SchecterAdvancedMath

Let's say you get a job helping someone with yardwork and you get $25 every Sunday for your efforts. After 4 Sundays you will have earned $25 x 4 = $100. If you worked for x Sundays then you would have earned $25 times x. In math we call rules like these functions. There is no good reason for the name function, and maybe we should just call these things rules, but function has stuck and that's what we use. A function is a rule that turns inputs into outputs. For example, if I put in 4, out comes 100; and if I put in 7, out comes 175; and if I put in x, out comes 25x, which means 25 times x. The only thing that must be true about a function is that it cannot change its mind. That means if 3 in gives 75 out, then when you put in 3 next time you still get 75. So if you got a raise you would need a new function to predict your earnings.

We usually give a function a name depending on what it is measuring. In t his example we might call the function Earnings, and we describe te function by writing:
Earnings(x) = 25x.
We read it this way: Earnings of x equals 25x.
It means simply that Earnings takes an input x and outputs 25x. A more interesting function is one that tells how many feet a rock falls x seconds after you drop it from a cliff on earth. We can call this function Distance, and physicists have discovered that Distance(x) = 16x². So after 1 second it falls 16 feet, and after 2 seconds it has fallen 64 feet, and after 3 seconds it has fallen 16 x 3 x 3 = 144 feet.

There are an infinite number of different functions but an important subset is the linear functions. Linear functions are ones where the output changes at a constant rate. That is for every unit change in the input, we should see the same change in the output. If you make a picture of a linear function, which we will discuss later, it will be a straight line, hence the name linear. For example, Earnings(x) = 25x is a linear function, becauyse every time you add an extra week, the Earnings go up by eaxctly $25.   Distance(x) = 16x² is not a linear function. When you increase the time from 1 second to 2 seconds, the distance increases from 16 to 64 feet, but when you go one more second from 2 to 3 seconds, the distance increases from 64 to 144. The first time period has the rock traveling 48 feet, while the second time period has the rock traveling 80 feet. The rock is speeding up or accelerating. This is not a linear function.

Let's say you started the summer with $530 in the bank, and you want a function that tells you what's in your bank account after x weeks of work. This is not quite the same as earnings because you started work with a bank account of $530. The Bankaccount function looks like this:
Bankaccount(x) = 530 + 25x. Note that the Bankaccount and Earnings functions both have the same rate of growth, namely 25 increase in output for every one increase in input. This constant rate of growth that is charcateristic of a linear function is called the slope.

Let's practice a few basic skills:

1. Given a linear function, calculate its slope.
2. Given the slope and one (input, output) pair, calculate the function.
3. Make a picture of a linear function.

If I give you any linear function there are dozens of ways to figure out the slope, and if you understand what slope means, then you can probably discover one yourself. Here is one way if you are having trouble. Let's say you have a function Mystery(x) = 3x - 9, and you want to know its slope. Recall that the slope of a linear function is how much it goes up for every unit increase of input. So one sure way to find the slope is just put in any two numbers you like that differ by one, and see how much the function goes up or down. For example, if you use 6 and 7, you get 3x6 - 9 and 3x7 - 9.   The difference between these two is 3. SO the slope is 3. That's it.

If you are the abstract type, you may have noticed that this works no matter what two consecutive numbers you use. If you put any number n into 3x-9, you get 3n-9, and if you put in n+1 you get 3(n+1) - 9 = 3n + 3 - 9 = 3n - 6. The difference between 3n - 6 and 3n - 9 is 3, and that is independent of n. If you are thinking -- ugh this abstract stuff is not for me, then hold up! Because the conclusion of this abstract detour is that the slope was hiding in the function all along! That is, if the function is written Mx ± N then the slope is M. The slope is just the number in front of the x. If you really understand this, it will save you a lot of time and avoid lots of dull arithmetic. Let's test and see if you get it.

What are slopes of each of the linear functions below:

8x + 10?      Answer: 8.
3x/2 - 9? Answer: 3/2. Slopes can be fractions. Be careful to identify all of the numbers in front of the x. 3x/2 is just (3/2)x in disguise.
15 - 2x?      Answer: -2. Slopes can be negative. This means that when you increase the input by 1, the output goes down by 2.
(8x-7)/8?   Answer: 1. If you rewrite the function in the way that reveals the slope Mx ± N, it looks like this 8x/8 - 7/8 = x - 7/8.
16x + 190    Answer: 16.

What if I tell you that a linear function has a slope of 3, do you know what the function is? No way! Because every linear function that starts with 3x has a slope of 3, including 3x, 3x + 1, 3x - 90. 3x + 8, ... There are an infinite number of linear functions whose slope is 3. You need to know more information to identify the function. If someone tells you one (input, output) pair for the function, that is enough information to uniquely identify the function. For example, let's say a function has slope 3, and when I input 4, out comes 36. Start out with the function looking like 3x. If you put in 4 into 3x you get 12. But you know you are supposed to get 36, so just fix it so that you do! Instead of 3x, use 3x + 24. Now just check to make sure that when you put in 4, out comes 36. Why not change the function to 9x instead of 3x + 24? That seems simpler, and when you put in 4, out still comes 36!   Well you tell me why! Answer: If we did that then the slope would not be correct.