Exponential Functions MM3A2e Investigate characteristics: domain and range, asymptotes, zeros, intercepts, intervals of increase and decrease, rate of change.

An exponential function is a function of the form where a is a positive real number (a > 0) and. The domain of f is the set of all real numbers.

(0, 1) (1, 3) (1, 6) (-1, 1/3) (-1, 1/6)

Summary of the characteristics of the graph of a >1 The domain is all real numbers. Range is set of positive numbers. No x-intercepts; y-intercept is 1. The x-axis (y=0) is a horizontal asymptote as a>1, is an increasing function and is one-to-one. The graph contains the points (0,1); (1,a), and (-1, 1/a).

(-1, 3) (-1, 6) (0, 1)(1, 1/3)(1, 1/6)

Summary of the characteristics of the graph of 0 <a <1 The domain is all real numbers. Range is set of positive numbers. No x-intercepts; y-intercept is 1. The x-axis (y=0) is a horizontal asymptote as 0<a<1, is a decreasing function and is one-to-one. The graph contains the points (0,1); (1,a), and (-1, 1/a). The graph is smooth continuous with no corners or gaps.

(0, 1) (1, 3)

(0, 1) (-1, 3)

(0, 3) (-1, 5) y = 2

Horizontal Asymptote: y = 2 Range: { y | y >2 } or Domain: All real numbers

Characteristics of Exponential Functions f(x) = a x Domain: (-∞,∞) y-intercept is the point (0,1) Continuous Increasing if a > 1 Range: (0,∞) No x-intercept Horizontal asymptote y=0 Decreasing if a < 1