Functions and Graphs Help
Introduction to Functions and Graphs
A function is a relationship in which one value depends upon another value. For example, if you are buying candy bars at a certain price, there is a relationship between the number of candy bars you buy and the amount of money you have to pay.
Think about functions as a machine—you put something into the machine, and it spits something back out. For example, when you enter quarters into a vending machine, you get a snack. (That is, unless it gets stuck on the ledge.) This is like a function. The input to the function was quarters, and the output of the function was a snack.
Basically, a function is a set of rules for using input and to produce output, and usually, this involves numbers.
Functions are written in the form beginning with the following symbol:
- f(x) =
For example, consider the function f(x) = 8x – 2. If you are asked to find f(3), you simply substitute the 3 into the given function equation.
f(x) = 8x – 2
f(3) = 8(3) – 2
f(3) = 24 – 2 = 22
So, when x = 3, the value of the function is 22.
You could also imagine functions that take more than one number as their input, like f(x,y) = x + y. That means that if you give the function the numbers 7 and 4 as input, the function spits out the number 11 as output.
Function tables portray a relationship between two variables, such as an x and a y. It is your job to figure out exactly what that relationship is. Let's look at a function table:
Notice that some of the data was left out. Don't worry about that! You can still figure out what you need to do to the x in order to make it the y. You see that x = 1 corresponds to y = 4; x = 2 corresponds to y = 5; and x = 4 corresponds to y = 7. Did you spot the pattern? Our y-value is just our x-value plus 3.
Functions and Coordinate Grids
X and Y Axis
Okay, here's a quick review of the coordinate grid.
In a coordinate grid, the horizontal axis is the x-axis, and the vertical axis is the y-axis. The place where they meet is the point of origin. Using this system, you can place any point on the grid if you give it an x-value and a y-value, conventionally written as (x,y). If you have two or more points, you have a line.
Now think about a basic function. If you input an initial value x, you get an f(x) value. If you call f(x) the y-value, you can see how a typical function can spit out a huge number of points that can then be graphed. Look back at f(x) = 8x – 2. If x = 2, y = 14. If x = 3, y = 22. So, we already have two points for this line: (2,14) and (3,22). Once you know two points of a function, you can draw a line connecting them. And guess what? You have now graphed a function!
The x-values are known as the independent variables. The y-values depend on the x-values, so the y-values are called the dependent variables.
Verticle Line Test
Potential functions must pass the vertical line test in order to be considered a function. The vertical line test is the following: Does a vertical line drawn through a graph of the potential function pass through only one point of the graph? If YES, then the vertical line passes through only one point, and the potential function is a function. If NO, then the vertical line passes through more than one point, and the potential function is not a function.