Problem 5 - College Algebra with Corequisite Support

Question

If a horizontal line has the equation \(f(x)=a\) and a vertical line has the equation \(x=a,\) what is the point of intersection? Explain why what you found is the point of intersection.

Step-by-Step Solution

Answer

The point of intersection is \((a, a)\).

Step 1: Understanding Horizontal Line

A horizontal line is of the form \( f(x) = a \). This means that for any value of \( x \), the value of \( f(x) \) (or \( y \) in terms of coordinates) is just \( a \). So every point on this line can be represented as \( (x, a) \).

Step 2: Understanding Vertical Line

A vertical line has the equation \( x = a \). This means that \( x \) is always \( a \), no matter what the value of \( f(x) \) or \( y \) is. Every point on this line can be represented as \( (a, y) \).

Step 3: Finding the Intersection

The point of intersection of a horizontal line and a vertical line would satisfy both equations simultaneously. That means we need the \( x \)-coordinate \( x = a \) from the vertical line and the \( y \)-coordinate \( y = a \) from the horizontal line.

Step 4: Combine the Coordinates

Since both lines have a constant coordinate \( a \) in their respective axes, the point of intersection of these two lines is where both coordinates are \( a \). Thus, the point of intersection is \( (a, a) \).

Step 5: Explanation of the Intersection Point

The horizontal line \( f(x) = a \) ensures that for any point on this line, \( y = a \). The vertical line \( x = a \) ensures that for any point on this line, \( x = a \). Hence, the only point that falls on both lines is \( (a, a) \), making it the correct point of intersection.

Key Concepts

Horizontal Line

A horizontal line is a line where all points have the same y-coordinate. This line is parallel to the x-axis. For a horizontal line represented by the equation \( f(x) = a \), the value of \( a \) is constant and determines the vertical position of the line on the graph. This means regardless of the value of \( x \), the y-coordinate will always be equal to \( a \).
For example, if \( a = 3 \), the line will pass through points like \((1, 3), (2, 3),\) and \((3, 3)\). The uniform y-coordinate across differing x-values indicates the horizontal span of the line.

• The equation \( f(x) = a \) means y is constant.
• Horizontal lines are always parallel to the x-axis.
• The slope of a horizontal line is 0, showing no vertical change as x increases.

Vertical Line

Vertical lines are defined by the equation \( x = a \). On these lines, the x-coordinate is constant and does not change regardless of the y-values. A vertical line runs parallel to the y-axis and essentially 'stands' tall on a graph.
For instance, if \( a = 2 \), the line will pass through points like \((2, 0), (2, 1),\) and \((2, 2)\). Each point has an identical x-coordinate, indicating the vertical consistency of the line.

• With \( x = a \), x remains unchanged.
• Vertical lines run parallel to the y-axis.
• The slope of a vertical line is undefined since there's no horizontal change.

Intersection Point

The concept of an intersection point is where two lines meet or cross each other on a graph. For our horizontal line \( f(x) = a \) and vertical line \( x = a \), the intersection occurs at the point \((a, a)\).
This is because the horizontal line dictates that anywhere on it, y equals \( a \), and the vertical line mandates that x equals \( a \). Therefore, the only combination that satisfies both equations simultaneously is \((a, a)\).

• Intersection points are where two graphs meet.
• Here, \((a, a)\) satisfies both \( y = a \) and \( x = a \).
• This point is crucial in determining overlap between the two lines.

Graphs of Equations

Graphs of equations give us a visual representation of mathematical relationships on a Cartesian plane. Each graph can uniquely depict the behavior of an equation depending on its structure.
For lines defined by simple linear equations like \( f(x) = a \) and \( x = a \), graphs can quickly convey their meaning through straightforward patterns.

• Horizontal and vertical lines are some of the simplest linear graphs.
• They illustrate constant values for y and x, respectively.
• Seeing graphs helps with understanding intersections and line behavior.

Understanding the visual aspects of the graph offers insight into how equations interact, such as where they might intersect or run parallel.