A FUNCTION OF BACON[br]y = 10x[br]f means "function"...[br]1. a function is a machine. [br]Example: You have a mechanical slicer machine. You put in one fresh slab o hog meat. Out comes lovely slices of bacon.[br][br] Input=x Output=y [br][br]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![br]That's why y = (is the result of) 10x (ten times x or however many slabs you put in the machine)[br][br]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 [math]\ast[/math] 4 = 40[br]In Math Talk: y = 10(4) --> y = 40.[br][br]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.[br][br]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.[br][br]A FUNCTION OF SAUSAGE[br]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...)[br][br]y = 5x - 1[br][br]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.[br][br]Input = x Output = y and [br][br]our machine works like this: [br][br]y = 5x - 1[br][br]So, let's start by putting in one hunk o hamburger.[br][br]y = 5(1) -1[br][br]y = 5 - 1[br][br]y = 4[br][br]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![br][br]But our friends don't believe us. So they buy 40 pounds of meat to make sausage.[br]Ok. Fine. [br]y = 5(40) - 1. (5 * 4 is 20, then add that extra zero from 40...200)[br]y = 200 - 1[br]y = 199[br][br]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.[br][br]Next...[br][br]A FUNCTION OF ICEMAKERS[br]Well, we need some water to wash down all that sodium.[br][br]y = [math]\frac{1}{2}[/math]x [br]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.[br][br]So if I put 47 gallons in (it would take me all day) and I'd get:[br]y = ½ x[br]with an input of 47, x = 47[br][br]y = ½ (47) [br][br](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 #$& of ^$#^#!)[br][br]so ½ (47) or ½ OF 47 (times=of) means ½ times 47/1[br][br]( 1 * 47 ) over ( 2 * 1 ) which is [br]47 over 2 which means 47 divided by 2.[br][br]y = 23 ½ [br][br]A note on fractions in algebra: try to avoid them...[br][br]If you can't, remember these key points:[br][br]1. multiplying fractions means you multiply the top numbers, put the line (aka disguised division symbol), then multiply the bottom numbers. [br]2. IF after this you can simplify it or make it shorter or it divides evenly, do that. [br]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.[br]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)[br][br]What does a function LOOK like??[br]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. [br][br]LOOK BELOW AT THE GRAPH[br]type in the function y=2x+2[br]-- (f) of y (output) = (is) 2x + 2[br][br]that makes a LINE on the graph, see?[br][br]When there is an input (x) and an output (y), it makes a line.[br][br]How? Ok. So. Look at the line. [br]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). [br]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)[br][br]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)[br][br]1. It looks like it crosses at (-1, 0) then (0, 2) then (1, 4) and 2 more points that I can see[br]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) )[br]3. That says "POINT A IS" and you'll see it put a dot on your line.[br]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.[br]5. Make sense? Good.[br][br]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...[br]TRY putting the input in the bacon machine (aka function equation) and solve it like we did in the beginning up top.[br]so y = 2 (0) + 2 (I subbed x, which in our point was zero, for the input)[br]answer?[br]y = ___[br]Look again at your ordered pair. WHAT???? THE Y IS 2?!?!?!! Yes.[br]WHEN you plug in an input of any x, and get any output, you can plot that point on a graph.[br][br]If I had y = 2x + 1 [br]and decided x would be 1 (the first half of my ordered pair will be 1)[br] y = 2 (1) + 1 I[br]y = 2 + 1 [br]y = 3[br][br]So, my first ordered pair is (1, 3)... [br][br]now, plug in a (-1) for x, and a 2 for x. [br]You'll have 2 more ordered pairs as answers[br][br]You'll write down 3 ordered pairs.[br][br]NOW comes the fun part. In the "input" box below, type in your function from this problem (the actual equation we used).[br][br]See the line it made on the graph?[br]Now look at your ordered pairs.[br]Notice anything?[br]Text me now (or Google Hangout Message me--gigglinggalaxy@ gmail and tell me what you notice.[br][br]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!!![br]NO HOMEWORK TOMORROW IF YOU CAN FIGURE OUT HOW THIS WORKS BEFORE THE LESSON.[br][br]