Problem 22 - College Algebra

Question

For the following exercises, determine whether each function is increasing or decreasing. $$ n(x)=-\frac{1}{3} x-2 $$

Step-by-Step Solution

Answer

The function is decreasing.

Step 1: Identify the Function Type

The given function is in the form of a linear equation. Linear equations are generally written in the form \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept.

Step 2: Determine the Slope

In the function \( n(x) = -\frac{1}{3} x - 2 \), the slope \( m \) is \(-\frac{1}{3}\). The slope tells us how the function behaves. A positive slope indicates an increasing function, while a negative slope indicates a decreasing function.

Step 3: Analyze the Slope

Since the slope of the function \( n(x) = -\frac{1}{3} x - 2 \) is negative, specifically \(-\frac{1}{3}\), this means that the function is decreasing. A negative slope means for every unit increase in \( x \), the value of \( n(x) \) decreases by a certain amount.

Key Concepts

Linear Equations

Linear equations are mathematical statements that describe the relationship between two variables, typically using the formula \( y = mx + b \). In this form, \( y \) represents the dependent variable, and \( x \) is the independent variable. The linear equation suggests a straight-line graph when plotted.
Linear equations are fundamental because they model real-world situations where one item changes at a constant rate with respect to another. This makes them incredibly useful in various fields such as physics, economics, and even everyday life situations.

• Simple structure: Only involves basic operations like addition, subtraction, and multiplication.
• Predictable behavior: The graph of a linear equation is always a straight line.

Understanding how to manipulate and interpret linear equations is crucial for analyzing trends and making predictions.

Slope

The slope of a line is a crucial concept in understanding linear equations. It indicates how steep the line is and how one variable changes in relation to the other. Mathematically, slope is represented by the letter \( m \) in the linear equation \( y = mx + b \).
Calculating slope involves determining the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. It is represented as:
\[ m = \frac{{\text{rise}}}{{\text{run}}} = \frac{{y_2 - y_1}}{{x_2 - x_1}} \]
A key point to remember is:

• A positive slope means the line is increasing, as \( x \) increases, \( y \) also increases.
• A negative slope means the line is decreasing, as \( x \) increases, \( y \) decreases.

The slope provides insight into how changes in one variable affect the other.

Function Behavior

Understanding the behavior of a function is essential when analyzing linear equations. The behavior can typically be described as either increasing, decreasing, or constant.
The slope of the linear equation plays a pivotal role in determining function behavior.

• If \( m > 0 \), the function is increasing — as \( x \) increases, \( y \) also increases, indicating a rising line when graphed.
• If \( m < 0 \), the function is decreasing — as \( x \) increases, \( y \) decreases, showing a falling line on a graph.
• If \( m = 0 \), the function is constant — \( y \) remains unchanged as \( x \) changes, resulting in a horizontal line.

For the function \( n(x) = -\frac{1}{3}x - 2 \), the negative slope of \(-\frac{1}{3}\) confirms that the function is decreasing. This means that for every unit increase in \( x \), the output \( n(x) \) decreases by \( \frac{1}{3} \). Understanding function behavior helps in predicting how variables interact.

Y-Intercept

The y-intercept is a vital component of the linear equation \( y = mx + b \). It is represented by \( b \) and signifies where the line crosses the y-axis. In practical terms, it shows the value of \( y \) when \( x \) is zero.
In our example, \( n(x) = -\frac{1}{3}x - 2 \), the y-intercept is \(-2\). This means that when \( x \) is zero, the value of the function is \(-2\).

• The y-intercept tells us the starting point of the graph on the y-axis.
• It gives a fixed point that helps in graphing the line quickly and accurately.

Working with the y-intercept in the context of the slope and function behavior enables a full understanding of the entire line and the relationship between the variables in the equation.