Problem 83 - Precalculus: Functions and Graphs

Question

Choose the equation that best describes the table of data. $$\begin{array}{|c|c|}\hline x & y \\\\\hline 1 & 2.1213 \\\\\hline 2 & 3.6742 \\\\\hline 3 & 4.7434 \\ \hline 4 & 5.6125 \\\\\hline 5 & 6.3640 \\\\\hline\end{array}$$ (1) \(y=1.5529 x+0.5684\) (2) \(y=\frac{3}{x}+x^{2}-1\) (3) \(y=3 \sqrt{x-0.5}\) (4) \(y=3 x^{1 / 3}+1.1213\)

Step-by-Step Solution

Answer

The equation that best describes the data is \( y=3 \sqrt{x-0.5} \).

Step 1: Recognize the pattern

Examine the data table to identify potential patterns or relationships in the values of \( x \) and \( y \). Notice that as \( x \) increases, \( y \) also increases in a non-linear fashion, suggesting a possible square root or cubic relationship.

Step 2: Evaluate Option 1

Substitute \( x = 1, 2, 3, 4, 5 \) into the equation \( y=1.5529 x+0.5684 \) to calculate predicted \( y \) values and compare them with the data table. Calculate and observe that the predicted values do not closely match the observed data.

Step 3: Evaluate Option 2

Substitute \( x = 1, 2, 3, 4, 5 \) into \( y=\frac{3}{x}+x^{2}-1 \). Calculate to find out that the predicted \( y \) values deviate significantly from the provided data.

Step 4: Evaluate Option 3

Substitute \( x = 1, 2, 3, 4, 5 \) into \( y=3 \sqrt{x-0.5} \). Calculate predicted values and observe that the results (approximately \( 2.1213, 3.6742, 4.7434, 5.6125, 6.3640 \)) closely match the data entries.

Step 5: Examine Option 4

Substitute \( x = 1, 2, 3, 4, 5 \) into \( y=3 x^{1 / 3}+1.1213 \) and calculate the predicted results. Realize these predictions do not match the data table closely.

Step 6: Conclusion

After evaluating all options, we find that Option 3 \( y=3 \sqrt{x-0.5} \) best describes the table of data, as its predicted values most closely match the given data.

Key Concepts

Patterns in Data

When examining data tables or sets of data, it is crucial to identify patterns. Patterns are tendencies or regularities that repeat over a set of data. They help us understand how two variables relate to each other.
For instance, by inspecting a data table, one can spot whether the association between variables is increasing or decreasing.
This primary identification helps us determine the type of mathematical equation that might fit the data best.

• Look for changes: As one variable increases, does the other consistently increase as well?
• Check for regularity: Are the changes between values of each variable steady or irregular?
• Increase specificity: Based on observed regularities, consider if the pattern suggests linear or non-linear relationships.

Recognizing these patterns allows for a more informed decision when choosing an appropriate model or equation to describe the data.

Non-Linear Relationships

A non-linear relationship means that the data does not form a straight line when graphed; instead, the graph could be a curve. These relationships are crucial because they capture more complex dynamics between variables.
In the example provided, the variable increases in a non-linear manner, hinting that a more sophisticated mathematical model, such as square root or cubic functions, may be suitable. Some key indicators of non-linear relationships include:

• Curbing in plotted points when represented graphically.
• Values not changing consistently - i.e., the rate of change itself changes.
• Use of functions like quadratic, cubic, or square root, as they offer more than straight-line fits.

Non-linear equations often better represent the data compared to simple linear ones. Being able to identify such relationships is fundamental to accurate data analysis.

Equation Substitution

Equation substitution is a technique used to test different equations by plugging in specific values from the dataset to see which equation most closely predicts or matches the data outcomes. This is especially useful when trying multiple equations to find the most suitable model for a given dataset.
In the provided exercise, each proposed equation is evaluated by inputting the values of the independent variable, typically denoted as \( x \). This results in calculated outcomes which are compared to the actual data points to determine consistency.
Steps to follow when substituting equations:

• Select equations to test.
• Substitute all given values of \( x \) into the selected equations.
• Compute and compare predicted \( y \) values with the observed values.
• Identify which equation delivers results closest to the observed data, therefore best fitting the patterns noted in the data tables.

Mastering equation substitution allows for efficient data modeling and better prediction of outcomes.

Data Analysis in Mathematics

Data analysis needs to be thorough and methodical. In mathematics, it involves critically looking at data to draw meaningful conclusions. This includes selecting and using mathematical models that best describe the relationships observed in the data.
For the original exercise, analyzing the data involved examining potential equations, performing substitutions, and evaluating the fit of each equation. It's important to ask:

• What is the nature of the data: linear or non-linear?
• Do the model's predictions align with the data points?
• Is there a mathematical explanation for any anomalies or deviations observed?

Analytical skills in mathematics are not just about finding the right model but understanding why the model is appropriate. This involves a cycle of testing, evaluating, and refining choices to better describe data trends and patterns.