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Chapter 3: Problem 49

Graph each equation. $$y=-2$$

Short Answer

Expert verified
The graph of the equation \( y = -2 \) is a horizontal line parallel to the x-axis crossing the y-axis at -2.

Step by step solution

01

Set up the coordinate system

First, set up a Cartesian coordinate system. Draw two perpendicular lines intersecting at a point called the origin. Label the horizontal line as \( x \) and the vertical line as \( y \). Ensure to mark the point -2 on the \( y \) line.
02

Identify the equation

Observe the equation to be graphed. Here, the equation \( y = -2 \) means that no matter what the value of \( x \) is, \( y \) will always be -2.
03

Graph the line

To graph the line, draw a straight horizontal line passing through the point where \( y \) equals -2. As a result, you should have a straight horizontal line parallel to the x-axis, passing through y=-2.

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Key Concepts

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

Understanding the Coordinate System
The coordinate system is a framework we use to locate points in a plane. It consists of two number lines, the x-axis (horizontal) and the y-axis (vertical), that intersect at a point called the origin, denoted as (0, 0). Every point in this system can be described by an ordered pair (x, y). This means for any point, you have a specific x-value that tells you how far along it is from the origin horizontally, and a y-value that tells you how far up or down it is vertically.

To set up a coordinate system yourself:
  • Draw two perpendicular lines on paper.
  • Label the horizontal line as the x-axis.
  • Label the vertical line as the y-axis.
  • Mark the intersection as the origin (0,0).
This system helps us easily graph equations and visualize the relationships between variables.
Graphing Horizontal Lines
A horizontal line in a graph has the same y-value for all x-values. This means it runs parallel to the x-axis. For instance, in the equation \( y = -2 \), no matter what x-values you choose, the y-value always remains at -2.

To graph a horizontal line:
  • Identify the constant y-value, which in this example is -2.
  • Draw a line that runs parallel to the x-axis, passing through the point where y is -2.
Horizontal lines are unique because their slope is 0. This signifies there is no vertical change as we move along the line.
Exploring the Cartesian Plane
The Cartesian plane, also known simply as the coordinate plane, is the area created by the intersection of the x-axis and y-axis. This plane is divided into four quadrants, each containing ordered pairs that describe specific locations on the plane.

The quadrants are typically labeled as:
  • Quadrant I: Here, both x and y values are positive.
  • Quadrant II: x-values are negative, and y-values are positive.
  • Quadrant III: Both x and y values are negative.
  • Quadrant IV: x-values are positive, and y-values are negative.
Understanding this layout helps us not only in graphing functions but also in comprehending their behaviors and relationships in different parts of the plane.

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. In each exercise, evaluate $$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$ for the given ordered pairs \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\). $$\left(x_{1}, y_{1}\right)=(1,3) ;\left(x_{2}, y_{2}\right)=(6,13)$$

If two lines are perpendicular, describe the relationship between their slopes.

Use a graphing utility to graph each equation in Exercises in a standard viewing rectangle, \([-10,10,1]\) by \([-10,10,1] .\) Then use the \([\text { TRACE }]\) feature to trace along the line and find the coordinates of two points. $$y=\frac{3}{4} x-2$$

Use a graphing utility to graph each equation. You will need to solve the equation for \(y\) before entering it. Use the graph displayed on the screen to identify the \(x\) -intercept and the \(y\) -intercept. $$3 x-y=9$$

Exercises \(82-84\) will help you prepare for the material covered in the next section. In each exercise, solve for \(y\) and put the equation in slope- intercept form. $$y-30.0=0.265(x-10)$$
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