Algebra II: Graphs and Functions

This can be explained with a formula: ((y2) - (y1))/((x2) - (x1)).  (The varialbes would be subscripted if text only browsers allowed for subscripted characters.)  To find the slope, you pick any two points on the line and find the change in y, and then divide it by the change in x.

Example:

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)          The variables would be sub-
             m = -----------          scripted if text only browsers
                 (x2) - (x1)         allowed for subscripted
                                      characters.

                 6 - 2
             m = -----
                 3 - 1

             After simplification, m = 2.
         

One way to tell if a graph is a function is the vertical line test, which says if it is possible for a vertical line to meet a graph more than once, the graph is not a function.  The accompanying figure is an example of a function.

Functions are usually denoted by letters such as f or g.  If the first coordinate of an ordered pair is represented by x, the second coordinate (the y-coordinate) can be represented by f(x).  In the accompanying figure, f(1) = -1 and f(3) = 2.

When a function is an equation, the domain is the set of numbers that are replacements for x that give a value for f(x) that is on the graph.  Sometimes, certain replacements do not work, such as 0 in the following function: f(x) = 4/x (you cannot divide by 0).  In that case, the domain is said to be x <> 0.

There are a couple of special functions whose graphs you should have memorized because they are sometimes hard to graph.  They are the absolute value function (example figure) and the greatest integer function (example figure).

The greatest integer function, y = [x] is defined as follows: [x] is the greatest integer that is less than or equal to x.

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Example:

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.
         

Example:

1.  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.
         

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.
         

Example:

1.  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 reciprocal 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.  The quiz is very useful for either review of to see if you've really got the topic down.