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, f(x) = b.
- Vertical lines are written in the form, x = b.
- 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.
- 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.
Glossary
- 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 x = a, 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