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Chapter 4: Problem 18

For the following exercises, determine whether each function is increasing or decreasing. $$ h(x)=-2 x+4 $$

Short Answer

Expert verified
The function is decreasing.

Step by step solution

01

Determine the slope of the function

The function given is a linear function of the form \( h(x) = -2x + 4 \). In a linear function, the coefficient of \( x \) is the slope. Here, the slope is \( -2 \).
02

Analyze the slope

The sign of the slope determines whether the function is increasing or decreasing. If the slope is positive, the function is increasing. If the slope is negative, the function is decreasing. Since the slope here is \( -2 \), which is negative, the function is decreasing.

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Key Concepts

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

Understanding Slope
The concept of slope is essential when dealing with linear functions. The slope of a line represents the rate at which one quantity changes compared to another. In a linear equation like \( y = mx + b \), the slope is indicated by \( m \). This slope describes how steep the line is:
  • If the line is rising from left to right, the slope is positive.
  • If the line is falling from left to right, the slope is negative.
  • A horizontal line has a slope of 0, indicating no change.
For the exercise at hand, the function is \( h(x) = -2x + 4 \), and the slope is \(-2\). A slope of \(-2\) tells us that for every unit increase in \( x \), the value of \( h(x) \) decreases by 2 units.
Exploring Increasing and Decreasing Functions
When analyzing functions, knowing whether they are increasing or decreasing is crucial. This analysis helps us understand the behavior of the function over a given interval.

  • **Increasing Function:** The function rises as \( x \) values increase. This occurs when the slope is positive. For instance, a function \( f(x) = 3x + 1 \) has a positive slope of 3, and is therefore increasing.
  • **Decreasing Function:** The function falls as \( x \) values increase. This happens when the slope is negative. Our example, \( h(x) = -2x + 4 \), has a negative slope of \(-2\), indicating that it is decreasing.
Understanding whether a function is increasing or decreasing helps in predicting outcomes and drawing graphs of functions effectively.
Basics of Linear Equations
Linear equations are fundamental in algebra and appear in the form \( y = mx + b \). Here, \( m \) represents the slope and \( b \) is the y-intercept, where the line crosses the y-axis. These equations graph straight lines, offering a simple representation of relationships between variables.

Characteristics of linear equations include:
  • They produce straight lines on a graph.
  • The largest exponent of a variable is 1.
  • They have constant rates of change, meaning the slope value remains the same for any two points on a line.
Understanding linear equations as seen in \( h(x) = -2x + 4 \), allows us to easily determine the slope and the behavior of the function along its path. These equations are quickly solvable and provide insight into a diverse range of real-world situations where linear relationships are present.

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Most popular questions from this chapter

For the following exercises, consider this scenario: The number of people afflicted with the common cold in the winter months steadily decreased by 205 each year from 2005 until 2010 . In \(2005,12,005\) , \(12,025\) people were afflicted. If the function \(C\) is graphed, find and interpret the \(x\) -and \(y\) -intercepts.

A phone company charges for service according to the formula: \(C(n)=24+0.1 n,\) where \(n\) is the number of minutes talked, and \(C(n)\) is the monthly charge, in dollars. Find and interpret the rate of change and initial value.

For the following exercises, use the median home values in Mississippi and Hawaii (adjusted for inflation) shown in Table 2 . Assume that the house values are changing linearly. $$ \begin{array}{lll}{\text { Year }} & {\text { Mississippi }} & {\text { Hawaii }} \\ {1950} & {\$ 25,200} & {\$ 74,400} \\ {2000} & {\$ 71,400} & {\$ 272,700}\end{array} $$ If we assume the linear trend existed before 1950 and continues after 2000 , the two states' median house values will be (or were) equal in what year? (The answer might be absurd.)

For the following exercises, which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data. $$ \begin{array}{|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 2 & 4 & 6 \\ \hline \boldsymbol{g}(\boldsymbol{x}) & 6 & -19 & -44 & -69 \\ \hline \end{array} $$

For the following exercises, consider this scenario: The weight of a newborn is 7.5 pounds. The baby gained one-half pound a month for its first year. Find a reasonable domain and range for the function \(W .\)
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