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Chapter 2: Problem 11

Determine if each function is increasing or decreasing $$ h(x)=-2 x+4 $$

Short Answer

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

Step by step solution

01

Identify the Type of Function

The function given is \( h(x) = -2x + 4 \). This is a linear function because it can be written in the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
02

Determine the Slope

For a linear function of the form \( f(x) = mx + b \), the coefficient of \( x \), which is \( m \), represents the slope of the line. In the function \( h(x) = -2x + 4 \), the slope \( m \) is \(-2\).
03

Analyze the Slope

The slope \( m = -2 \) is negative. For a linear function, if the slope is negative, the function is decreasing. Conversely, if the slope were positive, the function would be increasing.
04

Conclusion About the Function's Behavior

Since the slope of \( h(x) = -2x + 4 \) is \(-2\), the function is decreasing over its entire domain.

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Key Concepts

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

Understanding Linear Functions
Linear functions are the simplest type of functions characterized by a straight-line graph. They can be represented by the equation \( f(x) = mx + b \), where:
  • \( m \) is the slope of the line.
  • \( b \) is the y-intercept, the point where the line crosses the y-axis.
Linear functions are crucial in mathematics because they define a constant rate of change.
They are used to model many real-life scenarios such as distance over time, cost per item, and more.
Any time you see an equation of the form \( mx + b \), you're looking at a linear function. Understanding this base form helps to predict its graph and behavior.
The Importance of Slope Analysis
When dealing with linear functions, the slope \( m \) is a key determinant of the function's behavior.
It tells you about the direction and steepness of the line on the graph.
The slope is calculated as the "rise over run" between two points on the line, \[m = \frac{\text{change in y}}{\text{change in x}}\]In the function \( h(x) = -2x + 4 \), the slope is \(-2\).
This means for every 1 unit increase in \( x \), the value of \( h(x) \) decreases by 2 units.
  • A positive slope indicates an increasing function, where the graph goes upwards as you move from left to right.
  • A negative slope indicates a decreasing function, where the graph goes downwards as you move from left to right.
Being able to quickly determine the slope from a linear equation helps to predict how it behaves over its domain.
Deciphering Function Behavior
Function behavior tells us how a function behaves across its domain鈥攊n other words, over all possible \( x \) values.
With linear functions, this is mostly determined by the slope:
For \( h(x) = -2x + 4 \):
  • A negative slope of \(-2\) indicates that the function is decreasing.
  • This means as \( x \) increases, \( h(x) \) decreases, creating a downward slope on the graph.
  • The function consistently decreases at a uniform rate, reflecting the simplicity and predictability of linear functions.
In conclusion, understanding function behavior is crucial for visualizing and predicting the graph of the function.
This can aid in identifying solutions and analyzing real-world applications effectively.

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Most popular questions from this chapter

Determine if each function is increasing or decreasing $$ n(x)=-\frac{1}{3} x-2 $$

Find the slope of the line that passes through the two given points (1,5) and (4,11)

Find the value of \(m\) so the lines \(f(x)=m x+5\) and \(g(x)=x\) and the \(y\) -axis form a triangle with an area of 10 . [UW]

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Which of the following tables could represent a linear function? For each that could be linear, find a linear equation models the data. $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{g}(\boldsymbol{x}) \\ \hline 0 & 6 \\ \hline 2 & -19 \\ \hline 4 & -44 \\ \hline 6 & -69 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{h}(\boldsymbol{x}) \\ \hline 2 & 13 \\ \hline 4 & 23 \\ \hline 8 & 43 \\ \hline 10 & 53 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|} \hline x & f(x) \\ \hline 2 & -4 \\ \hline 4 & 16 \\ \hline 6 & 36 \\ \hline 8 & 56 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{k}(\boldsymbol{x}) \\ \hline 0 & 6 \\ \hline 2 & 31 \\ \hline 6 & 106 \\ \hline 8 & 231 \\ \hline \end{array} $$
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