Problem 37 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the given equation. Label the intercepts. $$x=-4$$

Step-by-Step Solution

Answer

A vertical line passing through x = -4 with an intercept at (-4, 0).

Step 1: Identify the Type of Equation

The given equation is in the form of a vertical line, indicating that the value of x is always -4, regardless of the value of y.

Step 2: Determine the Intercepts

For a vertical line such as this, there is no y-intercept since the line doesn't cross the y-axis. The x-intercept occurs at the point where the line crosses the x-axis. Since x is always -4, the x-intercept is at \((-4, 0)\).

Step 3: Sketch the Graph

Draw a vertical line that passes through -4 on the x-axis. Extend this line upwards and downwards, showing that it runs infinitely in both directions.

Step 4: Label the Intercepts

Label the intercept on the graph. The x-intercept at \((-4, 0)\).

Key Concepts

graphing

Graphing is the process of plotting points or lines on a coordinate plane. It is an essential skill in mathematics because it allows you to visually represent equations and better understand their behaviors. In this exercise, you are asked to graph a vertical line defined by the equation \(x = -4\). Here are the steps involved in graphing this type of line:

First, identify that the given equation represents a vertical line. This means that every point on this line will have the x-coordinate of -4.
Next, understand that a vertical line has no y-intercept. The line does not cross the y-axis, only the x-axis.
The x-intercept can be found by looking at the given equation. Since x is always -4, the intercept on the x-axis is at (−4, 0).

Then, draw a straight, vertical line through the point (−4, 0). Extend this line infinitely in both directions for clarity. Make sure to label the intercept point for a complete graph.
By following these steps, you can accurately sketch the graph of vertical lines and interpret similar equations easily.

x-intercept

The x-intercept of a graph is the point where the line crosses the x-axis. This is an important concept because it tells us where the graph meets the horizontal axis. In this exercise, the x-intercept is central to understanding the vertical line defined by \(x = -4\).

For a vertical line, identifying the x-intercept is very straightforward.
Since the equation is \(x = -4\), it means that for any value of y, x will always be -4.
Thus, the line will cross the x-axis exactly at the point where x is -4, and y is 0.
To summarize, the x-intercept in this case is at (−4, 0). This point indicates that the line intersects the x-axis at −4. Understanding x-intercepts can help you quickly sketch and interpret the graph of any linear equation.

equation of a line

The equation of a line is a mathematical statement that describes all the points along that line. There are different forms an equation can take, each suited to different types of lines. In this exercise, the equation \(x = -4\) represents a vertical line.

Here is what you need to know about the equation \(x = c\), where c is a constant:
1. This type of equation is always vertical and parallel to the y-axis.
2. The value of x remains constant (in this case, -4) irrespective of the value of y.
3. Because the line is vertical, it does not have a slope. The concept of slope applies to non-vertical lines.
4. It also means it does not have a y-intercept because it never crosses the y-axis.

Understanding the equation of a line is vital in graphing and analyzing different types of lines, including vertical, horizontal, and slanted lines. By mastering this, you can easily deal with any linear equation you encounter.