Problem 6 - College Algebra with Modeling and Visualization

Question

Exercises \(3-6:\) A function \(f\) is given. Determine whether \(f\) models the data exactly or approximately. $$ f(x)=13.3 x-6.1 $$ $$ \begin{array}{cccc} x & 1 & 2 & 5 \\ y & 7.2 & 20.5 & 60.4 \end{array} $$

Step-by-Step Solution

Answer

The function models the data exactly.

Step 1: Evaluate Function at Data Points

To check if the function models the data, we need to evaluate the function \( f(x) = 13.3x - 6.1 \) at each given \( x \) value from the dataset. Calculate \( f(1) \), \( f(2) \), and \( f(5) \).For \( x = 1 \):\[ f(1) = 13.3(1) - 6.1 = 7.2 \]For \( x = 2 \):\[ f(2) = 13.3(2) - 6.1 = 20.5 \]For \( x = 5 \):\[ f(5) = 13.3(5) - 6.1 = 60.4 \]

Step 2: Compare Results with Given Data

Compare the results calculated from the function \( f(x) \) with the \( y \) values in the data table.- For \( x = 1 \), \( f(1) = 7.2 \) matches with \( y = 7.2 \).- For \( x = 2 \), \( f(2) = 20.5 \) matches with \( y = 20.5 \).- For \( x = 5 \), \( f(5) = 60.4 \) matches with \( y = 60.4 \).

Step 3: Conclusion on Fit

Since the function values exactly match the given \( y \) values for all \( x \) in the table, the function \( f \) models the data exactly.

Key Concepts

Data Modeling

In mathematics, data modeling is the process of creating a model or representation of data within a given context. This allows us to understand how different variables relate to each other and find patterns within data sets. In this exercise, we are given a linear function and a set of data points. Our task is to determine if the function accurately models the data, either exactly or approximately. This involves assessing whether the function's outputs coincide with the actual data values.

To evaluate the data model, we

• Compute the function for each input (or x-value) from the dataset.
• Compare each result against the given data points (y-values).

An exact data model is where the function's output matches the actual measurement precisely. Our exercise required this exact matching to declare successful modeling. Understanding these core ideas simplifies determining how functions relate to data in mathematical and real-world applications.

Linear Equation

A linear equation is an equation of the first degree, meaning it has no variables raised to a power higher than one. These equations form a straight line when graphed. The general form of a linear equation is \[ y = mx + c \] where

• \(m\) represents the slope of the line
• \(c\) is the y-intercept (where the line crosses the y-axis)

In the exercise, we are given the linear function \[ f(x) = 13.3x - 6.1 \].
Here, 13.3 is the slope, indicating how much y changes with a unit increase in x. The y-intercept is -6.1, which is the value of the function when x equals zero. Understanding this format helps to interpret and solve linear equations easily and highlights their applicability in modeling relationships between data sets.

Function Evaluation

Evaluating a function involves calculating its output for specific input values. This crucial step is often employed to understand how well a function aligns with a given set of data, as demonstrated in our exercise. When evaluating the function \( f(x) = 13.3x - 6.1 \), we substitute each x-value from the data set into the function and simplify.

For example:

• Substitute \(x = 1\) to get \( f(1) = 13.3 \times 1 - 6.1 = 7.2 \).
• Substitute \(x = 2\) to get \( f(2) = 13.3 \times 2 - 6.1 = 20.5 \).
• Substitute \(x = 5\) to get \( f(5) = 13.3 \times 5 - 6.1 = 60.4 \).

The results show that the function's output matches the given y-values, confirming the function perfectly models the data set. Function evaluation reveals critical insights, making it an indispensable tool in analyzing mathematical relationships and data trends.