Problem 5 - College Algebra with Modeling and Visualization

Question

Exercises \(3-6:\) A function \(f\) is given. Determine whether \(f\) models the data exactly or approximately. $$ f(x)=3.7-1.5 x $$ $$ \begin{array}{cccc} x & -6 & 0 & 1 \\ y & 12.7 & 3.7 & 2.1 \end{array} $$

Step-by-Step Solution

Answer

The function models the data approximately.

Step 1: Substitute the First x Value

Substitute the first given data point, \(x = -6\), into the function to see if \(f(x)\) equals the corresponding \(y\) value, \(12.7\). \[f(-6) = 3.7 - 1.5(-6) = 3.7 + 9 = 12.7\] Since \(f(-6) = 12.7\), this data point matches the function.

Step 2: Substitute the Second x Value

Substitute the second given data point, \(x = 0\), into the function to see if \(f(x)\) equals the corresponding \(y\) value, \(3.7\). \[f(0) = 3.7 - 1.5(0) = 3.7\] Since \(f(0) = 3.7\), this data point matches the function.

Step 3: Substitute the Third x Value

Substitute the third given data point, \(x = 1\), into the function to see if \(f(x)\) equals the corresponding \(y\) value, \(2.1\). \[f(1) = 3.7 - 1.5(1) = 3.7 - 1.5 = 2.2\] Since \(f(1) = 2.2\) but \( y = 2.1 \), this data point does not match exactly with the function.

Key Concepts

Linear Equations

Linear equations are a fundamental part of algebra and describe a straight line when graphed on a coordinate plane. They are typically expressed in the form \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept.
The function given in the exercise, \(f(x) = 3.7 - 1.5x\), fits this structure with a slope of \(-1.5\) and a y-intercept of \(3.7\). The slope \(-1.5\) indicates that for every unit increase in \(x\), the value of \(f(x)\) decreases by \(1.5\).
Understanding these components allows us to visually predict how changes in \(x\) will affect \(f(x)\) and relate it to real-world contexts, such as calculating costs or predicting trends.

Mathematical Modeling

Mathematical modeling involves using mathematical functions to represent real-world data. The goal is either to find patterns within the data or to make predictions. In the given exercise, the function \(f(x) = 3.7 - 1.5x\) is used to model the relationship between \(x\) and \(y\) values provided.
The process of verifying if a model is exact involves substituting each \(x\) value into the function and checking if it yields the corresponding \(y\) value. For the given data:

• For \(x = -6\), the result matches, indicating the model fits the data perfectly at this point.
• The same is true for \(x = 0\).
• However, at \(x = 1\), a small discrepancy exists, showing the model approximates rather than represents this portion of the data precisely.

Such analysis helps determine the effectiveness and reliability of mathematical models in capturing data trends.

Data Comparison

Data comparison is the process of assessing whether two sets of data are equivalent. In the context of this exercise, it involves comparing the function-generated values of \(y\) with the given \(y\) values. This comparison helps determine the accuracy of the function as a model for the data.
To verify if the model is exact, substitute each \(x\) value into the equation to calculate \(f(x)\) and then compare it to the corresponding given \(y\):

• For \(x = -6\), the values align perfectly, confirming a match.
• This alignment is consistent at \(x = 0\).
• For \(x = 1\), the function yields \(2.2\) instead of the actual \(2.1\), highlighting a slight mismatch.

This assessment is crucial when you want to use mathematical models effectively in real-world applications, where precise data alignment can be essential.