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Chapter 2: Problem 21

For the following exercises, determine whether each function is increasing or decreasing. $$ j(x)=\frac{1}{2} x-3 $$

Short Answer

Expert verified
The function is increasing because the slope is positive.

Step by step solution

01

Understand the function

The function given is a linear function: \( j(x) = \frac{1}{2}x - 3 \). A linear function is generally expressed in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
02

Identify the Slope

The slope of the function \( j(x) = \frac{1}{2}x - 3 \) is \( \frac{1}{2} \). The slope, \( m \), in the form \( y = mx + b \), determines whether the function is increasing or decreasing.
03

Determine if the function is increasing or decreasing

If \( m > 0 \), the function is increasing. If \( m < 0 \), the function is decreasing. Since \( \frac{1}{2} > 0 \), the function \( j(x) = \frac{1}{2}x - 3 \) is increasing.

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Key Concepts

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

Slope Determination
When analyzing linear functions, understanding the slope is essential. The slope, represented by the variable \( m \), plays a crucial role in describing the function's behavior. For a linear function expressed as \( y = mx + b \), the slope \( m \) quantifies the steepness and direction of the line.

  • A positive slope \( m > 0 \) indicates that the function is increasing as it moves from left to right.
  • A negative slope \( m < 0 \) suggests the function is decreasing in the same direction.
  • A slope of zero \( m = 0 \) implies a horizontal line, where the value does not increase or decrease.

For instance, in the function \( j(x) = \frac{1}{2}x - 3 \), the slope \( m \) is \( \frac{1}{2} \). Since \( \frac{1}{2} > 0 \), the function is increasing.
Increasing and Decreasing Functions
Increasing and decreasing behavior of functions is primarily determined by the sign of the slope. By examining \( m \), you can quickly determine whether a function ascends or descends as you move along the x-axis.

  • An increasing function shows that as the input (x-values) rise, the output (y-values) also rises. This is the case when \( m > 0 \).
  • Conversely, a decreasing function means that as the input increases, the output decreases, occurring when \( m < 0 \).

For the function \( j(x) = \frac{1}{2}x - 3 \), we established that \( m = \frac{1}{2} \) is positive, thus confirming it as an increasing function.
Function Analysis
Function analysis involves examining various components and behaviors of a function to understand it better. In the case of a linear function like \( j(x) = \frac{1}{2}x - 3 \), key aspects include:
  • **Slope (\( m \)):** Dictates the rate of change or steepness and indicates whether the function is increasing or decreasing.
  • **Y-Intercept (\( b \)):** The point where the function crosses the y-axis. For \( j(x) \), \( b = -3 \) denotes the line crosses the y-axis at \( -3 \).
  • **General Equation (\( y = mx + b \)):** Helps in visualizing and understanding the graph's form and position.

By combining these insights, one can effectively discern and graph the function to visualize its increasing or decreasing nature and its position in the coordinate plane.

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Most popular questions from this chapter

For the following exercises, consider this scenario: The profit of a company decreased steadily over a ten-year span. The following ordered pairs shows dollars and the number of units sold in hundreds and the profit in thousands of over the tenyear span, (number of units sold, profit) for specific recorded years: $$(46,1,600), \quad(48,1,550), \quad(50,1,505), \quad(52,1,540), \quad(54,1,495).$$ Use linear regression to determine a function \(P\) where the profit in thousands of dollars depends on the number of units sold in hundreds.

Determine whether the following function is increasing or decreasing. \(f(x)=7 x+9\)

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 \((0,10)\)

Determine whether the following function is increasing or decreasing. \(f(x)=-2 x+5\)

Given below are descriptions of two lines. Find the slopes of Line 1 and Line 2. Is the pair of lines parallel, perpendicular, or neither? Line \(1 :\) Passes through \((-2,-6)\) and \((3,14)\) Line \(2 :\) Passes through \((2,6)\) and \((4,14)\)
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