Section+3.1

 polynomial function: f(x) = x^2  exponential function: f(x) = 2^x exponential function: f(x) = a x b^x where: a does not = 0, b>0, and b does not equal 1 "a" = initial value (when x=0) "b" = base Each "y" value is multiplied by 3 in order to get the next y value, therefore, the base (b) is 3. The "a" value is 4 because y=4 when x=0.
 * EXPONENTIAL AND LOGISTIC FUNCTIONS **
 * example**: Determine the formula for the given data
 * x || y ||
 * 0 || 4 ||
 * 1 || 12 ||
 * 2 || 36 ||
 * 3 || 108 ||
 * equation: y = 4 x 3^x

__Two Types (growth and decay):__ 1)** Exponential Growth : b>1 Graph: y=2^x After plugging these ordered pairs onto your graph, you will see what the exponential growth looks like. The "y" values will increase as the "x" values approach positive infinity.
 * X** **Y**
 * -2 || 1/4 ||
 * -1 || 1/2 ||
 * 0 || 1 ||
 * 1 || 2 ||
 * 2 || 4 ||

Graph y = 1/2^x After plugging these ordered pairs onto your graph, you will see what the exponential decay looks like. The "y" values will decrease as the "x" values approach positive infinity.
 * 2)** Exponential Decay: 0>b>1
 * x || y ||
 * -2 || 4 ||
 * -1 || 2 ||
 * 0 || 1 ||
 * 1 || 1/2 ||
 * 2 || 1/4 ||

__**PROPERTIES**__ 1. growth: b>1 decay 01 (growth), 0>b>1 (decay)
 * h**: shit left/right
 * k**: shift up/down

Your graph demonstrates growth (because of the 2) and it should shift 1 to the right. Find one other point to be sure of where your line is headed. This graph should look the same as the y=2^x graph, however it is shifted 1 to the right due to the x-1.
 * example:** g(x) = 2^x-1


 * example:** h(x) = 2^-x .......convert it to h(x) = 1/2^x This is a parent function, so it is much easier to work with. Use the graph discussed under the exponential decay section.

things to include in graph: growth, reflection over x-axis, vertical stretch by 3, left 1, down 5 NOTE: The horizontal asymptote was originally just y=0, however it is now also y= -5 because of the -5 in the equation. Using all of this information, find another point and draw the graph!
 * example:** k(x) = -3 x 2^(x+1) - 5

__**The Natural Base**__: "e" is approximately equal to 2.72 (Euler) f(x) = e^x (growth) (1, 2.72) - plot this point and draw the growth graph accordingly

__**Logistic Growth Function:**__ restricted growth bounded above and below by horizontal asymptotes let a, b, c, k be positive constants, with b>1 f(x) = (c) / (1+ a x b^x) or f(x) = (c) / (1x a x e^-x)
 * horizontal asymptotes:** y=0, y=c
 * range**: (0,c)

f(x) = (8) / (1+3 x 0.7^x) horizontal asymptotes: y=8, y=0 y-int: (0,2)
 * example:** Find the y-int and horizontal asymptotes


 *  HOMEWORK FOR THIS SECTION = **pg 286 #1-29 and pg 287 # 41-55