Summary so Far
In the previous exercises, you encountered your first function, a model of the height of a missile at various times. It's totally normal if you are feeling like the notation is a little over your head. Let's get some of it straight before moving forward.
Just to be clear, this is your first function from the previous activities:
g(x)=-4.92507*x^2+297.84768*x+616.05959
The above is the "code view" of the function which is how you'd type the function into Geogebra. You might also see functions (and other mathematical objects) written in "math view" which italicizes everything that isn't a number.
Most of the time in this book we use code view for mathematics since you will mostly learn by way of Geogebra.
The best way to think of this function, and in fact, any function, is as a computer program. This particular program takes a time as an input, and returns the estimated height of the missile at that time.
Let's talk about the notation that is used to talk about functions like this one.
- On the left of the equation,
g(x)is called the declaration of the function. It's only purpose is tell you the name of the function, and also to tell you "hey, we're about to talk about functions here!" It doesn't really do anything. This function is called g, but you can name a function anything you like (within reason... maybe don't call your functionfarts(x), for instance). Note: One thing that definitely is not happening, isgis not being multiplied byx. You might think this based on your study of algebra earlier in life, but try to bury that thinking right now if you can. - The letter
xrepresents an arbitrary input for the function. In the case of this function, inputs represent time after detection of the missile, measured in seconds. - On the right of the equation is the algebraic "code" for how to calculate the output of the function using the input,
x. In our case the output is the height of the missile at an input time ofxseconds.
x, use the code on the right to calculate the height. For instance at time 1 second after detection, the height of the missile is predicted by the model to be g(1):
g(1)=-4.92507*1^2+297.84768*1+616.05959
I expect you to be able to do the arithmetic above on your own in this class (with a calculator, of course), and calculate that this is about 908.9822. However, I also expect that you're lazy, and would rather let the computer do it whenever possible. That's fine. Me too.
Turns out, Geogebra is perfect for this type of stuff! I've switched off the Graphics mode, and turned on the "Algebra" mode below. Later we'll have both on at the same time.
For now, try out the algebraic mode: type g(1) in the applet below to do the above calculation. You can also try other inputs as well. For instance try g(5) to get the model's prediction of the height of the missile at time 5 seconds. You should get a number between 1982 and 1983 meters. You can also make more guesses about the maximum height of the missile (which we were doing in the last activity) by typing guesses like g(30.5). Quick Check: What's g(6)?
You can also try out creating your own function now if you like. Try copying and pasting this code snippet into the input bar above.
dianna(x)=3x
After declaring dianna with the above code, see what dianna does with code like dianna(4). This will create a number with the value of 12 (3 times 4). As you can see, dianna multiplies her input times 3, so she's a lot less complex than our model g(x) from above. Quick Check: What's dianna(5)?
By the way, yes, you can name your functions (almost) anything you like. You could just as well have named
dianna something else, like farts if you wanted, even though I told you that the name farts is a bad name for a function. It would be pretty funny.
A few names however are absolutely off limits. For instance you can't call your function "x" since x is reserved for the inputs of functions, and something like x(x) would just be too confusing. There are some other restrictions. You can't call your functions "y", "e" or "i", but there's no need to go into why these are not allowed.
It's worth noting though, capitalization does matter when you name functions or any other object in Geogebra. For instance, Farts(x)=2x and farts(x)=2 are two different functions to Geogebra because of the difference in a lower case f and capital F. Also "a" and "A" are different as well. This might seem annoying now, but later on this will be quite convenient.
Move forward to continue learning about functions.