Overview
On this page, we review some important facts about functions and the notation we use to describe them algebraically, and how to obtain (at least in principle) the graph of any function.
Basic Learning Objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
- Understand function notation and be able to use it to evaluate a given function at a specific input value.
- Understand how one can in principle obtain the graph of any function by evaluating (“sampling”) it at many input values.
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
- Use a graphing calculator or website (like Desmos.com) to draw the graph of a given function.
- Be aware that a picture of the graph of a function can be misleading and could hide important aspects, depending on the choice of the shown domain and range.
To prepare for class
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Watch the following videos (by Tyler Wallace, 3m51s, and patrickJMT, 5m30s, respectively) which explain function notation and how to evaluate a given function at specific \(x\)-values:
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Watch the following video (by Mashup Math, 5m27s) which shows one can sketch the graph of any function by evaluating (“sampling”) it at many input values:
After class
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Think about all the possible ways you might get an incorrect sketch of the graph when using the table or “sampling” method described in the previous video. In particular, ask yourself the following questions:
- Can you always connect neighbouring dots? If so, how should you connect them?
- How many sampling points do you need to obtain an accurate sketch of the graph? Is that number the same for all possible functions?
- Even if you use a graphing calculator to draw the graph, can you trust the obtained picture to be precise? What might you be missing?
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Watch the following video (by The Organic Chemistry Tutor, 1m02s) which shows how to use the online graphing calculator Desmos.com to draw the graph of any function:
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Experiment with Desmos.com and let it draw the graph of many functions, making sure to move around and zoom in and out (on both axes, or just on one). Try to invent your own functions, but when in doubt use the following as some inspiration (note that you can show them all in the same picture if you like, and that you can turn each graph on or off by clicking on the round symbol to the left of its box):
- \(f(x)=x^5+3x^4-3x^2+x-3\)
- \(g(x)=\sin(3x)\)
- \(h(x)=\frac{g(x)}{x^2}\)
- \(i(x)=e^x\)
- \(j(x)=e^{-x^2}\)
- \(k(x)=A\sec(Bx)+C\) (Desmos will ask if you want to create “sliders” for the constants \(A\), \(B\), and \(C\): click on “all” and play around with the sliders!)
