Horizontal and Vertical Lines

Idea

Two kinds of line do not need the point-slope formula at all.

A horizontal line never goes up or down, so its slope is $0$ and its $y$ value is the same everywhere. A vertical line never goes left or right, so it has no slope at all, and its $x$ value is the same everywhere. In both cases one coordinate is fixed, and that fixed coordinate is the equation.

Rule

A line parallel to the $x$ -axis (horizontal) freezes $y$ :

y = c

m = 0

A line parallel to the $y$ -axis (vertical) freezes $x$ :

x = c

m \text{ is undefined}

To find $c$ , read the matching coordinate off any point the line passes through. Nothing is computed.

How it is used

The only decision is which coordinate to freeze. The wording is where mistakes happen.

"Perpendicular to an axis" means parallel to the other axis. Picture the line before you write anything down.

The line is...	Direction	Equation through $(a, b)$
parallel to the $x$ -axis	horizontal	$y = b$
perpendicular to the $y$ -axis	horizontal	$y = b$
parallel to the $y$ -axis	vertical	$x = a$
perpendicular to the $x$ -axis	vertical	$x = a$

Here is a check that helps. The line is parallel to the axis that is not named in its equation.

So $y = 3$ is parallel to the $x$ -axis, and $x = 3$ is parallel to the $y$ -axis. This feels backwards at first, so test it with a point instead of trusting your instinct.

One more result to remember. A vertical line has no $y = mx + n$ form, because that form needs a slope, and a vertical line has none. You cannot write it in slope-intercept form at all.

Worked example

Find the equation of the line through $(4, -2)$ that is parallel to the $x$ -axis. Parallel to the $x$ -axis means the line is horizontal, so its equation has the form $y = c$ . A horizontal line holds its ordinate constant, and the given point's ordinate is $-2$ . So every point on the line has $y = -2$ , whatever its abscissa:

y = -2

Explore

Drag B until it is directly above or below A. The run becomes zero, the slope reads undefined, and the equation becomes $x = a$ . Now line the points up side by side. The slope becomes $0$ and the equation freezes $y$ . These are the only two lines whose equation uses just one letter.

A line through two points

Try this. Move the two points and compare the equation. Equal x values produce a vertical line; equal points cannot define one.

Point A · x-2Point A · y3Point B · x3Point B · y1

y = -0.4x + 2.2 · Slope = -0.4