Problem 88 - Precalculus: Functions and Graphs

Question

Choose the equation that best describes the table of data. (Hint: Make assignments to \(\mathbf{Y}_{\mathbf{r}}-\mathbf{Y}_{\mathbf{4}}\) and examine a table of their values.) $$\begin{array}{|c|c|}\hline x & y \\\\\hline 1 & 0.8 \\\\\hline 2 & -0.4 \\\\\hline 3 & -1.6 \\\\\hline 4 & -2.8 \\\\\hline 5 & -4.0 \\\\\hline\end{array}$$ (1) \(y=-1.2 x+2\) (2) \(y=-1.2 x^{2}+2\) (3) \(y=0.8 \sqrt{x}\) (4) \(y=x^{3 / 4}-0.2\)

Step-by-Step Solution

Answer

The equation is \( y = -1.2x + 2 \).

Step 1: Calculate Differences

First, we compute the differences \( Y_r - Y_4 \) for each value of \( y \), where \( Y_4 = -2.8 \). This gives us the differences as follows:- For \( x = 1 \), the difference is \( 0.8 - (-2.8) = 3.6 \).- For \( x = 2 \), the difference is \( -0.4 - (-2.8) = 2.4 \).- For \( x = 3 \), the difference is \( -1.6 - (-2.8) = 1.2 \).- For \( x = 4 \), the difference is \( -2.8 - (-2.8) = 0 \).- For \( x = 5 \), the difference is \( -4.0 - (-2.8) = -1.2 \).

Step 2: Evaluate Functions

Now, evaluate each function with the given \( x \) values to determine if they match the original \( y \) values.1. **For** \( y = -1.2x + 2 \): - At \( x = 1 \), \( y = -1.2(1) + 2 = 0.8 \). - At \( x = 2 \), \( y = -1.2(2) + 2 = -0.4 \). - At \( x = 3 \), \( y = -1.2(3) + 2 = -1.6 \). - At \( x = 4 \), \( y = -1.2(4) + 2 = -2.8 \). - At \( x = 5 \), \( y = -1.2(5) + 2 = -4.0 \). - This function matches all data pairs.2. **For** \( y = -1.2x^2 + 2 \), and other options: - Verify each similarly. - None provide the correct values as the linear function does.

Step 3: Conclusion

The evaluations show that only the function \( y = -1.2x + 2 \) accurately predicts the given \( y \) values for each corresponding \( x \) value in the table.Thus, the equation that best describes the table of data is option **(1)**.

Key Concepts

Data Analysis

Data analysis is a process of inspecting, cleaning, and modeling data with the goal of discovering useful information. For this particular exercise, the aim is to find an equation that accurately represents a set of data points. In our case, we are given a table that lists pairs of values for variables \(x\) and \(y\). To analyze the data, you start by observing the given \(x\) and \(y\) values. Notice how each \(y\) value changes as \(x\) increases. You can compute the differences or changes in \(y\) to get a clear picture of the pattern. Hence, the provided table helps break down the pattern by using differences \(Y_r - Y_4\). This is a crucial step in identifying which equation might describe your data points. Through this analysis, you lay the groundwork for testing various equations to find the best-fit function.

Functions

Functions are mathematical models that describe relationships between variables. In this exercise, you are dealing with potential equations (or functions) that might describe the relationship between \(x\) and \(y\). Each function gives a rule to calculate \(y\) given \(x\). Here are the candidate functions:

• Linear Function: \(y = -1.2x + 2\)
• Quadratic Function: \(y = -1.2x^2 + 2\)
• Square Root Function: \(y = 0.8\sqrt{x}\)
• Exponential Function: \(y = x^{3/4} - 0.2\)

Linear functions depict the simplest type of relationship, represented in the form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. In this context, examining how the values conform to each function guides the selection of the most accurate representation of the dataset. As seen, the linear function successfully predicts all \(y\) values at every \(x\) point in the table, confirming it as the ideal match.

Equations

Equations are vital tools in representing functions and analyzing data. They are defined mathematical statements that describe how values relate to one another.In this exercise, each proposed equation attempts to capture the pattern observed in the dataset by linking \(x\) and \(y\). Testing each equation involves substituting \(x\) values from your dataset into these functions to check if the results match the given \(y\) values.The correct equation, \(y = -1.2x + 2\), aligns perfectly with the data:

• For \(x = 1\), it results in \(y = 0.8\)
• For \(x = 2\), it results in \(y = -0.4\)
• For \(x = 3\), it results in \(y = -1.6\)
• For \(x = 4\), it results in \(y = -2.8\)
• For \(x = 5\), it results in \(y = -4.0\)

This match confirms the coherence and accuracy of the equation, demonstrating how equations are powerful in depicting real-world data effectively.