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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 and only 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 to form a right angle.

f ( x ) = m 1 x + b 1 and g ( x ) = m 2 x + b 2 are perpendicular if and only if
m 1 m 2 = 1 , 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 functions 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 equation shown.

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 ) . This type of problem is often described as a point-slope problem because we have a point and a slope. In our example, we know that the slope is 3. We need to determine which value of 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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Writing equations of perpendicular lines

We can use a very similar process to write the equation for a line perpendicular to a given line. Instead of using the same slope, however, we use the negative reciprocal of the given slope. Suppose we are given the function shown.

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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