Problem 9 - College Algebra with Modeling and Visualization

Question

Exercises \(7-10:\) Find the formula for a linear function \(f\) that models the data in the table exactly. $$ \begin{array}{rrrr} x & 1 & 2 & 3 \\ f(x) & 7 & 9 & 11 \end{array} $$

Step-by-Step Solution

Answer

The linear function is \( f(x) = 2x + 5 \).

Step 1: Understand What is Given

We are given pairs of input \(x\) and output \(f(x)\) for a linear function, which typically follows the form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Use Points to Calculate Slope

Use two points from the table, say \((1,7)\) and \((2,9)\), to find the slope \(m\). The formula for the slope is \( m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} = \frac{9 - 7}{2 - 1} = 2 \).

Step 3: Use Slope to Formulate Equation

With the slope \(m = 2\) known, use one of the points, say \((1,7)\), to solve for \(b\) in \(f(x) = mx + b\), starting with \(7 = 2(1) + b \).

Step 4: Solve for the Y-intercept

Rearrange the equation \(7 = 2 + b\) to find \(b\). Subtract 2 from both sides to get \(b = 5\).

Step 5: Write the Linear Function

The linear function that models the data is \(f(x) = 2x + 5 \).

Key Concepts

Slope Calculation

When dealing with linear functions, finding the slope is a crucial first step. The slope of a line is what tells us how steep the line is, and it is represented by the letter \( m \) in the equation \( f(x) = mx + b \). In a set of data points, we can calculate the slope using the formula:

• \( m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \)

This formula comes from the "rise over run" concept, where the "rise" is the change in \( y \)-values, and the "run" is the change in \( x \)-values. For example, given the points \((1, 7)\) and \((2, 9)\), we apply these to the formula:

• \( m = \frac{9 - 7}{2 - 1} = \frac{2}{1} = 2 \)

Thus, the slope \( m \) is 2, indicating that for every unit increase in \( x \), \( f(x) \) increases by 2 units. Recognizing the slope helps in understanding the rate of change of the function.

Y-intercept Determination

The y-intercept is the point where the line crosses the y-axis, and it is represented by \( b \) in the equation \( f(x) = mx + b \). Determining the y-intercept involves substituting a known point from the table into the linear equation and solving for \( b \).
Using the slope \( m = 2 \) and a point, for example \((1, 7)\), we substitute

• \( 7 = 2(1) + b \)

Here, we get \( 7 = 2 + b \). By subtracting 2 from both sides, we solve for \( b \):

• \( b = 5 \)

This means that when \( x = 0 \), \( f(x) \) equals 5. The y-intercept provides a starting value for the function when no other variables affect the outcome.

Algebraic Modeling

Algebraic modeling involves creating equations that represent real-world situations. For a linear function, this involves using the slope and y-intercept to write an equation in the form \( f(x) = mx + b \). With our calculated slope \( m = 2 \) and y-intercept \( b = 5 \), we construct the linear function:

• \( f(x) = 2x + 5 \)

This equation models the relationship between \( x \) and \( f(x) \) as observed in the data table. By inputting different values for \( x \), we can predict corresponding \( f(x) \) values, allowing us to understand and anticipate how data behaves.
In practical terms, this model helps us see patterns and make forecasts based on linear trends, which are common in various fields like economics, science, and engineering. With algebraic modeling, a simple equation can yield powerful insights into data.