Aim: What is the exponential function? Do Now: Given y = 2 x, fill in the table x /8 ¼ ½ y HW: Worksheet

y = 2 x

Graph the curves y = 2 x, y = 3 x, y = 4 x on the same coordinate system x y y = 2 x y = 3 x y = 4 x 1 Exponential Functions y=(1/2) x What about y=(1/2) x ? x2x2x (1/2) x -31/88 -21/44 1/ /4 381/8

Properties of the basic f(x) = abx function The domain consist of all real numbers x The range consists of all positive all real numbers y The function is increasing when b >1 and decreasing when 0 < b < 1 It is one to one function The x –axis is the horizontal asymptote to curve, toward the left when b>0 and toward the right for 0 < b <1

Exponential Function: Any equation in the form f(x) = Cb x. if 0 < b < 1, the graph represents exponential decay – the y-value is going down as x increases if b > 1, the graph represents exponential growth – the y-value is going up as x increases Examples: f(x) = (1/2) x f(x) = 2 x Exponential DecayExponential Growth We will take a look at how these graphs “shift” according to changes in their equation...

Take a look at how the following graphs compare to the original graph of f(x) = (1/2) x : f(x) = (1/2) x f(x) = (1/2) x + 1 f(x) = (1/2) x – 3 Vertical Shift: The graphs of f(x) = Cb x + k are shifted vertically by k units.

Take a look at how the following graphs compare to the original graph of f(x) = (2) x : f(x) = (2) x f(x) = –(2) x f(x) = –(2) x + 2 – 3 Notice that f(0) = 1 (0,1) This graph is a reflection of f(x) = (2) x. The graph is reflected over the x-axis. (0,-1) Shift the graph of f(x) = (2) x,2 units to the left. Reflect the graph over the x-axis. Then, shift the graph 3 units down (-2,-4)

What is the difference between and? f(x) = 2 x f(x) = 2 -x

e = …. This number is called the natural base. It is called this because the value seems to occur naturally in many situations. e is an irrational number that is similar to the property of . e is a very important in calculus since the derivative of e is itself. The function f(x) = e x is the natural exponential function.

f(x) = e x

Given f(x) = 3 x, evaluate f(–2) f(–1) f(0) f(1) f(2)

The population of the United States can be modeled by the function p(x) = e0.131x where x is the number of decades since 1900 and p(x) is the population in millions a. graph p(x) over the interval 0  x  15 b. If the population of United States continues to grow at this rate, predict the population in the years 2010 and 2020.

Sketch the graph of y = 3 x over -3  x  3

Sketch the graph of over -2  x  2