Given the equations of two lines, determine whether their graphs are parallel or perpendicular

The two lines in Figure 18 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.Figure 18. 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.Figure 19.


We can determine from their equations whether two lines are parallel by comparing their slopes. If the slopes are the same and the y-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 90-degree, angle. The two lines in Figure 20 are perpendicular.

Graph of two functions where the blue line is perpendicular to the orange line.Figure 20. 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 1. So, if

m1 and m2{m}_{1}\text{ and }{m}_{2}
are negative reciprocals of one another, they can be multiplied together to yield
1-1
.

m1m2=1{m}_{1}{m}_{2}=-1

To find the reciprocal of a number, divide 1 by the number. So the reciprocal of 8 is

18\frac{1}{8}
, and the reciprocal of
18\frac{1}{8}
is 8. 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, assuming that the lines are neither horizontal nor perpendicular. The slope of each line below is the negative reciprocal of the other so the lines are perpendicular.

{f(x)=14x+2negative reciprocal of14 is 4f(x)=4x+3negative reciprocal of4 is 14\begin{cases}f\left(x\right)=\frac{1}{4}x+2\qquad & \text{negative reciprocal of}\frac{1}{4}\text{ is }-4\qquad \\ f\left(x\right)=-4x+3\qquad & \text{negative reciprocal of}-4\text{ is }\frac{1}{4}\qquad \end{cases}

The product of the slopes is –1.

4(14)=1-4\left(\frac{1}{4}\right)=-1

A General Note: Parallel and Perpendicular Lines

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

f(x)=m1x+b1 and g(x)=m2x+b2 are parallel if m1=m2f\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}
.

If and only if

b1=b2{b}_{1}={b}_{2}
and
m1=m2{m}_{1}={m}_{2}
, we say the lines coincide. Coincident lines are the same line.

Two lines are perpendicular lines if they intersect at right angles.

f\left(x\right)={m}\text{\textunderscore}{1}x+{b}\text{\textunderscore}{1} and g\left(x\right)={m}\text{\textunderscore}{2}x+{b}\text{\textunderscore}{2}\text{ are perp\endicular if }{m}\text{\textunderscore}{1}{m}\text{\textunderscore}{2}=-1,\text{ and so }{m}\text{\textunderscore}{2}=-\frac{1}{{m}\text{\textunderscore}{1}}
.

Example 8: Identifying Parallel and Perpendicular Lines

Given the functions below, identify the functions whose graphs are a pair of parallel lines and a pair of perpendicular lines.

{f(x)=2x+3h(x)=2x+2g(x)=12x4j(x)=2x6\begin{cases}f\left(x\right)=2x+3\qquad & \qquad & h\left(x\right)=-2x+2\qquad \\ g\left(x\right)=\frac{1}{2}x - 4\qquad & \qquad & j\left(x\right)=2x - 6\qquad \end{cases}

Solutions

Parallel lines have the same slope. Because the functions

f(x)=2x+3f\left(x\right)=2x+3
and
j(x)=2x6j\left(x\right)=2x - 6
each have a slope of 2, they represent parallel lines. Perpendicular lines have negative reciprocal slopes. Because −2 and
12\frac{1}{2}
are negative reciprocals, the equations,
g(x)=12x4g\left(x\right)=\frac{1}{2}x - 4
and
h(x)=2x+2h\left(x\right)=-2x+2
represent perpendicular lines.

Analysis of the Solution

A graph of the lines is shown in Figure 21.

Graph of four functions where the blue line is h(x) = -2x + 2, the orange line is f(x) = 2x + 3, the green line is j(x) = 2x - 6, and the red line is g(x) = 1/2x - 4.

Figure 21. The graph shows that the lines

f(x)=2x+3f\left(x\right)=2x+3
and
j(x)=2x6j\left(x\right)=2x - 6
are parallel, and the lines
g(x)=12x4g\left(x\right)=\frac{1}{2}x - 4
and
h(x)=2x+2h\left(x\right)=-2x+2
are perpendicular.

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