On this page, we hope to clear up problems that you might have
with graphs and functions. Graphs are used to give a graphical
representation of equations, usually functions. Click any of the links
below or scroll down to start understanding graphs and functions!
Finding the slope of a line
Parallel and perpendicular lines
Quiz on Graphs and Functions
Graphing single ordered pairs is usually covered in most pre-algebra classes and that custom has been followed on this site. You can click here to learn about graphing points.
Graphing simple equations such as y = 2x - 3 is a topic usually covered in most elementary algebra (Algebra I) classes and that custom has been followed on this site. You can click here to learn about graphing simple equations.
When graphed, lines slope from left to right. However, some slope upward and others
slope downward. Some are really steep, while others have a gentle slope. The
slope of a line is defined as the change in y over the change in x, or the rise
over the run.
1. Problem: The points (1,2) and (3,6) are on a line. Find the line's slope. Solution: Plug the given points into the slope formula. y2 - y1 m = ------- x2 - x1 6 - 2 m = ----- 3 - 1 After simplification, m = 2
A function is a relation (usually an equation) in which no two
ordered pairs have the same x-coordinate when graphed.
If nonvertical lines have the same slope but different y-intercepts, they are parallel.
1. Problem: Determine whether the graphs of y = -3x + 5 and 4y = -12x + 20 are parallel lines. Solution: Use the Multiplication Principle to get the second equation in slope-intercept form. y = -3x + 5 y = -3x + 5 The slope-intercept equations are the same. The two equations have the same graph. 2. Problem: Determine whether the graphs of 3x - y = -5 and y - 3x = -2 are parallel. Solution: By solving each equation for y, you get the equations in slope-intercept form. y = 3x + 5 y = 3x - 2 The slopes are the same, and the y-intercepts are different, so the lines are parallel.
Sometimes, you will be asked to find the equation of a line parallel to another line. Not all the information to put the equation in slope-intercept form will always be given. Example:
3. Problem: Write an equation of the line parallel to the line 2x + y - 10 = 0 and containing the point (-1, 3). Solution: First, rewrite the given equation in slope-intercept form. y = -2x + 10 This tells us the parallel line must have a slope of -2. Plug the given point and the slope into the slope-intercept formula to find the y intercept of the parallel line. 3 = -2(-1) + b Solve for b. 1 = b The parallel line's equation is y = -2x + 1.
If two nonvertical lines have slopes whose product is -1, the lines are perpendicular. Example:
1. Problem: Determine whether the lines 5y = 4x + 10 and 4y = -5x + 4 are perpendicular. Solution: Find the slope-intercept equations by solving for y. y = (4/5)x + 2 y = -(5/4)x + 1 The product of the slopes is -1, so the lines are perpendicular.
Sometimes, you will be asked to find the equation of a line perpendicular to another line. Not all the information to put the equation in slope-intercept form will always be given. Example:
2. Problem: Write an equation of the line perpendicular to 4y - x = 20 and containing the point (2, -3). Solution: Rewrite the equation in slope-intercept form. y = .25x + 5 We know the slope of the perpendicular line is -4 because .25 * -4 = -1. (Notice that the slope of the perpendicular line is the re- ciprocal of the other line's slope.) Now plug the given point and the slope into a slope-intercept equation to find the y intercept. -3 = (-4)2 + b Solve for b. b = 5 Now, you have the information you need to write an equation for a line perpendicular to 4y - x = 20. The answer is the following equation: y = -4x + 5.
Take the Quiz on graphs and functions. (Very useful to review or to see if you've really got this topic down.) Do it!