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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 is decreasing.

Step by step solution

01

Identify the Function Type

The given function is a linear function of the form \( b(x) = mx + c \), where \( m \) represents the slope and \( c \) is the y-intercept. For \( b(x)=8-3x \), it can be rewritten as \( b(x) = -3x + 8 \).
02

Determine the Slope

The slope \( m \) of the function \( b(x) = -3x + 8 \) is represented by the coefficient of \( x \), which is \( -3 \).
03

Analyze the Slope

Since the slope \( m = -3 \) is negative, it indicates that the function is decreasing. In a linear function, a negative slope means that as \( x \) increases, \( b(x) \) decreases.

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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 creates a straight line when graphed. It is expressed in the general form of \( f(x) = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. The y-intercept is the point at which the line crosses the y-axis. For example, in the function \( b(x) = 8 - 3x \), we can rewrite it as \( b(x) = -3x + 8 \) to easily identify the slope and y-intercept. Here:
  • - The slope \( m \) is -3.
  • - The y-intercept \( c \) is 8.
Understanding the components of a linear function is essential as it helps us to predict the behavior of the function when graphed.
The Importance of Slope Analysis
The slope of a linear function is key in determining the direction and steepness of the line on a graph. In the function \( b(x) = -3x + 8 \), the slope \( m \) is -3. The slope tells us how much \( b(x) \) changes for every unit increase in \( x \).
  • If the slope is positive, the function is increasing, meaning that as \( x \) increases, \( b(x) \) also increases.
  • If the slope is negative, as in this case, the function is decreasing, so \( b(x) \) decreases with an increase in \( x \).
  • A zero slope means the function is constant and the line is horizontal.
Analyzing the slope is therefore essential for understanding how the function behaves as \( x \) changes.
Examining Function Behavior
The behavior of a function refers to how the output values change with respect to input values. When analyzing a linear function like \( b(x) = -3x + 8 \), understanding the behavior involves looking at the slope and intercept:
  • A negative slope tells us that the function is decreasing.
  • This means that as \( x \) increases, the output \( b(x) \) will decrease.
  • The y-intercept indicates the starting value on the y-axis when \( x = 0 \).
Knowing the function's slope helps predict whether the function will ascend or descend across the graph. Observing this behavior helps in graphing the function accurately, making it easier to visualize how changes in \( x \) affect \( b(x) \).

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Most popular questions from this chapter

Which of the following tables could represent a linear function? For each that could be linear, find a linear equation models the data. $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{g}(\boldsymbol{x}) \\ \hline 0 & 5 \\ \hline 5 & -10 \\ \hline 10 & -25 \\ \hline 15 & -40 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{h}(\boldsymbol{x}) \\ \hline 0 & 5 \\ \hline 5 & 30 \\ \hline 10 & 105 \\ \hline 15 & 230 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{f}(\boldsymbol{x}) \\ \hline 0 & -5 \\ \hline 5 & 20 \\ \hline 10 & 45 \\ \hline 15 & 70 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{k}(\boldsymbol{x}) \\ \hline 5 & 13 \\ \hline 10 & 28 \\ \hline 20 & 58 \\ \hline 25 & 73 \\ \hline \end{array} $$

Write an equation for a line parallel to \(g(x)=3 x-1\) and passing through the point (4,9)

Use algebra to find the point at which the line \(f(x)=-\frac{4}{5} x+\frac{274}{25}\) intersects the line \(h(x)=\frac{9}{4} x+\frac{73}{10}\)

A town's population has been growing linearly. In \(2003,\) the population was 45,000 , and the population has been growing by 1700 people each year. Write an equation, \(P(t),\) for the population \(t\) years after \(2003 .\)

Sketch a graph of each function $$ f(x)=3|x-2|-3 $$
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