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2.2 Graphs of linear functions  (Page 6/15)

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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 ) = 1 4 x + 2 negative reciprocal of 1 4  is  −4 f ( x ) = − 4 x + 3 negative reciprocal of − 4  is  1 4

The product of the slopes is –1.

− 4 ( 1 4 ) = − 1

Parallel and perpendicular lines

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

f ( x ) = m 1 x     +     b 1  and  g ( x ) = m 2 x     +     b 2  are parallel if  m 1     =     m 2 .

If and only if b 1 = b 2 and 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 ( x ) = m 1 x + b 1  and  g ( x ) = m 2 x + b 2  are perpendicular if  m 1 m 2 = − 1 ,  and so  m 2 = − 1 m 1 .

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 ) = 2 x + 3 h ( x ) = − 2 x + 2 g ( x ) = 1 2 x − 4   j ( x ) = 2 x − 6

Parallel lines have the same slope. Because the functions f ( x ) = 2 x + 3 and j ( x ) = 2 x − 6 each have a slope of 2, they represent parallel lines. Perpendicular lines have negative reciprocal slopes. Because −2 and 1 2 are negative reciprocals, the equations, g ( x ) = 1 2 x − 4 and h ( x ) = − 2 x + 2 represent perpendicular lines.

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Writing the equation of a line parallel or perpendicular to a given line

If we know the equation of a line, we can use what we know about slope to write the equation of a line that is either parallel or perpendicular to the given line.

Writing equations of parallel lines

Suppose for example, we are given the following equation.

f ( x ) = 3 x + 1

We know that the slope of the line formed by the function is 3. We also know that the y- intercept is ( 0 , 1 ) . Any other line with a slope of 3 will be parallel to f ( x ) . So the lines formed by all of the following functions will be parallel to f ( x ) .

g ( x ) = 3 x + 6 h ( x ) = 3 x + 1 p ( x ) = 3 x + 2 3

Suppose then we want to write the equation of a line that is parallel to f and passes through the point ( 1 ,  7 ) . We already know that the slope is 3. We just need to determine which value for b will give the correct line. We can begin with the point-slope form of an equation for a line, and then rewrite it in the slope-intercept form.

y − y 1 = m ( x − x 1 )     y − 7 = 3 ( x − 1 )     y − 7 = 3 x − 3          y = 3 x + 4

So g ( x ) = 3 x + 4 is parallel to f ( x ) = 3 x + 1 and passes through the point ( 1 ,  7 ) .

Given the equation of a function and a point through which its graph passes, write the equation of a line parallel to the given line that passes through the given point.

  1. Find the slope of the function.
  2. Substitute the given values into either the general point-slope equation or the slope-intercept equation for a line.
  3. Simplify.

Finding a line parallel to a given line

Find a line parallel to the graph of f ( x ) = 3 x + 6 that passes through the point ( 3 ,  0 ) .

The slope of the given line is 3. If we choose the slope-intercept form, we can substitute m = 3 , x = 3 , and f ( x ) = 0 into the slope-intercept form to find the y- intercept.

g ( x ) = 3 x + b       0 = 3 ( 3 ) + b       b = – 9

The line parallel to f ( x ) that passes through ( 3 ,  0 ) is g ( x ) = 3 x − 9.

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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