Problem 45 - Algebra 2

Question

Find the \(x\)-intercept and the \(y\)-intercept of the graph of each equation. Then graph the equation. \(x=1\)

Step-by-Step Solution

Answer

The \(x\)-intercept is at (1, 0); there is no \(y\)-intercept.

Step 1: Understanding the Problem

We need to find both the \(x\)-intercept and \(y\)-intercept for the equation \(x = 1\). Then, we'll graph it to visualize the intercepts.

Step 2: Identify the Equation Type

Recognize that the equation \(x = 1\) is a vertical line. Therefore, it only has an \(x\)-intercept and no \(y\)-intercept, as it does not cross the \(y\)-axis at any point.

Step 3: Finding the \(x\)-Intercept

The \(x\)-intercept is the point where the graph of the equation crosses the \(x\)-axis. For \(x = 1\), it crosses the \(x\)-axis at \((1, 0)\). This vertical line means it stays at \(x = 1\) across all values of \(y\).

Step 4: Understanding \(y\)-Intercept Absence

A\(x = 1\) vertical line indicates that it never crosses the \(y\)-axis. Therefore, there is no \(y\)-intercept for this equation.

Step 5: Graphing the Equation

To graph the equation \(x = 1\), draw a vertical line through the \(x\)-coordinate 1 on a coordinate plane. This line extends infinitely parallel to the \(y\)-axis, touching all points where \(x = 1\).

Key Concepts

Understanding the Y-Intercept

The y-intercept of an equation is the point where the graph crosses the y-axis. In other words, it's the value of y when x equals zero. It’s crucial to find the y-intercept as it provides insights into where the line begins in terms of vertical height. The y-intercept is represented as the point (0, y) on the coordinate plane. For most linear equations, you can identify the y-intercept by simply setting x to zero and solving for y.

However, for vertical lines like the equation x = 1, which is purely defined by a constant x-value without any dependency on y, it does not intersect the y-axis. This is because the condition for crossing the y-axis is when x equals zero, which is impossible for a line with a fixed x-coordinate such as x = 1.

So, vertical lines will always lack a y-intercept. This might seem perplexing, but it highlights the unique nature of vertical lines, emphasizing they stand straight up and down and never tilt or turn horizontally to interact with the y-axis.

Exploring Vertical Lines

Vertical lines are a special type of line in geometry where all points have the same x-coordinate. They extend infinitely in the up and down directions on the graph. The equation of a vertical line is always in the form x = a constant. In the example provided, x = 1 signifies that all points on this line will have an x-coordinate of 1, regardless of the y-value.

Key characteristics of vertical lines include:

• They are parallel to the y-axis and never intersect it, thereby lacking a y-intercept.
• Their slope is considered undefined because the change in x is zero, which would make dividing by this change impossible.
• They visually appear as straight up-and-down lines on a graph.

Understanding vertical lines is essential since they differ significantly from other types of lines which have both x and y-intercepts and a defined slope, making them an interesting exception in graphing techniques.

Basics of Graphing Equations

Graphing equations is a fundamental skill in mathematics, providing a visual representation of the relationship between variables. An equation is plotted by finding points on the graph that satisfy the equation, and then connecting these points to form a line or curve.

When graphing simpler equations like x = 1, the process is straightforward:

• Recognize the type of line (in this case, a vertical line).
• Identify key points like the x-intercept (here, (1,0)).
• Draw the vertical line through the x-coordinate, which will not change, indicating constant x-values.

By using these steps, graphing any equation becomes an organized task. It allows you to immediately see how the line behaves with respect to axes and other graphed elements. Accuracy in plotting is vital for interpreting mathematical relationships, especially in more complex scenarios beyond linear equations.