# Summary: Graphs of Linear Functions

## Key Concepts

- Linear functions may be graphed by plotting points or by using the
*y*-intercept and slope. - Graphs of linear functions may be transformed by using shifts up, down, left, or right, as well as through stretches, compressions, and reflections.
- 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. - Horizontal lines are written in the form, [latex]f(x)=b[/latex].
- Vertical lines are written in the form, [latex]x=b[/latex].
- Parallel lines have the same slope.
- Perpendicular lines have negative reciprocal slopes, assuming neither is vertical.
- 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, [latex]f\left(x\right)=mx+b[/latex], and using the*b*that results. Similarly, the point-slope form of an equation can also be used. - 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.

- The absolute value function is commonly used to measure distances between points.
- Applied problems, such as ranges of possible values, can also be solved using the absolute value function.
- The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction.
- In an absolute value equation, an unknown variable is the input of an absolute value function.
- If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable.
- An absolute value equation may have one solution, two solutions, or no solutions.
- An absolute value inequality is similar to an absolute value equation but takes the form [latex]|A|<B,|A|\le B,|A|>B,\text{ or }|A|\ge B[/latex]. It can be solved by determining the boundaries of the solution set and then testing which segments are in the set.
- Absolute value inequalities can also be solved graphically.

## Glossary

**absolute value equation**an equation of the form [latex]|A|=B[/latex], with [latex]B\ge 0[/latex]; it will have solutions when [latex]A=B[/latex] or [latex]A=-B[/latex]

**absolute value inequality**a relationship in the form [latex]|{ A }|<{ B },|{ A }|\le { B },|{ A }|>{ B },\text{or }|{ A }|\ge{ B }[/latex]

**horizontal line**a line defined by [latex]f\left(x\right)=b[/latex], where

*b*is a real number. The slope of a horizontal line is 0.

**parallel lines**two or more lines with the same slope

**perpendicular lines**two lines that intersect at right angles and have slopes that are negative reciprocals of each other

**vertical line**a line defined by [latex]x=a[/latex], where

*a*is a real number. The slope of a vertical line is undefined.

**the point on the graph of a linear function when the output value is 0; the point at which the graph crosses the horizontal axis**

*x*-intercept## Licenses & Attributions

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- Revision and Adaptation.
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- College Algebra.
**Provided by:**OpenStax**Authored by:**Abramson, Jay et al..**Located at:**https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites.**License:**CC BY: Attribution.**License terms:**Download for free at http://cnx.org/contents/[email protected].