Exponential Functions and Their Graphs

Exponential Function Families We’ve already learned about This is the parent function We’ll expand this to Note a=1 and h=0 in the parent function

Horizontal shift to the right Component Property + k Vertical shift up - k Vertical shift down - h Horizontal shift to the right + h Horizontal shift to the left a > 1 Stretch a < 1 Compression -a Reflection

Graph of Exponential Function (a > 1) The graph of f(x) = ax, a > 1 y 4 Range: (0, ) (0, 1) x 4 Horizontal Asymptote y = 0 Domain: (–, ) Graph of Exponential Function (a > 1)

Graph of Exponential Function (0 < a < 1) The graph of f(x) = ax, 0 < a < 1 y 4 Range: (0, ) Horizontal Asymptote y = 0 (0, 1) x 4 Domain: (–, ) Graph of Exponential Function (0 < a < 1)

Example: Sketch the graph of f(x) = 2x. x f(x) (x, f(x)) y x f(x) (x, f(x)) -2 ¼ (-2, ¼) -1 ½ (-1, ½) 1 (0, 1) 2 (1, 2) 4 (2, 4) 4 2 x –2 2 Example: Graph f(x) = 2x

Example: Translation of Graph Example: Sketch the graph of g(x) = 2x – 1. State the domain and range. y f(x) = 2x The graph of this function is a vertical translation of the graph of f(x) = 2x down one unit . 4 2 Domain: (–, ) x y = –1 Range: (–1, ) Example: Translation of Graph

Example: Reflection of Graph Example: Sketch the graph of g(x) = 2-x. State the domain and range. y f(x) = 2x The graph of this function is a reflection the graph of f(x) = 2x in the y-axis. 4 Domain: (–, ) x –2 2 Range: (0, ) Example: Reflection of Graph

Graph of Natural Exponential Function f(x) = ex The graph of f(x) = ex y x f(x) -2 0.14 -1 0.38 1 2.72 2 7.39 6 4 2 x –2 2 Graph of Natural Exponential Function f(x) = ex

The irrational number e, where e  2.718281828… is used in applications involving growth and decay. The number e

Pert

Continuously Compounded Interest Time Interest Rate (Annual) Final Amount Principal (beginning amount) Euler’s Number (2.71828…)

Pert Example Suppose you won a contest at the start of 5th grade that deposited $3000 in an account that pays 5% annual interest compounded continuously. How much will you have in the account when you enter high school 4 years later? Express the answer to the nearest dollar.