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Chapter 4: Problem 103

In the following exercises, graph each equation. . \(x=4\)

Short Answer

Expert verified
Graph a vertical line passing through x = 4.

Step by step solution

01

Identify the Type of Equation

The equation given is in the form of a vertical line. It specifies that for any value of y, x will always be 4.
02

Determine Points on the Graph

To graph the vertical line, choose several values for y (e.g., -2, 0, 3) and note that x will be 4 for all of these values. This gives points (4, -2), (4, 0), and (4, 3).
03

Plot the Points

On the coordinate plane, plot the points (4, -2), (4, 0), and (4, 3).
04

Draw the Line

Draw a straight vertical line through the points you plotted. This line represents the equation x = 4.

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Key Concepts

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

vertical lines
A vertical line in a coordinate plane occurs when the x-coordinate remains constant for all y-values. This means the equation of a vertical line will always be in the form of \(x = a\), where 'a' is a constant. In our example, the equation is \(x = 4\). This tells us that no matter what value y takes, x will always be 4.

To visualize a vertical line:

  • Pick any three points with the same x-coordinate.
  • For instance, select y-values like -2, 0, and 3.
  • The resulting points will be (4, -2), (4, 0), and (4, 3).
After selecting points, plot them on the graph. You'll notice they align vertically, making it easy to draw the line through them. This vertical alignment confirms that the equation \(x = 4\) dictates a vertical line on the coordinate plane.
coordinate plane
A coordinate plane is a two-dimensional plane where each point is defined by a pair of numerical coordinates. These coordinates are determined by the distance from two fixed perpendicular lines, typically known as the x-axis (horizontal) and y-axis (vertical).

In our exercise:

  • The x-axis represents all possible values of x.
  • The y-axis represents all possible values of y.
  • Where these axes intersect is called the origin, labeled as (0, 0).
Each point on this plane is written as (x, y). For example, (4, -2) means 4 units to the right from the y-axis and 2 units down from the x-axis. Understanding the coordinate plane is crucial for plotting points and graphing equations accurately.
plotting points
Plotting points on the coordinate plane is a fundamental step in graphing equations. To plot a point, you use an ordered pair \( (x, y) \) which tells you exactly where to place it on the plane.

Here鈥檚 how you can do it:

  • Start from the origin (0, 0).
  • Move horizontally to the x-coordinate.
  • From there, move vertically to the y-coordinate.
For example, to plot the point (4, 0), start at the origin. Move 4 units to the right (since x is 4) and 0 units up or down (because y is 0). The point (4, 0) lies exactly on the x-axis. Repeat this for other points like (4, -2) and (4, 3). Once the points are plotted, drawing the line through them will represent the given equation. This step-by-step approach ensures clarity and precision in graphing.

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Most popular questions from this chapter

The city mpg, \(x\), and highway mpg, \(y\), of two cars are given by the points (29,40) and (19,28) . Find a linear equation that models the relationship between city mpg and highway mpg.

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Find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope-intercept form. . line \(4 x+5 y=-3\), point (0,0)

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