Problem 54 - Algebra 1

Question

The table shows information about the top 7 winners of the 1997 Masters Tournament in Augusta, Georgia. Graph the relation. Is the money earned a function of the score? Explain. $$ \begin{array}{|l|c|c|c|c|c|c|c|}\hline \text { Score } & 270 & 282 & 283 & 284 & 285 & 285 & 286 \\\\\hline \text { Prize (s) } & 486,000 & 291,600 & 183,600 & 129,600 & 102,600 & 102,600 & 78,570 \\\\\hline\end{array} $$

Step-by-Step Solution

Answer

The money earned is not a function of the score because the same score (285) corresponds to two different prize money amounts ($102,600).

Step 1: Plot the Data

To begin, plot the data as a scatter plot using the scores on the x-axis and the corresponding prize money on the y-axis.

Step 2: Analyze the Graph

Once the data has been plotted, observe the relations and patterns within the graph. Are there any common trends? Do there appear to be any correlations between the scores and prize money?

Step 3: Determine if the Prize Money is a Function of Score

To determine if the prize money is a function of the score, see if there is one and only one prize money amount for each score. If the same score corresponds to multiple prize money amounts, then the prize money is not a function of score.

Key Concepts

Scatter Plot

A scatter plot is a type of graph that is used to visualize the relationship between two variables. It consists of points plotted on a horizontal and vertical axis, each representing a pair of data. In the context of the 1997 Masters Tournament data, a scatter plot helps to visually display the relationship between the players' scores and their corresponding prize money.

To create a scatter plot, each golfer's score is placed along the x-axis (independent variable), and the prize money earned is placed along the y-axis (dependent variable). Plotting these points provides a visual distribution of the data, making it easier to identify trends or patterns at a glance. Scatter plots are particularly useful for spotting outliers or for illustrating the strength and direction of a relationship between two variables.

Correlation

Correlation in statistics measures the strength and direction of a linear relationship between two variables. When analyzing a scatter plot, one can identify whether there is a positive correlation, negative correlation, or no correlation. A positive correlation means that as one variable increases, the other also increases, and vice versa for a negative correlation.

In the exercise, students are asked to look at the scatter plot of the Masters Tournament data and determine if there is a correlation between the scores and the prize money. If there's a downward trend, this would indicate a negative correlation, suggesting that lower scores (which represent better performance) result in higher prize money. It's essential to understand that correlation does not imply causation; it only indicates that there is a possible link between the two variables.

Algebraic Patterns

Algebraic patterns are relationships that follow a certain predictable rule, often represented by an equation or formula. In algebra, these patterns allow us to make predictions or conjectures about the behavior of the data. When dealing with problems like the Masters Tournament, looking for algebraic patterns can help students predict potential earnings based on scores or vice versa.

In practice, if the prize money amounts show a consistent decrease as the scores increase, this pattern can be described using a mathematical formula. However, it's important to recognize that the presence of identical prize amounts for different scores breaks the pattern and indicates that in this specific case, the prize money does not solely depend on the score, which leads to the next topic of whether or not the relationship is a function.

Graphing Relationships

Graphing relationships involves plotting two variables on a graph to discover how they interact with each other. In the case of the Masters Tournament data, students are asked to determine if the relationship between score and prize money is a function. A function is a special type of relationship where each input (score) is associated with exactly one output (prize money).

By graphing the relationship, students can visually assess whether any score on the x-axis corresponds to more than one prize amount on the y-axis. If there are multiple y-values for a single x-value, then the relationship is not a function. This concept is crucial for understanding how variables can depend on one another in different contexts, such as in mathematical modeling or real-world scenarios.