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The Function Machine
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Imagine you are going shopping, and you have to pay a 12% sales tax on everything you buy.
For each price, you want to know the tax you would pay. For a $50 item, 12% of $50 is 6 dollars tax. For a $100 item, 12% of $100 is $12 tax. ... 12% of $150 is $18 tax… and so on… This is an example of a function. In this case, the amount of tax paid is a function of the price … which is the same as saying “the tax depends on the price”. You can think of a function as an inputoutput machine. A function is a relation that describes how each input corresponds to a single output. So for each input (in this case the price), there is only one possible output (in this case the tax). In a function, any input produces only one output. Activity: Play with the input or price slider. Watch how the function works. It can only produce one output for each input. Adjust the slider so that the input is $240. What is the output? 
Teacher's Note: A friend of mine, Gary Tupper, wrote to me that: "if a function is exemplified by a 'real world' situation such as the 12% tax example, then that affords an opportunity to mention domain and range issues: the tax only starts at ? cents, and does the tax depend on the actual price (or rounded to the nickel?  in Canada only). These are good real contextual questions that should be discussed once students understand the basic concept of a function.

Three Ways to Write this Function
Let’s review 3 different notations for functions
… starting with the function we just discussed. We’ll enter a price of $350, and get a tax of $42. 1st way to represent this function We can represent this function as T(p) = 0.12 × p The tax, “T” for any given price “p” equals 0.12 times the price “p”. We can then substitute $350 as the price and get a tax of $42 for that price. 2nd way to represent this function We can also represent this function as f (x) = 0.12 × x. f tells us there is a function involving x which equals 0.12 times x. The tax, “f”, for any given price, x equals 0.12 times x.. We can then substitute $350 for x and get the $42 tax amount for that price. 3rd way to represent this function But we can also represent this function as “y = 0.12 × x” The output, ”y” equals 0.12 times the input “x “. The tax, y, equals 0.12 times the price, x. These THREE notations are three different ways of saying the same thing. Although we can choose any of these notations, once we choose a notation, it is important to remain consistent throughout our work. If we begin with T of p notation, we must first clearly define T and p, and then not change variables until we are at the end of the solution. Activity Play with the input or price slider. Watch how the notation works. 
Another Example of a Function
Now that we have represented this function as a table of values, let’s represent it as a graph:
We can start with the first pair of inputs and outputs from the table, and plot them as (x, y) coordinate pairs on the graph. Then plot the other pairs. As you plot more and more of the pairs of inputs and outputs, the points start to describe the function visually as a curve on a graph. Activity: Play with the input or speed slider. Watch the movement on the graph as the point traces the curve. Notice the function inputs and outputs. Adjust the slider to find the stopping distance at 55 mph. 
Classify the following polynomial functions
Activity
As you hover over each function on the left, you can see the corresponding graph. These functions can be organized into 4 types.

Quadratic Function Context
First, click on the picture to watch the video.
How fast was Simon Dumont going?
Use the graph of this quadratic function to help you. Simon Dumont launched from a 38foot quarter pipe that day at Sunday River. He maxed out at 35.5 feet above the quarter pipe.

See: this page for information about the development team. :)
A Short Summary
Can you guess the equations of each of the functions below? Click on them to interact with the applet.
A quadratic function has an inputvalue raised to the exponent two.
The graph of a quadratic function is always a parabola. (It changes direction once.) 
A constant function involves an output value that is always the same.
The graph of a constant function is always a horizontal line. 
A linear function has an inputvalue raised to the exponent one.
The graph of a linear function is always a an oblique line. 
A cubic function has an inputvalue raised to the exponent three.
The graph of a cubic function is sshaped. (It changes directions twice.) 
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How to cite this webpage (in APA 6th style):
Baron, L. M. (2016). Connected Mathematics Website. Retrieved from http://connectedmathbaron.weebly.com/
How to cite this webpage (in APA 6th style):
Baron, L. M. (2016). Connected Mathematics Website. Retrieved from http://connectedmathbaron.weebly.com/