Graphs of Linear Functions: Learn It 5

Determining Whether Lines are Parallel or Perpendicular

The two lines in the graph below are parallel lines: they will never intersect. Notice that they have exactly the same steepness which means their slopes are identical. The only difference between the two lines is the y-intercept. If we shifted one line vertically toward the y-intercept of the other, they would become the same line.

Graph of two functions where the blue line is y = -2/3x + 1, and the baby blue line is y = -2/3x +7. Notice that they are parallel lines.
Parallel lines.

The functions 2x plus 6 and negative 2x minus 4 are parallel. The functions 3x plus 2 and 2x plus 2 are not parallel.

We can determine from their equations whether two lines are parallel by comparing their slopes. If the slopes are the same and the [latex]y[/latex]-intercepts are different, the lines are parallel. If the slopes are different, the lines are not parallel.

Unlike parallel lines, perpendicular lines do intersect. Their intersection forms a right or[latex]90[/latex]-degree angle. The two lines below are perpendicular.

Graph of two functions where the blue line is perpendicular to the orange line.
Perpendicular lines.

Perpendicular lines do not have the same slope. The slopes of perpendicular lines are different from one another in a specific way. The slope of one line is the negative reciprocal of the slope of the other line.

The product of a number and its reciprocal is [latex]1[/latex]. If [latex]{m}_{1}\text{ and }{m}_{2}[/latex] are negative reciprocals of one another, they can be multiplied together to yield [latex]-1[/latex].

[latex]{m}_{1}*{m}_{2}=-1[/latex]

To find the reciprocal of a number, divide [latex]1[/latex] by the number. So the reciprocal of [latex]8[/latex] is [latex]\frac{1}{8}[/latex], and the reciprocal of [latex]\frac{1}{8}[/latex] is [latex]8[/latex]. To find the negative reciprocal, first find the reciprocal and then change the sign.

As with parallel lines, we can determine whether two lines are perpendicular by comparing their slopes. The slope of each line below is the negative reciprocal of the other so the lines are perpendicular.

[latex]\begin{array}{ll}f\left(x\right)=\frac{1}{4}x+2\hfill & \text{negative reciprocal of }\frac{1}{4}\text{ is }-4\hfill \\ f\left(x\right)=-4x+3\hfill & \text{negative reciprocal of }-4\text{ is }\frac{1}{4}\hfill \end{array}[/latex]

The product of the slopes is [latex]–1[/latex].

[latex]-4\left(\frac{1}{4}\right)=-1[/latex]

parallel and perpendicular lines

Two lines are parallel lines if they do not intersect. The slopes of the lines are the same.

[latex]f\left(x\right)={m}_{1}x+{b}_{1}\text{ and }g\left(x\right)={m}_{2}x+{b}_{2}\text{ are parallel if }{m}_{1}={m}_{2}[/latex].

If and only if [latex]{b}_{1}={b}_{2}[/latex] and [latex]{m}_{1}={m}_{2}[/latex], we say the lines coincide. Coincident lines are the same line.
[latex]\\[/latex]
Two lines are perpendicular lines if they intersect at right angles.

[latex]f\left(x\right)={m}_{1}x+{b}_{1}\text{ and }g\left(x\right)={m}_{2}x+{b}_{2}\text{ are perpendicular if }{m}_{1}*{m}_{2}=-1,\text{ and }{m}_{2}=-\frac{1}{{m}_{1}}[/latex].
Given the functions below, identify the functions whose graphs are a pair of parallel lines and a pair of perpendicular lines.

[latex]\begin{array}{l}f\left(x\right)=2x+3\hfill & \hfill & h\left(x\right)=-2x+2\hfill \\ g\left(x\right)=\frac{1}{2}x - 4\hfill & \hfill & j\left(x\right)=2x - 6\hfill \end{array}[/latex]