Aim: What is an exponential function? Do Now: Solve/evaluate: 64 = 23n + 1 Poll: n = 5/3 x = 6561 Rewrite in exponential form

Properties of Exponents Product of Powers Property am • an = am+n Power of Power Property (am)n = am•n Power of Product Property (ab)m = ambm Negative Power Property a-n = 1/an, a 0 Zero Power Property a0 = 1 Quotients of Powers Property Power of Quotient Property

beware! -23/2 is not the same as (-2)3/2 Types of exponents Positive Integer Exponent an = a • a • a • • • • a n factors Zero Exponent a0 = 1 Rational Exponent 1/n Rational Exponent m/n Negative Exponent - m/n beware! -23/2 is not the same as (-2)3/2

Exponential Function - Outbreak Disease Control - “Outbreak” y = 2x

y = a • bx y = 1 • 2x y = 2x Exponential Function . . have variables as exponents, and where a  0, base b > 0, and also b  1. When a = 1, graph always goes through (0,1) (0,1) The domain (x) is the set of real numbers: (-, ) The range (y) is the set of positive real numbers: (0, ) The x-axis is a horizontal asymptote: ax  0 as x  - If b > 1, the graph is increasing and continuous What happens as b increases in value? y = 1 • 2x y = 2x As b increases in value from 1, the slope of the graph gets steeper

y = a • bx The b Affect y = (1/2)x y = 2x y = 1 (0,1) If b > 1, the graph is increasing - Growth If 0 < b < 1 the graph is decreasing - Decay If b is a positive number other than 1, the graphs of y = bx and y = (1/b)x are reflections through the y-axis of each other What if b = 1? horizontal line: y = 1

if b < 0, no longer the exponential function The b Affect: b < 0 y = a • bx What if b < 0? Let b = (-2)? y = 1 • (-2)x if b < 0, no longer the exponential function graph: table: x = .1 table: x = 1

y = a • bx The a Affect y = 2x y = (2)2x y = (4)2x Graph: y = 2x; y = (2)2x; y = (4)2x a = 4 (0,4) (0,2) a = 2 (0,1) a = 1

The a Affect y = a • bx (0,-1) Graph f(x) = -2x = -1 • 2x Graph f(x) = -(1/2)x = -1 • (1/2)x Graph f(x) = -6x = -1 • (6)x

Graph the exponential functions The a affect Graph the exponential functions

substitute & solve for a Model Problem Write an exponential function y = abx for a graph that includes (2, 2) and (3, 4) general form substitute (2,2) for x & y solve for a substitute for a substitute (3,4) for x & y b = 2 simplify & solve for b a = 1/2 substitute & solve for a substitute for a and b

y  2.7183 is asymptotic to f(x). Where’d e Come From? Graph y  2.7183 Poll for asymptote: Use graphing calculator to find to nearest thousandth y  2.7183 is asymptotic to f(x). x = 1 x = 100 x = 10000

Natural Exponential Function f(x) = ex f(x) = (2.71828. . .)x Evaluate e2 e-1 e0.48 = 7.389056099 = 0.3678794412 = 1.616074402 Graph f(x) = 2e0.24x Graph f(x) = (1/2)e-0.58x

Transforming Functions If k and h are positive numbers and f(x) is a function, then f(x) + k shifts f(x) up k units f(x) – k shifts f(x) down k units f(x + h) shifts f(x) left h units f(x – h) shifts f(x) right h units f(x) = (x + h)2 + k - parabolic f(x) = |x + h| + k - absolute value ex. f(x) = (x – 4)2 + 4 is the image of g(x) = x2 after a shift of 4 units to the right and four units up or a translation of T4,4. NOTE: k is not the y-intercept. what is the y-intercept for f(x) = (x – 4)2 + 4?

Transforming the Exponential Function Graph the exponential functions y = 4x, y = 4x + 2, y = 4x – 3

Transforming the Exponential Function Graph the exponential functions y = 2x+3, y = 2x-2, y = 2x–1 – 2 (1, 1) (2, 1) (-3, 1) (0, 1)

Transforming the Exponential Function Write the equation and graph the exponential function f(x) = 2x after a reflection in both the x- & y-axes, and a vertical translation of 4. Graph y = 2x Graph y = (1/2)x Activity Center: send graph and equation Reflect thru x-axis: y = -(1/2)x Shift up 4 units y = -(1/2)x + 4

Transforming the Exponential Function Write the equation and graph the exponential function after a reflection in the x-axis, vertically stretched by a factor of 2, a vertical translation of 2 and a horizontal of 2. Graph y = 2x Reflect thru x-axis: y = -2x Activity Center: send equation and graph Shift right 2 units y = -(1/2)x –2 Shift up 2 units y = -(1/2)x –2 + 2

Transforming the Exponential Function The graph of y = (1/4)x is translated 5 units upward, 8 units to the right and then reflected in the x-axis. What’s the equation? Activity Center: send equation and graph f(x) = -f(x)

Transforming the Exponential Function The graph of y = (1/4)x is translated 5 units upward, 8 units to the right and then reflected in the x-axis. What’s the equation? Activity Center: send equation and graph f(x) = -f(x)