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Chapter 2: Problem 16

Determine if each function is increasing or decreasing $$ m(x)=-\frac{3}{8} x+3 $$

Short Answer

Expert verified
The function is decreasing, as the slope is negative.

Step by step solution

01

Identify the Function Type

The given function is in the form of a linear equation, specifically, it's given as \( m(x) = -\frac{3}{8} x + 3 \). In a linear function, the rate at which the function increases or decreases is determined by its slope.
02

Determine the Slope

In the linear function \( y = mx + b \), \( m \) represents the slope. Here, the function is \( m(x) = -\frac{3}{8} x + 3 \), so the slope \( m \) is \(-\frac{3}{8}\).
03

Analyze the Slope

The slope of the function \( m(x) = -\frac{3}{8} x + 3 \) is \(-\frac{3}{8}\). Since the slope is negative, the function is decreasing. A negative slope implies that as \( x \) increases, \( m(x) \) decreases.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope Analysis
When diving into linear functions, understanding the concept of slope is crucial. The slope is essentially the measure of how steep a line is. In a linear equation such as \( y = mx + b \), the term \( m \) represents the slope.
This value tells us how the function behaves as we move along the x-axis. Here's what you need to keep in mind about slope analysis:
  • If the slope \( m \) is positive, it means the line rises as it moves from left to right.
  • If the slope \( m \) is zero, the line is flat, indicating no increase or decrease as \( x \) changes.
  • If the slope \( m \) is negative, the line falls as it moves from left to right.
Slope analysis helps in predicting the direction of the line on a graph. For the given function \( m(x) = -\frac{3}{8}x + 3 \), the slope is \(-\frac{3}{8}\), clearly indicating a negative slope. Thus, the function decreases as \( x \) increases.
Increasing and Decreasing Functions
An understanding of increasing and decreasing functions allows us to predict the behavior of linear equations. Simply put:
  • An increasing function is one where the output or y-value grows as the input or x-value increases. This occurs when the slope \( m \) is greater than zero.
  • A decreasing function is one where the output or y-value decreases as the input or x-value increases. This happens when the slope \( m \) is less than zero.
In context, for the function \( m(x) = -\frac{3}{8}x + 3 \), because \( m = -\frac{3}{8} \), the function decreases. This characteristic of decreasing functions can often be seen graphically as a line descending from left to right.
Linear Equations
Linear equations are the backbone of many algebra problems and are fundamental in forming straight lines when graphed. A general linear equation can be written as \( y = mx + b \), where:
  • \( m \) is the slope, determining the inclination or steepness.
  • \( b \) is the y-intercept, where the line crosses the y-axis.
Linear equations depict a constant rate of change, which means for any two points on the line, the rate of change remains the same. This constant slope characteristic makes linear equations straightforward to analyze, as once the slope is known, the behavior of the entire line on a graph can easily be predicted.
In the exercise given, \( m(x) = -\frac{3}{8}x + 3 \), the equation clearly shows a linear formation with a negative slope, determining its behavior completely on a graph.

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