# 2.1 Graphs of linear functions  (Page 9/15)

 Page 9 / 15

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, $f\left(x\right)=b.$ 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, $f\left(x\right)=mx+b,$ 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] .

## 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.

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

If a horizontal line has the equation $f\left(x\right)=a$ 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 $\left(a,a\right).$ 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.

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

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 $y=mx+b$ and solve for $b.$ Then write the equation of the line in the form $y=mx+b$ by substituting in $m$ and $b.$

## Algebraic

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

$\begin{array}{l}4x-7y=10\hfill \\ 7x+4y=1\hfill \end{array}$

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Source:  OpenStax, Essential precalculus, part 1. OpenStax CNX. Aug 26, 2015 Download for free at http://legacy.cnx.org/content/col11871/1.1
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