Algebra and Funcions

An important part of algebra is the study of functions, since functions often appear in equations that we are trying to solve. A function is like a box you can put a number or numbers into and get a certain number out. When using functions, graphs can be powerful tools in helping us to study the solutions to equations.

A graph is a picture that shows all the values of the variables that make the equation or inequality true. Usually this is easy to make when there are only one or two variables. The graph is often a line, and if the line does not bend or go straight up-and-down it can be described by the basic formula y = mx + b. The variable b is the y-intercept of the graph (where the line crosses the vertical axis) and m is theslope or steepness of the line. This formula applies to the coordinates of a graph, where each point on the line is written (x, y).

In some math problems like the equation for a line, there can be more than one variable (x and y in this case). To find points on the line, one variable is changed. The variable that is changed is called the "independent" variable. Then the math is done to make a number. The number that is made is called the "dependent" variable. Most of the time the independent variable is written as x and the dependent variable is written as y, for example, in y = 3x + 1. This is often put on a graph, using an x axis (going left and right) and a y axis (going up and down). It can also be written in function form: f(x) = 3x + 1. So in this example, we could put in 5 for x and get y = 16. Put in 2 for x would get y=7. And 0 for x would get y=1. So there would be a line going thru the points (5,16), (2,7), and (0,1) as seen in the graph to the right.

If x has a power of 1, it is a straight line. If it is squared or some other power, it will be curved. If it uses an inequality (< or >), then usually part of the graph is shaded, either above or below the line.