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Chapter 1: Problem 4

Find the slopes and \(y\)-intercepts of the following lines. $$y=6$$

Short Answer

Expert verified
Slope: 0, Y-intercept: 6

Step by step solution

01

Understanding the Equation

First, recognize that the equation is in the form of a horizontal line. The equation is written as \( y = 6 \), which means that for any value of \( x \), the value of \( y \) is always 6.
02

Identify the Slope

For a horizontal line, the slope is always 0. This means there is no rise over run, as the line does not go up or down.
03

Identify the Y-intercept

The y-intercept is the value of \( y \) when \( x \) is 0. Here, the equation \( y = 6 \) tells us that when \( x \) is any value, \( y \) is always 6. Hence, the y-intercept is 6.

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Key Concepts

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

slope
In the context of linear equations, the slope is a crucial concept. It indicates the steepness or incline of a line. Mathematically, slope is defined as the change in the y-coordinate divided by the change in the x-coordinate, often represented as 'rise over run'. It is typically denoted by the letter 'm' in the slope-intercept form of a linear equation, which is expressed as:\[y = mx + b\]Here, 'm' represents the slope, and 'b' represents the y-intercept.

The slope determines the direction of the line:
  • A positive slope means the line rises as you move from left to right.
  • A negative slope means it falls as you move from left to right.
  • A zero slope indicates a horizontal line, where there is no rise or fall, as illustrated in our example equation, \(y = 6\).
  • An undefined slope occurs in vertical lines, where the run is zero.
y-intercept
The y-intercept of a line is the point at which the line crosses the y-axis. This is a fundamental aspect of the slope-intercept form of a linear equation, \(y = mx + b\), where 'b' represents the y-intercept.

Understanding the y-intercept helps to graph equations quickly and interpret them more easily. To find the y-intercept, observe the value of y when x equals 0. For a horizontal line such as \( y = 6 \), regardless of the value of x, y remains constant at 6. Thus, the y-intercept is 6.

Key points to remember about y-intercepts include:
  • It is always expressed as a coordinate \((0, b)\).
  • It simplifies plotting a line, allowing you to mark a starting point on the graph before applying the slope.
  • Knowing the y-intercept is particularly useful in understanding real-world applications of linear equations, such as in economics and different branches of science.
horizontal line
A horizontal line is a straight line that runs parallel to the x-axis. In linear equations, it takes the form of \( y = c \), where 'c' is a constant. This form signifies that the value of 'y' stays the same, regardless of the x-value.

Important characteristics of horizontal lines include:
  • The slope is zero, as there is no vertical change.
  • The y-intercept is the value of 'c', which is the point where the line crosses the y-axis.
  • They represent situations where a particular variable remains constant.
In our example, \(y = 6\), the line is horizontal at y-value 6, meaning for any x-value, y remains 6. This simplicity can make understanding linear equations easier, especially when starting with basic concepts.

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Most popular questions from this chapter

Let \(y\) denote the percentage of the world population that is urban \(x\) years after 2014. According to data from the United Nations, 54 percent of the world's population was urban in 2014 , and projections show that this percentage will increase to 66 percent by 2050. Assume that \(y\) is a linear function of \(x\) since 2014. (a) Determine \(y\) as a function of \(x.\) (b) Interpret the slope as a rate of change. (c) Find the percentage of the world's population that is urban in 2020. (d) Determine the year in which \(72 \%\) of the world's population will be urban.

Estimate how much the function $$f(x)=\frac{1}{1+x^{2}}$$ will change if \(x\) decreases from 1 to .9.

Examine the graph of the function and evaluate the function-atlarge values of \(x\) to guess the value of the limit. $$\lim _{x \rightarrow \infty} \frac{-8 x^{2}+1}{x^{2}+1}$$

Use limits to compute \(f^{\prime}(x) .\) $$f(x)=-x+11$$

Let \(C(x)\) be the cost (in dollars) of manufacturing \(x\) bicycles per day in a certain factory. Interpret \(C(50)=5000\) and \(C^{\prime}(50)=45\).
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