Problem 43 - Algebra 2

Question

Find the \(x\)-intercept and the \(y\)-intercept of the graph of each equation. Then graph the equation. \(x=8\)

Step-by-Step Solution

Answer

The x-intercept is 8; there is no y-intercept. The graph is a vertical line through x = 8.

Step 1: Identify the Equation Type

The given equation is in the form of a vertical line, which can be identified as it contains only the variable x and is set equal to a constant, 8.

Step 2: Finding the x-intercept

For vertical lines, the x-intercept is simply the constant value provided in the equation. Here, the x-intercept is x = 8.

Step 3: Finding the y-intercept

Vertical lines do not have a y-intercept because they are parallel to the y-axis and do not cross it, so there is no y-intercept.

Step 4: Graphing the Vertical Line

On the Cartesian plane, draw a straight vertical line passing through x=8. This line will be parallel to the y-axis and extend infinitely in the positive and negative y-directions.

Key Concepts

x-intercept

The x-intercept is the point where a graph crosses the x-axis. In simpler terms, it's the value of x where the graph touches the horizontal line that represents zero on the y-axis. To find the x-intercept of an equation, you set the y-value to zero and solve for x. For vertical lines, like in our equation \(x = 8\), the x-intercept is the given constant value, which means the graph crosses the x-axis at the point (8, 0). This is because, on the x-axis, y is always zero. Remember, each point on the x-axis has the form (x, 0). In this case, the line reaches the x-intercept directly at the value x = 8.

y-intercept

The y-intercept is where the graph of an equation crosses the y-axis. This is the point where the value of x is zero. For most equations, you can find the y-intercept by setting x to zero and solving for y.

• For many linear equations, the y-intercept gives us a starting point to plot the graph.
• Vertical lines, however, do not have a y-intercept because they run parallel to the y-axis and never touch or cross it.

In the case of the equation \(x = 8\), the line doesn't intersect the y-axis at any point, so there's simply no y-intercept. If you picture this line on a graph, you'll see it standing tall along the line x = 8, never crossing the y-axis.

vertical line

A vertical line is a straight line that goes up and down on the graph, but not left or right. It's defined by an equation that looks like \(x = a\), where \(a\) is any constant number (like 8 in our exercise). Vertical lines have some specific features:

• They do not have a slope because their steepness is infinite.
• They cross the x-axis at exactly one point, which is \(x = a\).
• They never intersect the y-axis, which means no y-intercept.

On the graph, a vertical line through \(x = 8\) is represented as a line parallel to the y-axis. It extends infinitely in both the upward and downward directions. Understanding vertical lines is crucial for quickly identifying their intercepts and behavior in any given equation.

Cartesian plane

The Cartesian plane is a two-dimensional plane used in mathematics to draw and analyze graphs. It's named after the mathematician René Descartes, who popularized its use. The plane consists of two axes:

• The x-axis, which runs horizontally, representing the independent variable x.
• The y-axis, which runs vertically, representing the dependent variable y.

Where these axes intersect is the origin, designated by the point (0, 0). Each point on the plane is described by a pair of numbers \((x, y)\), known as coordinates, indicating its position relative to the origin.Visualizing our line \(x = 8\) on the Cartesian plane is straightforward. This line will be a vertical line located eight units to the right of the origin, illustrating how the Cartesian plane helps us understand the position and interaction of various graphs.