Problem 40 - Algebra A Combined Function

Question

Graph each linear equation. See Examples 4 through \(7 .\) \(y=2 \frac{1}{2}\)

Step-by-Step Solution

Answer

Graph a horizontal line at \(y = 2.5\).

Step 1: Identify the Equation Type

The given equation is a linear equation in the form of a horizontal line since it is only composed of the constant term, which is the y-coordinate.

Step 2: Interpret the Equation

The equation is written as \(y = 2\frac{1}{2}\). This means for every point on the line, the y-value is \(2\frac{1}{2}\), or equivalently, \(y = 2.5\).

Step 3: Determine Constant Y-value

Since the equation states \(y = 2.5\), every point on the line will have the y-coordinate of 2.5. Thus, the line is horizontal and parallel to the x-axis.

Step 4: Graph the Line

To graph the equation, draw a horizontal line across the graph at the y-coordinate \(2.5\). This line will extend infinitely in both the left and right directions across the graph.

Key Concepts

Horizontal Line

A horizontal line is a straight line that runs from left to right or right to left across the coordinate plane. Unlike other lines that might have a tilted slope, horizontal lines have a slope of zero. This means they do not rise or fall as they move across the graph, and thus remain completely level.
To identify a horizontal line when looking at an equation, you can check if the equation is in the form of \(y = c\), where \(c\) is a constant number. This defines a line where all y-values along the line are equal, which visually appears as a flat line across the graph.

Y-Coordinate

The y-coordinate is one part of an ordered pair \((x, y)\) that specifies the position of a point on the Cartesian plane. It tells you how far up or down a point is from the x-axis, which is the horizontal axis.

• In the equation of a horizontal line, such as \(y = 2.5\), the y-coordinate is the constant value of 2.5 for every point on the line.
• This means that no matter the x-value, the position vertically never changes; it remains at 2.5.
• This is a key characteristic of horizontal lines, where the y-coordinate is consistently the same.

Constant Term

In the context of linear equations, the constant term in an equation like \(y = b\) dictates the horizontal line's position on the graph. This constant term is the y-value where the line consistently sits. It does not rely on the changing x-values as it remains fixed.

• The constant term provides the specific y-coordinate that all points on the horizontal line share.
• It's what makes the line horizontal, defining a flat, unvarying bin in y-values across all x.
• Understanding this helps in quickly and easily drawing horizontal lines on graphs without confusion.

Parallel to X-Axis

When a line is described as parallel to the x-axis, it means the line runs alongside the axis, never crossing it. Horizontally aligned, these lines keep a consistent distance from the x-axis.

• For example, with the line given by \(y = 2.5\), every point on the line maintains a fixed height of 2.5 units above the x-axis.
• This ensures that the line stays flat and does not incline or decline in relation to the axis.
• The concept of parallel lines is useful for understanding why horizontal lines do not vary in slope—they remain equidistant from the x-axis at all times.