The First Sizzling of Baconian Algebra
Sausage is for Sissies
A FUNCTION OF BACON
y = 10x
f means "function"...
1. a function is a machine.
Example: You have a mechanical slicer machine. You put in one fresh slab o hog meat. Out comes lovely slices of bacon.
Input=x Output=y
So, in our case, slab o hog meat (x) goes in the machine. The machine does something (slices into 10 pieces or multiplies by ten in math talk). A nice 10 piece portion of bacon (y) comes out!
That's why y = (is the result of) 10x (ten times x or however many slabs you put in the machine)
2. If we put 4 slabs o meat in, we'd get 40 pieces of bacon. Why? Well, 4 input means our "x" will be replaced by the number 4. The function says 10 times x. So, 10 4 = 40
In Math Talk: y = 10(4) --> y = 40.
note: when a letter is close-talking a number (like 10x or 4y--math people call that close talking # a coefficient, fyi)) it means to multiply those as soon as you know what number goes in for the letter. In our case, as soon as you knew you were putting IN 4 slabs, you knew x (the Input) would be replaced by the number 4.
note about the note: Anytime you replace a letter with a number, put the number in parentheses (10x becomes 10(4) and means 10 times 4). It's a long story why, but the main reason is if you have negative numbers as inputs, it gets confusing if you don't use the parentheses.
A FUNCTION OF SAUSAGE
Now, we also have in our kitchen, a beautiful sausage maker. We loathe it. But to prove to our friends how useless it is, we are going use it for that "function." (haha see what I did there? play on words?...yeah...anyways...)
y = 5x - 1
1. In this case, our sausage maker will make 5 pieces of sausage for every hunk o hamburger we put in, but it ruins one every round because it doesn't even like sausage, and because it is going to help us prove how stupid sausage is.
Input = x Output = y and
our machine works like this:
y = 5x - 1
So, let's start by putting in one hunk o hamburger.
y = 5(1) -1
y = 5 - 1
y = 4
Input of 1 hunk o hamburger results in only 4 pieces of sausage because we had to subtract the one that got ruined. SEE? SAUSAGE MAKING IS DUMB!
But our friends don't believe us. So they buy 40 pounds of meat to make sausage.
Ok. Fine.
y = 5(40) - 1. (5 * 4 is 20, then add that extra zero from 40...200)
y = 200 - 1
y = 199
Ah. Maybe they were right. They only lost a tiny fraction of their 40lbs of sausage in the machine. The more the buy, the less significant that (- 1) becomes. DARNIT.
Next...
A FUNCTION OF ICEMAKERS
Well, we need some water to wash down all that sodium.
y = x
This tells me that when I put a gallon of water in the ice cube trays, after my super freezing machine processes it, I only have a half gallon of ice cubes after.
So if I put 47 gallons in (it would take me all day) and I'd get:
y = ½ x
with an input of 47, x = 47
y = ½ (47)
(a fraction means you multiply the top # by the value of x, then divide by the bottom number. ½ is actually 1 divided by 2...that fraction line? It's a division symbol in disguise. Lying #{{SSR}}amp; of ^$#^#!)
so ½ (47) or ½ OF 47 (times=of) means ½ times 47/1
( 1 * 47 ) over ( 2 * 1 ) which is
47 over 2 which means 47 divided by 2.
y = 23 ½
A note on fractions in algebra: try to avoid them...
If you can't, remember these key points:
1. multiplying fractions means you multiply the top numbers, put the line (aka disguised division symbol), then multiply the bottom numbers.
2. IF after this you can simplify it or make it shorter or it divides evenly, do that.
3. IF one multiplier is NOT a fraction, make it one by simply putting 1 as the denominator--or bottom #-- because anything over one is itself. like 2 is the same as 2/1 because 2 divided by 1 is 2. Once you have a # on the top and bottom you can multiply the fractions.
4. look out for + or - signs in the numerator or denominator. IF you see them, and letters too, WATCH OUT!! It works differently (not relevant now but you'll need this later)
What does a function LOOK like??
Well, when you make bacon it SMELLS good. But if you had to see if your bacon making is improving or getting worse, you could graph your function.
LOOK BELOW AT THE GRAPH
type in the function y=2x+2
-- (f) of y (output) = (is) 2x + 2
that makes a LINE on the graph, see?
When there is an input (x) and an output (y), it makes a line.
How? Ok. So. Look at the line.
Find 3 points where it is crossing at an exact intersection on the graph (like the points happen on the corner of any of the little squares).
These points can be "mapped" by naming them. A mapped point is an "ordered pair" because it's 2 numbers (x distance on horizontal axis & y distance on vertical axis) and it's in alphabetical order (x, then y)
This is an ordered pair: (x,y) or where your dot is will go right or left x spaces, and will go up or down y spaces. If you go up three and left three it would be (-3, 3) (always in x,y order and negatives go down or left)
1. It looks like it crosses at (-1, 0) then (0, 2) then (1, 4) and 2 more points that I can see
2. So, where it says "input" below, type one of the ordered pairs above where the line crosses an intersection (type just as you see it: (0,2) )
3. That says "POINT A IS" and you'll see it put a dot on your line.
4. Now, let's take (0,2) and think about it. 0 is the x value of that point on the line, right? It's how far right or left you went from the origin (or 0,0, or the middle) before you hit the line. Then y is 2. That y is always how far up or down you go from the origin.
5. Make sense? Good.
WHAT IF...we remembered x is the input? So, our point's ordered pair we thought was on the line was (0,2)--that means the input was 0 (it's always x) and the output was 2 (always y). Here's the gem...
TRY putting the input in the bacon machine (aka function equation) and solve it like we did in the beginning up top.
so y = 2 (0) + 2 (I subbed x, which in our point was zero, for the input)
answer?
y = ___
Look again at your ordered pair. WHAT???? THE Y IS 2?!?!?!! Yes.
WHEN you plug in an input of any x, and get any output, you can plot that point on a graph.
If I had y = 2x + 1
and decided x would be 1 (the first half of my ordered pair will be 1)
y = 2 (1) + 1 I
y = 2 + 1
y = 3
So, my first ordered pair is (1, 3)...
now, plug in a (-1) for x, and a 2 for x.
You'll have 2 more ordered pairs as answers
You'll write down 3 ordered pairs.
NOW comes the fun part. In the "input" box below, type in your function from this problem (the actual equation we used).
See the line it made on the graph?
Now look at your ordered pairs.
Notice anything?
Text me now (or Google Hangout Message me--gigglinggalaxy@ gmail and tell me what you notice.
Lesson 2 tomorrow...we got the bacon from the slab (aka the output from the input) but now we shall get the function equation JUST from looking at our line!!!
NO HOMEWORK TOMORROW IF YOU CAN FIGURE OUT HOW THIS WORKS BEFORE THE LESSON.