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Chapter 2: Problem 19

Sketch the graph of each equation. $$ x=3 $$

Short Answer

Expert verified
It's a vertical line at \(x = 3\) on the Cartesian plane.

Step by step solution

01

Understand the Equation

The equation is given as \(x = 3\). This means that for all values of \(y\), \(x\) remains constant at \(3\). It represents a vertical line on the Cartesian plane where every point on the line has the x-coordinate equal to 3.
02

Identify Key Characteristics

This line will be parallel to the y-axis because it is defined such that for every y-value, the x-value is constant (\(x = 3\)).
03

Plot Points

To plot points, choose any values for \(y\) (e.g., \(y = -2, 0, 2\)) and pair them with \(x = 3\). Example points are \((3, -2)\), \((3, 0)\), and \((3, 2)\). All these points will lie on the vertical line \(x = 3\).
04

Draw the Line

Using a graph paper or a plotting tool, place a point at each of these coordinates from Step 3. Connect these points to form a vertical line. This line will extend infinitely in both the positive and negative direction along the y-axis.
05

Label the Graph

Label the vertical line as \(x = 3\) to ensure clarity, especially if there are multiple lines on the same graph.

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Key Concepts

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

Cartesian plane
The Cartesian plane is a two-dimensional surface defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). This system allows us to locate points using pairs of coordinates, indicating their positions related to these axes.

When you visualize the Cartesian plane, think of it as a kind of map grid.
  • The intersection point of the x-axis and y-axis is called the origin, denoted as \( (0, 0) \)
  • The plane is divided into four quadrants by the axes.
  • Each point on the plane is described by an ordered pair \( (x, y) \), where \( x \) is the horizontal distance from the origin, and \( y \) is the vertical distance.

By understanding and using this system, we can label locations precisely, making it a fundamental skill for graphing any mathematical function or equation, such as a vertical line like \( x = 3 \).
vertical line equation
A vertical line equation in the Cartesian plane is expressed in the form \( x = a \), where \( a \) is a specific constant that indicates the line's x-value for all y-values. This means:
  • No matter how far up or down you move along the y-axis, the x-coordinate remains the same at \( a \).
  • Vertically oriented lines do not have a slope, which distinguishes them from other lines that can be described by a slope-intercept form.
  • This line will always run parallel to the y-axis.

For example, the equation \( x = 3 \) describes a vertical line where every point on the line has \( x = 3 \). It stretches infinitely upwards and downwards on the plane, maintaining its parallel alignment to the y-axis. Visualizing vertical lines aids in the understanding of coordinate directions and relationships.
plot points on a graph
Plotting points on a graph involves marking specific locations determined by coordinates on the Cartesian plane. Here鈥檚 how you can plot the points for a vertical line like \( x = 3 \):
  • First, recognize that for this vertical line, \( x \) will always be \( 3 \), while you can choose any value for \( y \).
  • Pick several \( y \) values, such as \( -2, 0, \) and \( 2 \), and pair each with \( x = 3 \).
  • These pairs form the points \( (3, -2), (3, 0) \), and \( (3, 2) \).

Once you have a set of points, mark them on your graph at their corresponding locations. After plotting, connect the points. In this case, the points will form a vertical line, consistent with our equation \( x = 3 \). This graphical representation helps us see the structure and characteristics of various mathematical relationships.

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Most popular questions from this chapter

Determine if each function is increasing or decreasing $$ m(x)=-\frac{3}{8} x+3 $$

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