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Chapter 1: Problem 66

Find the slope and \(y\)-intercept of the line, and draw its graph. \(y=-2\)

Short Answer

Expert verified
Slope: 0, y-intercept: -2. The graph is a horizontal line at y = -2.

Step by step solution

01

Identify the equation type

The given equation is \(y = -2\). This is a special type of linear equation known as a constant equation. In this type of equation, the value of \(y\) is constant and doesn't depend on \(x\).
02

Determine the slope

In a constant equation like \(y = -2\), the slope is 0. This is because the line is horizontal, and there is no change in \(y\) as \(x\) changes. In general, if the equation of a line is \(y = b\), then the slope \(m = 0\).
03

Determine the y-intercept

The \(y\)-intercept of a line \(y = b\) is simply the value of \(b\). Here, \(b = -2\). This means the line crosses the \(y\)-axis at \(y = -2\).
04

Sketch the graph

To graph \(y = -2\), draw a horizontal line that passes through \(y = -2\) on the \(y\)-axis. This line is parallel to the \(x\)-axis and doesn't intersect it at any point other than at \(y = -2\).

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Key Concepts

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

Slope of a Line
In geometry, the slope of a line is a measure of its steepness and direction. It tells us how much the line goes up or down as we move from left to right along the x-axis. To compute the slope, you often use two points on the line
  • Calculate the change in y-values.
  • Calculate the change in x-values.
  • Divide the change in y-values by the change in x-values, \( m = \frac{{\text{{change in }} y}}{{\text{{change in }} x}} \).
However, in some cases, like with constant equations such as \( y = -2 \), the slope is zero. This is because the line is horizontal and does not rise or fall as it extends along the x-axis. Horizontal lines have no vertical change, leading to no steepness.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. It indicates the value of \( y \) when \( x \) is 0. For any linear equation in the form \( y = mx + b \), the y-intercept is the constant \( b \). In the specific case of our equation \( y = -2 \), this implies the y-intercept is \( -2 \). This means if you graph this equation, the line intersects the y-axis at the point \( (0, -2) \). Understanding the y-intercept helps to quickly identify how the graph of the equation begins at the y-axis, no matter what the value of \( x \) might be.
Graphing Linear Equations
Graphing linear equations involves representing a linear relationship on a two-dimensional plane. In a typical linear equation such as \( y = mx + b \):
  • \( m \) represents the slope of the line, indicating its steepness.
  • \( b \) is the y-intercept, showing where the line crosses the y-axis.
For the equation \( y = -2 \), the process of graphing is straightforward:
  • Draw a horizontal line across the graph.
  • Ensure this line crosses the y-axis at the point \( y = -2 \).
This graphing technique shows how simple it can be to visualize horizontal lines, where the line is parallel to the x-axis and lacks any slope. With such constant equations, you can predict its graph by recognizing it will always yield a flat, horizontal line passing through the given y-intercept.

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Most popular questions from this chapter

Complete the following tables. What happens to the \(n\) th root of 2 as \(n\) gets large? What about the \(n\) th root of \(\frac{1}{2} ?\) $$\begin{array}{|r|r|}\hline n & 2^{1 / n} \\\\\hline 1 & \\\2 & \\\5 & \\\10 & \\\100 & \\\\\hline\end{array}$$ $$\begin{array}{|r|r|} \hline n & \left(\frac{1}{2}\right)^{1 / n} \\\\\hline 1 & \\\2 & \\\5 & \\\10 & \\\100 & \\\\\hline\end{array}$$ Construct a similar table for \(n^{1 / n}\). What happens to the \(n\) th root of \(n\) as \(n\) gets large?

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If \(a

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Use scientific notation, the Laws of Exponents, and a calculator to perform the indicated operations. State your answer rounded to the number of significant digits indicated by the given data. $$\frac{1.295643 \times 10^{9}}{\left(3.610 \times 10^{-17}\right)\left(2.511 \times 10^{6}\right)}$$
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