Problem 4 - College Algebra with Modeling and Visualization

Question

Exercises \(3-6:\) A function \(f\) is given. Determine whether \(f\) models the data exactly or approximately. $$ f(x)=1-0.2 x $$ $$ \begin{array}{ccccc} x & 5 & 10 & 15 & 20 \\ y & 0 & -1 & -2 & -4 \end{array} $$

Step-by-Step Solution

Answer

The function models the data approximately, but not exactly.

Step 1: Substitute x-values into the Function

First, we will substitute the given values of \( x \) into the function \( f(x) = 1 - 0.2x \) to find the corresponding \( y \) values according to the function definition.

Step 2: Calculate f(x) for x=5

Substitute \( x = 5 \) into the function: \[ f(5) = 1 - 0.2 \times 5 = 1 - 1 = 0 \] Compare with the table value: \( y = 0 \). The calculated and table values match.

Step 3: Calculate f(x) for x=10

Substitute \( x = 10 \): \[ f(10) = 1 - 0.2 \times 10 = 1 - 2 = -1 \] Compare with the table value: \( y = -1 \). The calculated and table values match.

Step 4: Calculate f(x) for x=15

Substitute \( x = 15 \): \[ f(15) = 1 - 0.2 \times 15 = 1 - 3 = -2 \] Compare with the table value: \( y = -2 \). The calculated and table values match.

Step 5: Calculate f(x) for x=20

Substitute \( x = 20 \): \[ f(20) = 1 - 0.2 \times 20 = 1 - 4 = -3 \] Compare with the table value: \( y = -4 \). The calculated value does not match the table value, as \( f(20) = -3 \) but \( y = -4 \).

Step 6: Conclusion on Model Accuracy

Since all but one point from the table matches with the function, the function \( f(x) = 1 - 0.2x \) does not model the data exactly. There is a discrepancy at \( x = 20 \). Therefore, the function models the data approximately, but not exactly.

Key Concepts

Function Evaluation

To understand the relationship between the function and the given data, we perform function evaluation by plugging in the given x-values into the function equation. This is a crucial step for deriving corresponding y-values according to the function.

• Identify the function: Here, the function is given by \( f(x) = 1 - 0.2x \).
• Substitute known x-values: Using each x-value from the data, such as 5, 10, 15, and 20, we substitute these into the function one by one.
• Calculate y-values: After substitution, we compute for \( f(x) \) to see what values the function predicts.

Function evaluation helps us compare the derived y-values with those provided in the table to check for match or discrepancies. This comparison is key for making conclusions about the function’s accuracy in modeling the data.

Data Modeling

Data modeling involves using a mathematical function to represent the given data points. Here, we are assessing whether our function \( f(x) = 1 - 0.2x \) produces the same results as the data table.

• Exact model: If the function calculates y-values that match exactly with given data, it becomes an indicated perfect fit.
• Approximate model: If there’s a mismatch in some points while others align well, like in this case at \( x = 20 \), we recognize the function as approximate.

Analyzing how well the function fits the data allows us to decide if adjustments or new functions may be required for better modeling. Consistency with table values across multiple points increases our confidence in the model's precision.

Error Analysis

When evaluating a function that doesn't perfectly align with given data, conducting error analysis becomes necessary. Error analysis helps identify and measure discrepancies between expected and actual outcomes.

• Identify discrepancies: In our example, at \( x = 20 \), the calculated function value \( f(20) = -3 \) differs from the table value \( y = -4 \).
• Evaluate the impact: Even though one point doesn’t match, the overall pattern (trend) of accurate approximations may still be informative.
• Calculate and interpret errors: Quantify the size of the error (e.g., \( y_{error} = -4 - (-3) = -1 \)) to comprehend its severity.

Error analysis isn't merely about identifying single discrepancies but understanding the implications of deviations in the function, and deciding if adjustments are worthwhile for improved accuracy and reliability.