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Chapter 4: Problem 34

Write the equation in slope-intercept form. Then graph the equation. $$ y=-2 $$

Short Answer

Expert verified
The given equation is already in slope-intercept form with slope 0 and y-intercept -2. It represents a horizontal line crossing the y-axis at (0, -2).

Step by step solution

01

Recognize the equation's form

The given equation is \(y = -2\). This is already in slope-intercept form of a line, which is \(y = mx + b\) where m is the slope of the line and b is the y-intercept.
02

Determine the slope and y-intercept

Since there's no x-term in the equation, the slope m is 0 and the y-intercept is -2. This means the line is horizontal and crosses the y-axis at the point (0,-2).
03

Graph the equation

Sketch a horizontal line that passes through the y-axis at the point (0, -2). The line represents all the solutions to the equation.

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Key Concepts

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

horizontal line
A horizontal line is one of the simplest types of lines you can encounter in graphing. This type of line runs parallel to the x-axis and has a unique characteristic: it has a slope of zero.
The equation of a horizontal line can often be written in the form of \(y = c\), where \(c\) is a constant value.
This means that no matter which point on the line you look at, the y-value remains the same. For example, in the equation \(y = -2\), every point on this line has a y-coordinate of -2, such as (1, -2), (-3, -2), and (5, -2).
  • A horizontal line has an undefined x-intercept, except in a rare case where it lies along the x-axis.
  • The slope (\[m = 0\]) indicates it's neither rising nor falling.
  • It's always parallel to the x-axis regardless of its y-value.
Recognizing a horizontal line from its equation is straightforward when the equation lacks an x-term.
graphing linear equations
Graphing linear equations is an essential skill in algebra, helping you visualize the relationships between variables.
There are different forms in which a linear equation can appear, but a common one is the slope-intercept form, \(y = mx + b\). Here, \(m\) represents the slope, while \(b\) is the y-intercept.
When graphing such an equation, begin by plotting the y-intercept on the y-axis. This point is where the line crosses the y-axis, and its coordinates are (0, b).
  • For the equation \(y = -2\), the graph starts at (0, -2).
  • Use the slope, \(m\), to find additional points. However, with a horizontal line like \(y = -2\), the slope is 0, leading to no vertical change as you move along the x-axis.
  • Draw the line through these points, maintaining its parallel alignment with the x-axis.
Graphing provides a visual representation of all possible solutions to the equation, determined by infinite points along the line.
y-intercept
The y-intercept of a line is the point at which the line crosses the y-axis. It is a crucial element in the slope-intercept form of a linear equation, \(y = mx + b\), where \(b\) denotes the y-intercept.
For the equation \(y = -2\), the y-intercept is clearly \(-2\), showing that the graph intersects the y-axis at the point (0, -2).
  • The y-intercept is significant, especially for evaluating and sketching the graph of the equation.
  • It gives a definite starting point when drawing the line on a coordinate plane.
  • Identifying the y-intercept helps understand the vertical position of the line in relation to the origin.
  • A change in the y-intercept shifts the entire line up or down, without altering its slope.
Recognizing the y-intercept helps solidify the understanding of linear equations and their graphical depictions.

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Most popular questions from this chapter

Find the diffe$$ 5-9 $$rence.

Solve the equation if possible. $$ 3 n=5 n-12 $$

Find the difference. $$ -8-9 $$

Find the \(x\) -intercept and the \(y\) -intercept of the line. Graph the equation. Label the points where the line crosses the axes. $$ 0.5 y=-2 x+8 $$

Solve the equation if possible. $$ 3 b+8=9 b-7 $$
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