Problem 3 - College Algebra with Modeling and Visualization

Question

Exercises \(3-6:\) A function \(f\) is given. Determine whether \(f\) models the data exactly or approximately. $$ f(x)=5 x-2 $$ $$ \begin{array}{ccccc} x & 1 & 2 & 3 & 4 \\ y & 3 & 8 & 13 & 18 \end{array} $$

Step-by-Step Solution

Answer

The function models the data exactly.

Step 1: Evaluate the Function at Each x Value

We have the function \( f(x) = 5x - 2 \). We will first evaluate this function at each of the given \( x \) values to calculate the corresponding \( y \) values and compare them with the data provided in the table.When \( x = 1 \), \( f(1) = 5(1) - 2 = 3 \).When \( x = 2 \), \( f(2) = 5(2) - 2 = 8 \).When \( x = 3 \), \( f(3) = 5(3) - 2 = 13 \).When \( x = 4 \), \( f(4) = 5(4) - 2 = 18 \).

Step 2: Compare Computed Values with Table Values

We will now compare the computed \( y \) values with those in the provided table:- For \( x = 1 \), computed \( y = 3 \) and table \( y = 3 \).- For \( x = 2 \), computed \( y = 8 \) and table \( y = 8 \).- For \( x = 3 \), computed \( y = 13 \) and table \( y = 13 \).- For \( x = 4 \), computed \( y = 18 \) and table \( y = 18 \).In each case, the computed \( y \) values from \( f(x) \) exactly match the \( y \) values from the table.

Step 3: Conclusion

Since the values computed using \( f(x) = 5x - 2 \) exactly match the values in the provided data table for each \( x \) value, we conclude that the function \( f \) models the data exactly.

Key Concepts

Linear Functions

A linear function is a mathematical equation that models a straight line when plotted on a graph. It's characterized by the formula \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
Linear functions are fundamental in algebra and are used to represent relationships with a constant rate of change.

• The **slope (m)** indicates how steep the line is and the direction it goes (upwards or downwards).
• The **y-intercept (b)** represents the point where the line crosses the y-axis.

Understanding linear functions helps in predicting outcomes and understanding trends in data, which is why it's crucial in many math and real-world applications. In this context, our function is \( f(x) = 5x - 2 \), where the slope is 5, meaning for each unit increase in \( x \), \( y \) increases by 5. The y-intercept \(-2\) indicates the starting value of \( y \) when \( x \) is zero.

Data Analysis

Data analysis involves evaluating data to uncover useful information that can aid decision-making. In this exercise, we use a linear function to analyze the data points given in a table.
This involves checking if our function correctly predicts the y-values for different x-values from the table.

• First, we apply the function to calculate hypothetical y-values.
• Then, we compare these values with those provided in the table to see if there's a match.

Through this process, we can determine how well the function represents the data. This is a basic form of data analysis where we assess the fit of a model to actual data, which is essential for statistical modeling and forecasts.

Exact Match

An exact match between a model and data indicates that the model perfectly predicts the outcomes represented by the data. In our context, when we say a function models data exactly, every computed y-value must match the corresponding y-value in the dataset without any discrepancies.
Here’s how we check for an exact match in our exercise:

• Compute the y-values using the function for each given x-value.
• Compare these computed values with actual y-values from the dataset.

In this exercise, the function \( f(x) = 5x - 2 \) provided y-values that were precisely identical to those in the data table for each x-value tested. Thus, the function models the data exactly, providing a perfect correspondence between the function's output and the actual data. This highlights the function's accuracy in representing the relationship within the dataset.