# Algebra II: Graphs and Functions

On this page we hope to clear up problems that you might have with graphs and functions.  Graphs are used to give graphical representation of equations, usually functions.  Read on or follow any of the links below to start understanding graphs and functions!

Graphing points
Graphing lines
Finding the slope of a line
Functions
Parallel and perpendicular lines
Quiz on Graphs and Functions

## Graphing Points

Graphing single ordered pairs is usually covered in most pre-algebra classes and that custom has been followed on this site.  You can follow this link to learn about graphing points.

## Graphing Lines

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 follow this link to learn about graphing simple equations.

## Finding the Slope of a Line

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.

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.
```

## Functions

A function is a relation (usually an equation) in which no two ordered pairs have the same x-coordinate when graphed.

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.

## Parallel and Perpendicular Lines

If nonvertical lines have the same slope but different y-intercepts, they are parallel.

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.
```
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:

```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.
```
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:

```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.

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Math for Morons Like Us -- Algebra II: Graphs and Functions
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