Problem 37 - College Algebra with Modeling and Visualization

Question

Exercises \(33-38:\) Write a formula for a linear function f whose graph satisfies the conditions. Slope 0.5, passing through \((1,4.5)\)

Step-by-Step Solution

Answer

The linear function is \( f(x) = 0.5x + 4 \).

Step 1: Understand the Linear Function Formula

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

Step 2: Identify Given Information

From the problem, we are given that the slope \( m = 0.5 \) and the line passes through the point \((1, 4.5)\).

Step 3: Substitute the Slope Into the Formula

We first substitute the slope into the formula, giving us \( f(x) = 0.5x + b \).

Step 4: Solve for the Y-Intercept (b)

Use the point \((1, 4.5)\) to find \( b \). Substitute \( x = 1 \) and \( f(x) = 4.5 \) into the equation: \[ 4.5 = 0.5 imes 1 + b \]Solving this gives:\[ 4.5 = 0.5 + b \]\[ b = 4.5 - 0.5 = 4 \]

Step 5: Write the Final Linear Function

Now with \( b = 4 \), we can write the linear function as:\[ f(x) = 0.5x + 4 \]

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of expressing linear equations using a simple and standardized formula. It's written as \( f(x) = mx + b \). Here, \( m \) represents the slope of the line, which indicates how steep the line is. The slope tells us how much the value of \( y \) changes when \( x \) increases by one unit.
The \( b \) in the formula signifies the y-intercept. The y-intercept is the point where the line crosses the y-axis. This point is vital because it gives us a starting position when sketching the graph of the line.
This form is beneficial because it provides direct insight into the characteristics of a line. It easily shows both how steep the line is and exactly where it begins on the y-axis. It is often used when quickly sketching graphs or analyzing linear equations.

Graphing Linear Equations

Graphing linear equations can be straightforward once you understand the slope-intercept form \( f(x) = mx + b \). The formula gives us all the information needed to plot a line accurately.
First, identify the y-intercept \( b \). This point is plotted directly on the y-axis. From this starting point, use the slope \( m \) to determine the next point. The slope is a ratio often expressed as a fraction \( \frac{rise}{run} \).

• If \( m = 0.5 \), interpret this as \( \frac{1}{2} \). For every 2 units you move along the x-axis, go up 1 unit on the y-axis.
• Plot the second point using this method.

After you have at least two points, draw a straight line through them. This line is the graph of your linear equation, extending infinitely in both directions. Graphing provides a visual representation of relationships in the equation, helping to visualize the effect of changing any part of it.

Solving for Y-Intercept

Solving for the y-intercept is an important step in determining the full equation of a line. When you have the slope \( m \) and a single point such as \((1, 4.5)\) through which the line passes, the y-intercept \( b \) can be determined using the formula \( f(x) = mx + b \).
To find \( b \), substitute the given point into the equation. Place \( x = 1 \) and \( f(x) = 4.5 \) into the equation: \[ 4.5 = 0.5 \times 1 + b \]
From here, solve the equation by isolating \( b \).

• Calculate \( 0.5 \times 1 \), which equals 0.5.
• Subtract 0.5 from 4.5 to solve for \( b \).
• Therefore, \( b = 4 \).

With this, you now know that the line intersects the y-axis at \( 4 \), and you have all the information required to write the complete equation \( f(x) = 0.5x + 4 \). Solving for the y-intercept gives the final piece needed to describe the line and allows for easy graphing of the equation.