**dan's precalculus lesson**

**dan's precalculus lesson**

*©1997-2016 - all rights reserved - dan bach @dansmath*

**Functions and Graphs **

**What's a function? **

Ok, I'm going out on a limb here, but I'd say the concept of "function" is THE most important in mathematics. Any arguments from math teachers or anyone?

A **function** is a way of dealing with an **input**, applying some **rule** (the function), and then getting an **output**. What's the catch? There can be at most one output for every input. The inputs that "make sense" form the **domain **of the function, and the answers or outputs form the **range.**

Function Notation: We can call the inputx, the rulef, and then the output isf(x).

This DOES NOT mean "f times x" , it's just a notation device to record the input and output.

For example, if f(x) = x^2, then f(3) = 3^2 = 9, not f times 3 (meaningless).

Think of f(x) = x^2 as **f( ) = ( )^2** ; that way you can safely plug in negative numbers or even other expressions:

f(-5) = (-5)^2 = 25 ; f(x+h) = (x+h)^2.

**The graph of a function**

Graphs are a visual way of displaying numerical information.

The **graph** of a function y = f(x) is the set of all points (x, y) with y = f(x), so this means: *put a dot at*** (x, f(x))** *for every x that makes sense* (the domain).

Below are **four functions and their graphs:***(plotted in Mathematica)*

**Moving graphs around**

If you have the graph of y = f(x) you can *move it to the right* c units by graphing **y = f(x – c)**. For example, to move the parabola** y = x^2** to the right by 3 units, you'd graph y = f(x – 3) or **y = (x – 3)^2** .

To **move a graph up or down**, you'd change the y-coordinates, so graphing **y = x^2 + k **puts the "vertex" at (0,k). In general to move the graph up by k, do **y = f(x) + k **.

By doing a combination of both, we can move the graph anywhere in the plane:**y = f(x – c) + k**, or in the case of the parabola,**y = (x – c)^2 + k**.

My *danimation* shows both effects:

**Four-way functions **

Part of the "reform" movement in precalculus and calculus is to look at functionsfour ways:(1) Verbally, (2) Numerically, (3) Symbolically, (4) Graphically.

**(1) "The squaring function" **would **describe** what's being done to the input to get the output.

**(2) **Make a **table of values** for the inputs and outputs:

** x x^2**

1.0 1.00

1.1 1.21

1.2 1.44

1.3 1.69

1.4 1.96

1.5 2.25

**(3) **Describe the output **algebraically **in terms of the input: **f(x) = x^2**, where f(x) is the function notation described above. *I like f( ) = ( )^2.*

**(4) ** Draw a **graph** of y = f(x) on a pair of coordinate axes; here's **y = x^2.**

**Exponential and Logarithm Functions**

If you get a job in June that pays $1 the first day, $2 the second day, $3 the third day, etc, earning one more dollar per day each day, you'd have a total for the month of:

1 + 2 + 3 + . . . + 30 = 30(30 + 1) / 2 = $465.

This daily salary grows **linearly**, and the running total (465 is a "triangular number") grows** ****quadratically**.* (See above red graphs)*

But what if your salary **doubled** every day for a month? Think about it first.

Ok, you'd have 1 + 2 + 4 + 8 + . . . + 2^29 = 2^30 – 1 = $1,073,741,823 (a billion bucks)*(The 30th day is 2^29 because you start with 1 = 2^0)*

This is called **exponential** **growth**, and you can see its rapid rise just for three days in the graph. Some more realistic examples are compound interest, bacteria or population growth, and spread of disease or rumors in a large group.

*(The graph of y = 2^x appears at the right)*

**Simple and Compound Interest**

Say you deposit some money, initiallyPdollars (or your favorite unit of currency), into a bank. The bank pays youinterestto keep it there; it can pay eithersimpleorcompoundinterest.

**Simple interest **is simple to figure out; hence the name! If you make, say, 8% interest for 5 years, that makes 40%. The total amount of money **A** that you'll have after **t** years at interest rate **r** is given by

** A = P** **(1 + r** **t)**

So if you put in $2000 for 5 years at 8% interest, you get** A** = 2000(1 + (0.08)(5)) = 2000(1 + 0.40) = 2000(1.40) =** $2,800.**

Notice the factor of 1.40 means you've made a profit of 40%.

