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Chapter 2: Problem 24

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

Short Answer

Expert verified
The function is decreasing.

Step by step solution

01

Identify the Slope

The function given is \( m(x) = -\frac{3}{8}x + 3 \). This is a linear function in the form of \( mx + b \), where \( m \) is the slope. Here, the slope \( m = -\frac{3}{8} \).
02

Analyze the Slope

Since the slope \( m = -\frac{3}{8} \) is negative, this indicates that the function decreases. In a linear function, a negative slope always corresponds to a decreasing function, as the value of the function will diminish as \( x \) increases.

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Key Concepts

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

Slope
In the context of linear functions, the slope is a crucial concept. It indicates how steep a line is and in which direction it moves across the coordinate plane. A slope can be seen as the "rise over run," where it measures the change in the vertical direction ('rise') over the change in the horizontal direction ('run'). In the linear function formula \[ y = mx + b \] 's' represents 'y,' 'm' is the slope, and 'b' is the y-intercept. The slope \( m \) defines how much \( y \) increases or decreases as \( x \) increases by 1 unit. For example, a slope of \( \frac{1}{2} \) means that for each unit \( x \) increases, \( y \) increases by 0.5 units.
Increasing and Decreasing Functions
When looking at a linear function, understanding whether it is increasing or decreasing over its domain is important. The sign of the slope in a linear equation helps determine this.
  • If the slope \( m \) is positive, the function is increasing. This means as \( x \) gets larger, so does \( y \).
  • If the slope \( m \) is negative, the function is decreasing. Here, as \( x \) gets larger, \( y \) becomes smaller.
  • If the slope \( m \) is zero, the line is flat, and the function is constant across all values of \( x \).
The slope therefore directly influences whether the function rises, falls, or stays constant as you move along the x-axis.
Negative Slope
A negative slope specifically means the function decreases as you move from left to right on a graph. For the function in the exercise\[ m(x) = -\frac{3}{8}x + 3 \]the negative slope of \(-\frac{3}{8}\) indicates the following:
  • As \( x \) increases, the value of \( y \) decreases. This is consistent with any negative slope.
  • The line moves downward from left to right, showing a decrease in the output value of the function.
Understanding negative slope helps predict how the function behaves across its graph, which is an essential concept in analyzing linear functions.

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