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Special Cases — Horizontal and Vertical Lines

Horizontal lines have equations of the form y = k, where y stays constant. Vertical lines have equations of the form x = h, where x stays constant.

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Student-friendly explanation

In y = k, every point on the line has the same y-coordinate, so the line is parallel to the x-axis. In x = h, every point has the same x-coordinate, so the line is parallel to the y-axis. These special cases are important because one variable is fixed while the other can change.

How to write this in exams

  1. 1

    Start with the exact idea

    Horizontal lines have equations of the form y = k, where y stays constant. Vertical lines have equations of the form x = h, where x stays constant.

  2. 2

    Then show how to use it

    Check which variable is fixed, list points by keeping that coordinate constant, plot at least two points, and draw the line parallel to the correct axis.

  3. 3

    Add one concrete example

    The graph of y = 3 passes through (0, 3), (2, 3), and (-4, 3). The graph of x = -2 passes through (-2, 0), (-2, 5), and (-2, -3).

  4. 4

    Avoid this incomplete answer

    For x = 4, students may draw a horizontal line through 4 on the y-axis, but x = 4 must be vertical.

Definition

Horizontal lines have equations of the form y = k, where y stays constant. Vertical lines have equations of the form x = h, where x stays constant.

Example

The graph of y = 3 passes through (0, 3), (2, 3), and (-4, 3). The graph of x = -2 passes through (-2, 0), (-2, 5), and (-2, -3).

Rule to remember

Horizontal line: y = k. Vertical line: x = h. Here k and h are constants.

Memory hook

y fixed means flat; x fixed means standing.

Examples and method

Worked example

Plot y = -2. Choose points (-3, -2), (0, -2), and (4, -2). Since all have the same y-coordinate, joining them gives a horizontal line.

Method to apply

Check which variable is fixed, list points by keeping that coordinate constant, plot at least two points, and draw the line parallel to the correct axis.

Diagram support

Draw y = k parallel to the x-axis and x = h parallel to the y-axis, marking two or three points on each line.

How CBSE asks it

Students may be asked to identify the graph from the equation, write the equation of a horizontal or vertical line, or match points to the line.

Avoid common mistakes

Common confusion

Students often think y = 3 is a vertical line because it touches the y-axis at 3, but it is horizontal.

Common wrong answer

For x = 4, students may draw a horizontal line through 4 on the y-axis, but x = 4 must be vertical.

Exam tip

If y is constant, the line is horizontal. If x is constant, the line is vertical.

Quick check

What type of line is x = 5 on the Cartesian plane?

x = 5 is a vertical line because every point on it has x-coordinate 5 while y can take different values.

Answer writing and exam use

1-mark answer

Horizontal lines have equations of the form y = k, where y stays constant. Vertical lines have equations of the form x = h, where x stays constant.

2-mark answer

Horizontal lines have equations of the form y = k, where y stays constant. Vertical lines have equations of the form x = h, where x stays constant. Horizontal line: y = k. Vertical line: x = h. Here k and h are constants. The graph of y = 3 passes through (0, 3), (2, 3), and (-4, 3). The graph of x = -2 passes through (-2, 0), (-2, 5), and (-2, -3).

3-mark answer

In y = k, every point on the line has the same y-coordinate, so the line is parallel to the x-axis. In x = h, every point has the same x-coordinate, so the line is parallel to the y-axis. These special cases are important because one variable is fixed while the other can change. Horizontal line: y = k. Vertical line: x = h. Here k and h are constants. Plot y = -2. Choose points (-3, -2), (0, -2), and (4, -2). Since all have the same y-coordinate, joining them gives a horizontal line. Students may be asked to identify the graph from the equation, write the equation of a horizontal or vertical line, or match points to the line. For x = 4, students may draw a horizontal line through 4 on the y-axis, but x = 4 must be vertical.
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