideal 100%. This graphical insight reflects the effect of cumulative accuracy over many attempts, clearly depicting the concept of diminishing returns in percentage growth.

Asymptotes

Asymptotes are lines that a graph approaches but never actually reaches. In the context of rational functions, they offer clues about the long-term behavior of the function.

For Zonta’s free-throw function \( P(x) = \frac{6+x}{10+x} \), identifying the asymptotes helps us understand how her percentage behaves with increasing free throw success. The horizontal asymptote occurs as \( x \) becomes very large and \( P(x) \) approaches 1. This happens because as \( x \) increases, the term \( x/x \) in the numerator and denominator tends to 1, making the influence of the initial values (6 and 10) negligible.

• The presence of a horizontal asymptote at \( y = 1 \) illustrates the ceiling effect - Zonta's percentage gets closer and closer to 100% but never quite reaches it.

It's important to note there are no vertical asymptotes here, as the denominator \( 10 + x \) is never zero for \( x \geq 0 \). This ensures that the function is defined for all non-negative \( x \), reflecting every possible consecutive make streak Zonta could have. Understanding asymptotes provides a deeper insight into how changes affect performance over time, especially in sports statistics like free-throw percentages.