Problem 40 - College Algebra with Modeling and Visualization

Question

Determine if \(f\) is a linear or nonlinear function. If \(f\) is a linear function, determine if \(f\) is a constant function. Support your answer by graphing \(f\). $$ f(x)=3 x-2 $$

Step-by-Step Solution

Answer

The function \( f(x) = 3x - 2 \) is a non-constant linear function.

Step 1: Identify Linear Function Characteristics

A linear function is of the form \( f(x) = mx + b \), where \( m \) and \( b \) are constants. If \( f(x) = mx + b \) with \( m eq 0 \), it is a non-constant linear function. If \( m = 0 \), \( f(x) \) is a constant function.

Step 2: Analyze the Given Function

The function provided is \( f(x) = 3x - 2 \). In this equation, \( m = 3 \) and \( b = -2 \). Since \( m eq 0 \), this is a linear function and not a constant function.

Step 3: Verify by Graphing

Graph the function \( f(x) = 3x - 2 \). The graph is a straight line, which confirms that \( f(x) \) is a linear function. The slope of the line is \( 3 \), indicating it is not constant as the output changes with \( x \).

Key Concepts

Constant Function

A constant function is a special type of linear function where the slope is zero. This means that no matter what value of \( x \) you use, the output, or \( f(x) \), remains the same. Essentially, graphing a constant function results in a horizontal line.

• **Equation Form**: For a constant function, the equation is of the form \( f(x) = b \), where \( b \) is a constant value.
• **Graph Representation**: The line is horizontal, indicating that there is no change in \( y \) as \( x \) increases or decreases.
• **Real-world Example**: Consider the function \( f(x) = 5 \). No matter the \( x \)-value, the output is always 5.

Constant functions have an important significance in mathematics, highlighting situations where changes in input do not affect the output. Understanding how they look both in the equation and on a graph is crucial for distinguishing them from other types of functions.

Graphing Linear Equations

Graphing linear equations involves plotting a straight line on a coordinate plane based on the linear equation. For any linear function in the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept, the graph will always be a straight line. To graph such a function:

• **Identify the Slope**: The coefficient of \( x \), \( m \), represents the slope, which indicates how steep the line is and the direction in which it slants.
• **Determine the Y-intercept**: This is the value \( b \) and it represents where the line crosses the y-axis.
• **Plot the Y-intercept**: Start by marking the point \( (0, b) \) on the graph.
• **Use the Slope**: From the y-intercept, count up or down based on the slope. For example, a slope of \( \frac{3}{1} \) tells you to go up three units for every one unit right.
• **Draw the Line**: Connect the points with a straight line that extends indefinitely in both directions.

Understanding how to graph these equations helps visualize the relationship between variables, and it is critical in identifying how changes in \( x \) affect \( f(x) \). The ability to graph correctly is foundational in analyzing linear behaviors in a variety of contexts.

Slope-Intercept Form

The slope-intercept form of a linear function is a way of writing the equation of a linear line so that you can easily identify the slope and the y-intercept. The general formula is \( y = mx + b \). Understanding the slope-intercept form is key to quickly sketching a graph or understanding a function's characteristics.

• **Slope (\( m \))**: This tells you the rise over run, or the angle of the line. A positive slope means the line rises as it moves from left to right, and a negative slope indicates it falls.
• **Y-intercept (\( b \))**: This is where the line crosses the y-axis. It gives you the starting point for the line when \( x = 0 \).
• **Application**: Given any equation, converting it to slope-intercept form allows you to immediately understand how changes in \( x \) will affect \( y \). For example, \( f(x) = 3x - 2 \), tells you the line has a slope of 3 and intersects the y-axis at -2.

By mastering the slope-intercept form, students can more easily handle tasks such as graphing lines, predicting function behavior, and solving real-world problems involving linear relationships.