Problem 23 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether each function is increasing or decreasing. $$ m(x)=-\frac{3}{8} x+3 $$

Step-by-Step Solution

Answer

The function is decreasing.

Step 1: Identify the Slope

To determine whether the function is increasing or decreasing, we need to look at the slope of the linear function. In the function \( m(x) = -\frac{3}{8} x + 3 \), the slope is \( -\frac{3}{8} \).

Step 2: Interpret the Slope

If the slope is positive, the function is increasing; if the slope is negative, the function is decreasing. In this case, the slope \( -\frac{3}{8} \) is negative.

Step 3: Conclusion

Since the slope of the function \( m(x) = -\frac{3}{8} x + 3 \) is negative, we can conclude that the function is decreasing.

Key Concepts

Slope Interpretation

The slope of a linear function is a critical component in determining the behavior of the function, specifically whether it increases or decreases. In a linear equation of the form \( y = mx + b \), the \( m \) represents the slope.
The slope indicates how the function value changes as the input changes. Here’s what to keep in mind:

• A positive slope means the function increases, i.e., as \( x \) increases, \( y \) also increases.
• A negative slope indicates the function decreases, i.e., as \( x \) increases, \( y \) decreases.
• A slope of zero signifies a constant function, not increasing or decreasing.

For example, the function \( m(x) = -\frac{3}{8}x + 3 \) has a slope of \( -\frac{3}{8} \).
This negative slope means that for every unit increase in \( x \), the value of \( m(x) \) decreases by \( \frac{3}{8} \), resulting in a downward direction on a graph.

Increasing and Decreasing Functions

Understanding whether a function is increasing or decreasing helps to predict its behavior over different inputs.
To determine this, we focus on the slope of the function. Let's examine what this means:

• An increasing function is one where as \( x \) becomes larger, the output \( y \) continues to grow.
• Conversely, a decreasing function has outputs that shrink as \( x \) rises.
• If the function has a flat slope, its behavior remains constant regardless of changes in \( x \).

Consider the linear function \( m(x) = -\frac{3}{8}x + 3 \).
Since we determined earlier that the slope \( -\frac{3}{8} \) is negative, this function is decreasing.
It moves downwards, showing reduced values of \( y \) as \( x \) increases.

Linear Equations

Linear equations form straight lines when graphed and are defined by their standard form \( y = mx + b \).
Here:

• \( m \) is the slope of the line, dictating its steepness and direction.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

The simplicity of linear equations makes them easy to graph and analyze.
For the equation \( m(x) = -\frac{3}{8}x + 3 \), the graph would cross the y-axis at \( y = 3 \) and fall as \( x \) increases due to the negative slope.
Linear equations are useful in many real-life situations where constant rates of change are involved, like calculating speed or cost over time.
Understanding how to extract the slope and y-intercept from an equation enables you to quickly visualize the function's behavior and graph its path.