Exponential Functions and Their Graphs Sec. 5.1 Exponential Functions and Their Graphs

Algebraic Functions Polynomial functions and rational functions

Transcendental functions Exponential functions and logarithmic functions

Definition of Exponential Function f(x) = ax a > 0, a ≠ 1, and x is a real number a is the base x is the exponent

Natural Base Represented by e e ≈ 2.71828… (Irrational) f(x) = ex is called the Natural Exponential Function

Ex. 1 2-3.1 .1166291 2-π .1133147

Ex. 2 Evaluate and round to nearest thousandth 3.92.1 2-3π 4√5291

Ex. 5 a) e-2 b) e-1 c) e1 d) e2

Graph of Exponential Functions f(x) = 2x g(x) = 4x x 2x 4x -2 -1 1 2 3

F(x) = 2-x G(x) = 4-x x 2-x 4-x -3 -2 -1 1 2

Note: the negative exponent reflects about the y-axis Find for each Domain Range Intercept Direction Horizontal Asymptote Continuous?

Graph on the calculator h(x) = (½)x It is the same as f(x) = 2-x

What will happen to the graph of f(x) = 2x Which means 2x+2 f(x-3) f(x) + 3 Which means 2x + 3 f(x) – 2 -f(x)

Summary Given f(x) = ax f(-x) = a-x f(x+b) = ax+b f(x-b) = ax-b Reflects about y axis Horizontal left b units Horz. Right b units Vertical up b units Vertical down b units Reflects over x axis