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Chapter 2: Problem 33

For each equation, find the slope. If the slope is undefined, state this. $$ f(x)=-2 $$

Short Answer

Expert verified
The slope is 0.

Step by step solution

01

Identify the Equation Type

The given function is in the form of a constant function: \( f(x) = -2 \).
02

Understand the Graph

For a constant function, the graph is a horizontal line. In this case, the line is horizontal at \( y = -2 \).
03

Define Slope of Horizontal Line

The slope of a horizontal line is always equal to zero.
04

State the Slope

Since the function \( f(x) = -2 \) represents a horizontal line, the slope is \( 0 \).

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Key Concepts

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

Constant Function
Understanding a constant function is crucial for finding the slope in certain types of equations. A constant function has the form: \( f(x) = c \), where \( c \) is a constant value that does not change.
This means no matter what value \( x \) takes, \( f(x) \) will always be \( c \).
For example, in the exercise, we have \( f(x) = -2 \).
This is a constant function because \( f(x) \) will always equal \( -2 \) regardless of \( x \) value.
Constant functions are essential in understanding horizontal lines and their slopes.
Horizontal Line
A horizontal line is the graphical representation of a constant function.
In general, a horizontal line can be represented by an equation of the form \( y = c \), where \( c \) is a constant.
In the context of the exercise, our horizontal line is \( y = -2 \).
This means that for every \( x \) coordinated along the x-axis, our y-value is always \( -2 \).
The characteristic feature of a horizontal line is that it runs parallel to the x-axis.
This feature will help us understand the important concept of slope when dealing with horizontal lines.
Slope of Zero
The slope is a measure of steepness or the rate at which a line rises or falls.
For a horizontal line, the slope is always \( 0 \).
To understand why, consider the formula for slope: \( m = \frac{螖y}{螖x} \), where \( 螖y \) is the change in y-coordinates, and \( 螖x \) is the change in x-coordinates.
In a horizontal line, the y-coordinates never change, so \( 螖y = 0 \). Thus, \( m = \frac{0}{螖x} = 0 \).
This is why the slope of a horizontal line representing a constant function is zero. In our example with \( f(x) = -2 \), the slope is correctly identified as zero.
Knowing this will help you easily identify and compute slopes for other horizontal lines you may encounter.

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Most popular questions from this chapter

Determine whether each equation is linear. Find the slope of any nonvertical lines. $$ 3 x+5 y+15=0 $$

Solve. One angle of a triangle measures twice the second angle. The third angle measures three times the second angle. Find the measures of the angles.

Write equations for two functions \(f\) and \(g\) such that the domain of \(f+g\) is \(\\{x | x \text { is a real number and } x \neq-2 \text { and } x \neq 5\\}\). Answers may vary.

Explain the difference between finding \(f(0)\) and finding the input \(x\) for which \(f(x)=0\)

Determine whether each equation is linear. Find the slope of any nonvertical lines. $$ \frac{y}{3}=x $$
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