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Chapter 2: Problem 10

Determine if each function is increasing or decreasing. $$ b(x)=8-3 x $$

Short Answer

Expert verified
The function \( b(x) = 8 - 3x \) is decreasing.

Step by step solution

01

Identify the Function Type

The function provided is a linear function of the form \( b(x) = mx + c \), where \( m \) is the coefficient of \( x \). In this function, \( b(x) = 8 - 3x \), \( m = -3 \).
02

Determine the Sign of the Slope

The slope of the function \( b(x) = 8 - 3x \) is \( -3 \). A slope tells us the direction of the line; here, it is negative.
03

Analyze the Slope

If the slope \( m \) is positive, the function is increasing. If \( m \) is negative, the function is decreasing. In this case, \( m = -3 \), which is negative.
04

Conclusion on the Function's Behavior

Since the slope is negative, the function \( b(x) = 8 - 3x \) is decreasing.

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Key Concepts

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

Understanding Linear Functions
A linear function is a type of function that graphs to a straight line. It follows the standard form \( y = mx + c \), where:\
    \
  • \( m \) is the slope of the line, determining its steepness and direction.
    • \
  • \( c \) is the y-intercept, where the line crosses the y-axis.
    • \
\The function given, \( b(x) = 8 - 3x \), can be seen as \( b(x) = -3x + 8 \). In this case, the slope \( m \) is \(-3\), and the y-intercept \( c \) is \(8\).
This linear equation will graph as a straight line on a coordinate plane, guiding you to evaluate its behavior easily by using the slope and intercept. Understanding these elements allows you to predict how the function behaves graphically and mathematically.
Slope Analysis: Direction and Steepness
The slope \( m \) of a linear function is crucial in determining how the line behaves. It reflects the line's tilt and placement on the graph.

Given the equation \( b(x) = 8 - 3x \):
  • A negative slope, like \(-3\), indicates the line moves downwards as you move from left to right.
  • The magnitude of the slope tells you the steepness. A larger absolute value implies a steeper slope, so \(-3\) is somewhat steep.
By understanding the slope, you can predict the line's orientation without graphing it. A negative slope clearly directs us to anticipate a decreasing function.
Increasing and Decreasing Functions
Function behavior鈥攚hether it is increasing or decreasing鈥攊s directly influenced by the slope of a linear function.

If the slope \( m \) is positive, the function is increasing, moving upwards as x increases. Conversely, a negative slope results in a decreasing function, indicating a downward movement on the graph.

Let's consider the function \( b(x) = 8 - 3x \):
  • Here, the slope is \(-3\), a negative value.
  • Therefore, the function is decreasing; as \( x \) increases, the value of \( b(x) \) decreases.
This interpretation allows for quick evaluations of linear functions, emphasizing the impact of slope on function direction and helping in understanding the existing behavior of the graph.

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