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Access these online resources for additional instruction and practice with graphs of linear functions.

Key concepts

  • Linear functions may be graphed by plotting points or by using the y -intercept and slope. See [link] and [link] .
  • Graphs of linear functions may be transformed by using shifts up, down, left, or right, as well as through stretches, compressions, and reflections. See [link] .
  • The y -intercept and slope of a line may be used to write the equation of a line.
  • The x -intercept is the point at which the graph of a linear function crosses the x -axis. See [link] and [link] .
  • Horizontal lines are written in the form, f ( x ) = b . See [link] .
  • Vertical lines are written in the form, x = b . See [link] .
  • Parallel lines have the same slope.
  • Perpendicular lines have negative reciprocal slopes, assuming neither is vertical. See [link] .
  • A line parallel to another line, passing through a given point, may be found by substituting the slope value of the line and the x - and y -values of the given point into the equation, f ( x ) = m x + b , and using the b that results. Similarly, the point-slope form of an equation can also be used. See [link] .
  • A line perpendicular to another line, passing through a given point, may be found in the same manner, with the exception of using the negative reciprocal slope. See [link] and [link] .
  • A system of linear equations may be solved setting the two equations equal to one another and solving for x . The y -value may be found by evaluating either one of the original equations using this x -value.
  • A system of linear equations may also be solved by finding the point of intersection on a graph. See [link] and [link] .

Section exercises

Verbal

If the graphs of two linear functions are parallel, describe the relationship between the slopes and the y -intercepts.

The slopes are equal; y -intercepts are not equal.

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If the graphs of two linear functions are perpendicular, describe the relationship between the slopes and the y -intercepts.

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If a horizontal line has the equation f ( x ) = a and a vertical line has the equation x = a , what is the point of intersection? Explain why what you found is the point of intersection.

The point of intersection is ( a , a ) . This is because for the horizontal line, all of the y coordinates are a and for the vertical line, all of the x coordinates are a . The point of intersection will have these two characteristics.

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Explain how to find a line parallel to a linear function that passes through a given point.

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Explain how to find a line perpendicular to a linear function that passes through a given point.

First, find the slope of the linear function. Then take the negative reciprocal of the slope; this is the slope of the perpendicular line. Substitute the slope of the perpendicular line and the coordinate of the given point into the equation y = m x + b and solve for b . Then write the equation of the line in the form y = m x + b by substituting in m and b .

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Algebraic

For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:

4 x 7 y = 10 7 x + 4 y = 1

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Practice Key Terms 5

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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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