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Chapter 3: Problem 52

Find the intercepts and graph them. $$ x=-2 $$

Short Answer

Expert verified
The x-intercept is at (-2, 0); the line has no y-intercept.

Step by step solution

01

Understand the equation

The equation given is \( x = -2 \). This represents a vertical line where every point on the line has an \( x \)-coordinate of \(-2\).
02

Find the x-intercept

The x-intercept is the point where the line intersects the x-axis. For vertical lines like \( x = -2 \), this is exactly the point \((-2, 0)\) because it directly specifies \( x \).
03

Determine y-intercepts

Vertical lines parallel to the y-axis, such as \( x = -2 \), do not intersect the y-axis. Thus, there are no y-intercepts for this line.
04

Graph the line

To graph the line \( x = -2 \), draw a straight vertical line crossing the x-axis at \( x = -2 \). Ensure the line does not tilt; it remains entirely parallel to the y-axis across all y-values.

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Key Concepts

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

Understanding the X-Intercept
Let's dive into the concept of the x-intercept, a crucial part of graphing linear equations. The x-intercept is the point where a line touches the x-axis.
This means that the height, or the y-value, is zero at this point.

To find it, you substitute zero for the y-coordinate and solve the equation for x.
However, for the equation \(x = -2\), things are simpler, because x is already specified.
The x-intercept is the point
  • Location: Where the line meets the x-axis
  • Notation: \((a,0)\), where \(a\) is the x-coordinate
  • Example: If \(x = -2\), the x-intercept is at \((-2, 0)\)
Understanding the x-intercept helps us see where a line crosses the horizontal axis and is key to accurately graphing the line.
Grasping the Y-Intercept
The y-intercept is a point where the line intersects the y-axis, meaning the x-value on this point is zero.
It provides a starting reference when plotting the graph.
But, there's a twist when it comes to vertical lines like \(x = -2\).

Vertical lines are parallel to the y-axis and never cross it.
This means they have no y-intercept.
  • Location: Where a line crosses the y-axis
  • Notation: \((0, b)\), where \(b\) is the y-coordinate
  • Special Case: Vertical lines like \(x = -2\) have no y-intercept as they run parallel to the y-axis
Understanding this concept helps differentiate between various types of lines and their characteristics on a graph.
Recognizing a Vertical Line Graph
A vertical line graph is unique and easy to recognize.
These lines run straight up and down parallel to the y-axis and have a constant x-value.
For our example, the equation \(x = -2\) defines such a line.

Key characteristics include:
  • Equations: Always in the form \(x = \text{constant}\)
  • Graph Shape: A straight line parallel to the y-axis
  • Intercepts: An x-intercept exists at \(a, 0\) but no y-intercept
Graphing a vertical line involves drawing a line through the x-axis point, such as \(-2\), extending both upwards and downwards, keeping it straight.

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