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Chapter 1: Problem 14

Find the slope and the \(y\) -intercept (if possible) of the line. $$ y=-1 $$

Short Answer

Expert verified
The slope of the line is 0 and the y-intercept is -1.

Step by step solution

01

Identify the form of the equation

The equation is given in slope-intercept form, which is \(y = mx + c\) where \(m\) is slope and \(c\) is the y-intercept. But in this equation, it is given as \(y = -1\), which is the form of a horizontal line \(y = c\). There is no \(x\) term in the equation, so its coefficient (which would be the slope) is zero.
02

Determine the slope

In the equation \(y = -1\), the slope (m) equivalent is the coefficient of \(x\), but \(x\) is not present in the equation, so the slope is 0. This makes sense geometrically since a horizontal line does not rise or fall.
03

Identify the y-intercept

The y-intercept is defined as the point where the line intersects the \(y\)-axis. Since our equation does not contain an \(x\) value this simply means that for every \(x\), \(y\) will be -1. Thus, the y-intercept is -1.

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Key Concepts

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

Slope of a line
The slope of a line is a measure of how steep the line is. In mathematics, it is represented by the letter \( m \). When we talk about slope, we are essentially talking about the rate of change between any two points on a line.
You can think of it as "rise over run," which means how much the line goes up or down (rise) for each step it moves left or right (run). If a line moves upward from left to right, it has a positive slope. Conversely, if it moves downward from left to right, it has a negative slope.
  • A zero slope means the line is perfectly horizontal, which means there is no rise at all, only run.
  • An undefined slope corresponds to a vertical line, where there is rise without any run.

In the equation \( y = -1 \), the slope \( m \) is 0 because the line is completely horizontal. There's no change in its \( y \) value as \( x \) changes.
Y-intercept
The \( y \)-intercept of a line is the point where the line crosses the \( y \)-axis. In slope-intercept form, which is \( y = mx + c \), the \( y \)-intercept is denoted by \( c \). This value tells us exactly where the line will touch the vertical \( y \)-axis.
  • It is a specific point in the form of \((0, c)\), meaning the value of \( y \) when \( x = 0 \).
  • The \( y \)-intercept can be negative, positive, or even zero, depending on how the line is situated.

In the equation \( y = -1 \), the \( y \)-intercept is -1. This means the line crosses the \( y \)-axis exactly at the point \( (0, -1) \). This point serves as a starting reference for drawing the line.
Horizontal line equation
A horizontal line equation is unique because it is simply expressed as \( y = c \), where \( c \) is a constant. This tells us that no matter what the \( x \)-value is, the \( y \)-value remains constant.
The equation \( y = -1 \) is a perfect example of a horizontal line equation. This means:
  • Every point on the line has the same \( y \)-value, in this case, -1.
  • The slope of a horizontal line is 0 because there is no rise over the run.
  • The line runs parallel to the \( x \)-axis.

Horizontal lines are easy to identify and graph because they maintain the same elevation, hugging the coordinate plane's width at a fixed height.

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Most popular questions from this chapter

Explain why the function has a zero in the given interval. $$ \begin{array}{lll} \text { Function } & \text { Interval } \\ \hline f(x)=x^{2}-4 x+3 & {[2,4]} \\ \end{array} $$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f(x)=x^{n}\) where \(n\) is odd, then \(f^{-1}\) exists.

The signum function is defined by \(\operatorname{sgn}(x)=\left\\{\begin{array}{ll}-1, & x<0 \\ 0, & x=0 \\ 1, & x>0\end{array}\right.\) Sketch a graph of \(\operatorname{sgn}(x)\) and find the following (if possible). (a) \(\lim _{x \rightarrow 0^{-}} \operatorname{sgn}(x)\) (b) \(\lim _{x \rightarrow 0^{+}} \operatorname{sgn}(x)\) (c) \(\lim _{x \rightarrow 0} \operatorname{sgn}(x)\)

In your own words, explain the Squeeze Theorem.

Prove that if \(\lim _{x \rightarrow c} f(x)=0\) and \(|g(x)| \leq M\) for a fixed number \(M\) and all \(x \neq c,\) then \(\lim _{x \rightarrow c} f(x) g(x)=0\).
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