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 shifting the graph up, down, left, or right as well as using 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, 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 the 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.
- x-intercept
- 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
