Aim: What is the exponential function? Do Now: Given y = 2x, fill in the table x -3 -2 -1 0 1 2 3 1/8 ¼ ½ 1 2 4 8 y HW: Worksheet

y = 2x

Exponential Functions Graph the curves y = 2x, y = 3x, y = 4x on the same coordinate system What about y=(1/2)x ? x 2x 3x 4x -3 1/8 1/27 1/64 -2 1/4 1/9 1/16 -1 1/2 1/3 1 2 3 4 9 16 27 64 y = 4x (1/2)x 8 4 2 1 1/2 1/4 1/8 y = 3x x y y = 2x y=(1/2)x 1

a = constant b = base x = variable All base exponential functions are similar because they all go through the point (0,1), regardless of the size of their base number Exponential Functions are written in the form: f(x) = abx a = constant b = base x = variable

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

if b > 1, the graph represents Exponential Function: Any equation in the form f(x) = Cbx. 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) = 2x Exponential Decay Exponential Growth We will take a look at how these graphs “shift” according to changes in their equation...

f(x) = (1/2)x f(x) = (1/2)x + 1 f(x) = (1/2)x – 3 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) = Cbx + k are shifted vertically by k units.

f(x) = (2)x f(x) = –(2)x f(x) = –(2)x + 2 – 3 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 (0,1) (0,-1) (-2,-4) Notice that f(0) = 1 This graph is a reflection of f(x) = (2)x . The graph is reflected over the x-axis. 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

e = 2.71828…. 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) = ex is the natural exponential function.

f(x) = ex

The population of the United States can be modeled by the function p(x) = 80.12e0.131x where x is the number of decades since 1900 and p(x) is the population in millions graph p(x) over the interval 0  x  15 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 = 3x over -3  x  3

Sketch the graph of over -2  x  2