A *graph* of your money A as a function of t would look like a *straight line;* you always make 8% of just $2000, or $160, each year, forever. The amount **A** is a **linear function **of **t**.

**Compound interest **is more comp-licated. The difference is that you make interest on the interest.

After 1 year your $2000 would grow by 8% (which puts it at 108% of the original), so multiply by 1.08 to include the original deposit:** A(1) = 2000 (1.08) = $2,160.**

But the second year you'd want to make interest on the $2160, not just on the $2000. So the second year balance should be 2160(1.08) = $2332.80.

But this equals 2000(1.08)(1.08) = 2000 (1.08)^2. After 5 years you'd have a total of:** A(5) = 2000 (1.08)^5** = 2000 (1.4693280768) = $2938.65615** **≈** $2,938.65**** ***(the bank rounds down)*

Note the factor of 1.469... means your effective profit is now** 46.9% **(compared to 40% for simple interest.)

The general formula (if interest is "**compounded yearly**" like this) for your balance at time **t** yrs is:** A = P** **(1 + r)^t** (where the ^ means *to the power of*)

Since the variable **t** is in the exponent, this time **A** is an **exponential function** of **t**., and your balance would grow at an ever-increasing rate (dollars per year), but a constant percentage rate. The graph would be an exponential curve like the y = 2^x example above* (only increasing a lot more slowly.)*

If the interest is **compounded n times per year** (n=4 for quarterly, for example), then this means you get paid that fraction (1/n) of the interest n times per year.

With your 8% annual rate, each quarter you'd gain 8% / 4 = 2% of your money, so you multiply your balance by 1.02, four times a year.

In** t **years this makes **4t** quarters (3-month periods); after 5 years you'd have 20 quarters, and so:

** A(5) **= 2000 (1.02)^20 = 2000 (1.*4859*47396) ≈ **$2,971.89. **This represents a *48.6%* profit.

The general formula (if interest is "**compounded yearly**" like this) for your balance at time t yrs is:

** A = P** **(1 + r/n)^(nt) ** *(remember to do the r/n first!)*

It seems to be better when the interest is compounded more often! Let's make a table of how your final balance after 5 years depends on n, the number of compounding periods per year.

** n**** (1 + r/n)^(nt)**** = growth factor**** A = P*(growth)**

1* (yearly) * (1 + .08/1)^(1*5) 1.08^5 1.46933 2938.65

2 *(semiann) * (1 + .08/2)^(2*5) 1.04^10 1.48024 2960.48

4 *(quarterly) * (1 + .08/4)^(4*5) 1.02^20 1.48595 2971.89

12* (monthly)* (1 + .08/12)^(12*5) 1.00667^60 1.48985 2979.69

52 *(weekly) * (1 + .08/52)^(52*5) 1.001538^260 1.49137 2982.73

365* (daily) *(1 + .08/365)^(365*5) 1.000219^1825 1.49176 2983.51

Apparently the total grows, but not very much, as n goes to infinity. In the **limit**, we say the interest is **compounded continuously**, and the amount is given by the easy 'pert formula':** A = P** **e^(rt)** where e = base of natural logs; e ≈ 2.71828...

Going back one last time to our example, we just put in r = 0.08 and t = 5:** A = 2000 e^(0.08 * 5) **= 2000 e^0.4 = 2000 (1.49182) = **$2,983.65.**

This is the best we can do in 5 years with the given interest rate; you get a 49.18% profit; it's not much stronger than the weekly compounding (hehe).

**Logarithms **

What arelogarithms? And why are they misunderstood? Here's an easy way to deal:LOGS ARE EXPONENTS.(I have a t-shirt with this slogan.)

**Example**: You know **2^3 = 8** because 2^3 means 2 * 2 * 2. (3 factors.)

What's the base? 2. What's the exponent? 3. What's the answer? 8.

So we record the exponent, 3, that you need to put onto a base of 2 to get an answer of 8.

Say this as **"****log, ****base 2, of 8, equals 3****"** because the exponent (the log) is 3.

This is written as** log2(8) = 3. ***(The 2 should be smaller and below, like a subscript.)*

I'm too cheap to put a separate **graph for y = log****2(x)**, so just flip the above y = 2^x graph across the line y = x to get the log graph; this is because they're inverse functions.