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Chapter 2: Problem 17

For the following exercises, determine whether each function is increasing or decreasing. $$ a(x)=5-2 x $$

Short Answer

Expert verified
The function is decreasing.

Step by step solution

01

Identify the Type of Function

First, identify the function type. The function given, \( a(x) = 5 - 2x \), is a linear function because it can be expressed in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
02

Determine the Slope

Next, determine the slope \( m \) of the function. In the expression \( a(x) = 5 - 2x \), the slope \( m \) is \( -2 \).
03

Analyze the Slope

Examine the sign of the slope. A function is increasing if the slope \( m > 0 \) and decreasing if \( m < 0 \). Since the slope \( m = -2 \), which is less than 0, the function is decreasing.

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Key Concepts

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

Increasing Functions
Increasing functions are functions where the output, or dependent variable, grows as the input, or independent variable, increases. In simpler terms, as you move from left to right along the x-axis, the line or curve rises.
Here鈥檚 what you should know about increasing functions:
  • **Positive Slope**: For a linear function described by the equation \( y = mx + b \), the function is increasing if the slope \( m \) is greater than zero, \( m > 0 \).
  • **Graph Behavior**: Graphically, increasing functions will have a line or curve that moves upwards as it progresses from left to right.
  • **Real-World Examples**: Consider scenarios like a business's profits over time or a distance traveled over time if the speed is consistent.
Understanding the nature of increasing functions can help you quickly analyze real-world situations and mathematical problems that describe growth or ascent.
Decreasing Functions
Decreasing functions are a type of function where the output diminishes as the input increases. Simply put, as you travel from left to right across the x-axis, the graph will slope downwards.
Let鈥檚 dig into the characteristics of decreasing functions:
  • **Negative Slope**: Like in the example \( a(x) = 5 - 2x \), a function is decreasing if the slope \( m \) is negative, \( m < 0 \). For this example, the slope \( m = -2 \), which means the line descends as you move rightward.
  • **Graph Dynamics**: Visually, if you plot the function on a graph, a decreasing function will slope downwards, indicating a drop as the x-value increases.
  • **Common Scenarios**: Decreasing functions can represent various scenarios, such as decreasing temperatures throughout the day or reduced quantities of a product in a store over time.
Decreasing functions are crucial for understanding situations displaying decline or reduction, whether in real-life events or abstract mathematical concepts.
Slope of a Line
The slope of a line is a crucial concept in understanding how lines behave on a graph. It tells us about the line's direction and steepness.
Here's what you need to know about slopes:
  • **Definition**: In a linear equation \( y = mx + b \), the \( m \) represents the slope. It indicates the change in \( y \) (vertical change) for a unit change in \( x \) (horizontal change).
  • **Positive vs. Negative**: A positive slope means the line rises, and a negative slope means the line falls. Zero also plays a role as it suggests a perfectly horizontal line, reflecting no change in \( y \) regardless of \( x \) changes.
  • **Calculation**: The slope can be calculated as \( \frac{\Delta y}{\Delta x} \), which stands for the change in \( y \) divided by the change in \( x \). In simpler terms, it's how much the line goes up or down for each step to the right.
  • **Interpretation**: Understanding the slope helps in predicting and analyzing trends, efficiencies, and rates of change. It's fundamental in both academia and practical applications such as physics (velocity) and economics (cost functions).
Grasping the concept of slope is like holding a key to unlocking the behavior of straight lines. This knowledge will enable you to interpret and predict various models effectively.

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Most popular questions from this chapter

A phone company has a monthly cellular plan where a customer pays a flat monthly fee and then a certain amount of money per minute used on the phone. If a customer uses 410 minutes, the monthly cost will be \(\$ 71.50 .\) If the customer uses 720 minutes, the monthly cost will be \(\$ 118\) . a. Find a linear equation for the monthly cost of the cell plan as a function of \(x\) , the number of monthly minutes used. b. Interpret the slope and \(y\) -intercept of the equation. c. Use your equation to find the total monthly cost if 687 minutes are used.

Does the following table represent a linear function? If so, find the linear equation that models the data. $$\begin{array}{|c|c|c|c|c|c|}\hline x & {6} & {8} & {12} & {26} \\ \hline g(x) & {-8} & {-12} & {-18} & {-46} \\ \hline\end{array}$$

Given each set of information, find a linear equation that satisfies the given conditions, if possible. \(x\) -intercept at \((6,0)\) and \(y\) -intercept at \((0,10)\)

Suppose that average annual income (in dollars) for the years 1990 through 1990 through 1999 is given by the linear function: \(I(x)=1054 x+23,286,\) where \(x\) is the number of years after \(1990 .\) Which of the following interprets the slope in the context of the problem? a. As of 1990 , average annual income was \(\$ 23,286\) . b. In the ten-year period from \(1990-1999\) , average annual income increased by a total of \(\$ 1,054\) . c. Each year in the decade of the 1990 s, average annual increased by \(\$ 1,054\) . d. Average annual income rose to a level of \(\$ 23,286\) by the end of 1999 .

Determine whether the function is increasing or decreasing. $$f(x)=7 x-2$$
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