1. Exponential Functions Evaluate Exponential Functions Graph Exponential Functions Define the number e Solve Exponential Equations

2. Laws of Exponents If s, t, a and b are real numbers with a>0 and b>0 then, a s ∙ a t = a s+t a -s = (a s ) t = a st (ab) s = a s ∙b s a 0 =1 1 s =1

3. Exponential Functions An exponential function is a function of the form f(x) = a x where a is a positive real number (a>0) and a ≠1. The domain of f is the set of all real numbers

4. Activity Evaluate f(x) = 2 x at x= -2,-1,0,1,2, and 3 Evaluate g(x) = 3x+2 at x= -2,-1,0,1,2, and 3 Comment on the pattern that exists in the values of f and g

5. In words For an exponential function f(x) = a x, for 1 unit changes in the input x, the ratio of consecutive outputs is the constant a.

6. Theorem For an exponential function f(x)= a x, a>0, a ≠1, if x is any real number, then

7. Graphing an Exponential Function Graph f(x) = 2 x What is the domain What is the range What are the x intercepts

8. Properties of the Exponential Function f(x) = a x, a>1 1.The domain is the set of all real numbers; the range is the set of positive real numbers 2.There are no x intercepts, the y intercept is 1 3.The x axis (y=0) is the horizontal asymptote as x →- ∞ 4.f(x) = a x, a>1 is an increasing function and is one to one 5.The graph of f contains the points (0,1) (1,a) and (-1, ) 6.The graph of f is smooth and continuous, with no corners or gaps

9. Properties of the graph of an Exponential function f(x) = ax, 0<a<1 1.The domain is the set of all real numbers; the range is the set of positive real numbers 2.There are no x intercepts; the y intercept is 1 3.The axis (y=0) is a horizontal asymptote as x →∞ 4.f(x)= a x, 0<a<1, is decreasing function and is one to one 5.The graph of f contains the points (0,1), (1,a) and (-1, )

10. The base e The number e is defined as the number that the expression

11. Graphing exponential functions using transformations Graph compare this graph to The graph (The e button is shift ln)

12. Classwork/ Homework Page 255 examples 21-28 even, 29-36 and 53-70