Problem 10 - 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. $$ \begin{array}{rrrr} x & 15 & 30 & 45 \\ f(x) & 40 & 30 & 20 \end{array} $$

Step-by-Step Solution

Answer

The linear function is \( f(x) = -\frac{2}{3}x + 50 \).

Step 1: Identify the form of a linear function

A linear function can be written in the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Calculate the slope (m)

To find the slope \( m \), use the formula \( m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \). Let's choose two points from the table, say \((15, 40)\) and \((30, 30)\). Plug these into the slope formula:\[m = \frac{30 - 40}{30 - 15} = \frac{-10}{15} = -\frac{2}{3}\]

Step 3: Calculate the y-intercept (b)

We can find the y-intercept \( b \) using one of the points and the slope we found. Let's use point \((15, 40)\). Substitute \( m = -\frac{2}{3} \), \( x = 15 \), and \( f(x) = 40 \) into the linear equation:\[40 = -\frac{2}{3} \cdot 15 + b\]Solve for \( b \):\[40 = -10 + b \b = 40 + 10 = 50\]

Step 4: Write the linear function

Substitute the slope \( m = -\frac{2}{3} \) and the y-intercept \( b = 50 \) into the linear equation:\( f(x) = -\frac{2}{3}x + 50\)

Step 5: Verify the linear function with remaining data points

Check the function \( f(x) = -\frac{2}{3}x + 50 \) with the other points from the table to ensure it fits all data:1. For \( x=45, f(x) = -\frac{2}{3} \cdot 45 + 50 = -30 + 50 = 20 \) which matches the table value.Confirmation means the function is correct.

Key Concepts

Slope Calculation

Slope is an important part of any linear equation. It's what tells us how steep a line is; imagine it's like the "tilt" of the line. To find the slope, mathematical language uses the letter \( m \).
This tells us how much \( y \) (the vertical line) changes for every change in \( x \) (the horizontal line).
If you have two points, like \((x_1, y_1)\) and \((x_2, y_2)\), you can find the slope with this easy formula:

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

You can try it out with real numbers like those in the exercise above. By picking points \((15, 40)\) and \((30, 30)\), the formula works out as:

• \( m = \frac{30 - 40}{30 - 15} = -\frac{10}{15} = -\frac{2}{3} \)

This tells us that for every 15 units \( x \) changes, \( y \) will decrease by 10 units.

Y-Intercept

The y-intercept is where our line crosses the y-axis. Think of it as the starting point of a line when \( x \) is zero.
In a linear equation \( f(x) = mx + b \), \( b \) is the y-intercept. It's like the starting value of the line before it starts to "tilt" by the amount of the slope.
To find \( b \), use one of the points from your data and the slope you calculated.
For example, with \((15, 40)\) and using the slope \(-\frac{2}{3}\), plug into the equation:

• \( 40 = -\frac{2}{3} \cdot 15 + b \)

Where you'll calculate:

• \( 40 = -10 + b \)

Adding 10 to both sides gives you \( b = 50 \).
This means the line starts at a value of 50 on the y-axis.

Linear Equation Form

Now that we have both the slope and the y-intercept, we can form the linear equation. A linear equation models the relationship between two variables with a straight line.
It's written as:

• \( f(x) = mx + b \)

Here, \( m \) and \( b \) are already known from the steps above. Thus, the equation becomes:

• \( f(x) = -\frac{2}{3}x + 50 \)

This equation describes how \( f(x) \) changes with \( x \) consistently. Whenever you have a new x-value, just drop it into \( f(x) \) and solve it to know what \( y \) should be.
Linear equations are simple but powerful as they precisely describe data with straight lines.

Data Modeling

Data modeling with a linear function involves using math to describe real-world data with an equation
By identifying patterns in the data, we can use a linear function to predict, explain, or understand relationships.
The linear equation we formed, \( f(x) = -\frac{2}{3}x + 50 \), professionally models the data from the table.
In this context, it tells us that as \( x \) increases by a certain amount, \( f(x) \) will decrease consistently based on the slope.
You can test this model by checking if the outcome matches other data points. For example, with \( x = 45 \):

• Calculate \( f(x) = -\frac{2}{3} \times 45 + 50 \)
• The answer is 20, which matches the given table, confirming the accuracy

Data modeling is a critical skill for predicting outcomes and making informed decisions based on trends we uncover in datasets.