Vertical and Horizontal Lines

Most of the lines we have worked with so far have been slanted, or oblique. In other words, they were neither horizontal nor vertical lines. The equations of vertical and horizontal lines do not require any of the preceding formulas, although we can use the formulas to prove that the equations are correct. The equation of a vertical line is given as

where [latex]c[/latex] is a constant. The slope of a vertical line is undefined, and regardless of the [latex]y[/latex]–value of any point on the line, the [latex]x[/latex]–coordinate of the point will be [latex]c[/latex].

The equation of a horizontal line is given as

where [latex]c[/latex] is a constant. The slope of a horizontal line is zero, and for any [latex]x[/latex]–value of a point on the line, the [latex]y[/latex]–coordinate will be [latex]c[/latex].

Find the equation of a line containing the following points:

1. [latex]\left(-3,-5\right),\left(-3,1\right),\left(-3,3\right)[/latex], and [latex]\left(-3,5\right)[/latex].
2. [latex]\left(-2,-2\right),\left(0,-2\right),\left(3,-2\right)[/latex], and [latex]\left(5,-2\right)[/latex].

Show Answer

1. Let’s use the point-slope form: [latex]y-y_1 = m(x-x_1)[/latex].
   First, we need to find the slope:
   [latex]m=\frac{5 - 3}{-3-\left(-3\right)}=\frac{2}{0}[/latex]Zero in the denominator means that the slope is undefined and, therefore, we cannot use point-slope form. However, we can plot the points. Notice that all of the x-coordinates are the same and we find a vertical line through [latex]x=-3[/latex].
2. Let’s use the point-slope form again: [latex]y-y_1 = m(x-x_1)[/latex].
   First, we need to find the slope:
   [latex]\begin{array}{l}m=\frac{-2-\left(-2\right)}{0-\left(-2\right)}\hfill \\ =\frac{0}{2}\hfill \\ =0\hfill \end{array}[/latex]Use any point for [latex]\left({x}_{1},{y}_{1}\right)[/latex] in the formula, or use the y-intercept.
   [latex]\begin{array}{l}y-\left(-2\right)=0\left(x - 3\right)\hfill \\ y+2=0\hfill \\ y=-2\hfill \end{array}[/latex]The graph is a horizontal line through [latex]y=-2[/latex]. Notice that all of the y-coordinates are the same.