1. Exploring Exponential Functions Using a Graphing Calculator

2. Objective – To be able to graph exponential Functions. Asymptote – is the line that a graph approaches as you move away from the origin.

3. Properties of Exponents Let a and b be real numbers and let m and n be integers. Product of powers property Power of power property Power of product property Negative exponent property Zero exponent property Quotient of powers property Power of a quotient property

4. Exponential Functions – involves the expression y = ab x base – is the value b, with the exponent Exponential Growth If a > 0 and b > 1 –1 –5–4–3–2–112543 4 1 2 3 5 6 9 8 7 10 y = 4(3) x Exponential Decay If a > 0 and 0 < b < 1 –1 –5–4–3–2–112543 4 1 2 3 5 6 9 8 7 10 y = 3(½) x

5. Example 1 Graph the function a) y = 2 / 3 (2) x 1) First draw an x/y box and find 2 points: –1 –5–4–3–2–112543 4 1 2 3 5 6 9 8 7 10 x y 0101 2/32/3 4/34/3

6. Example 2 Graph the function a)y = 4 ( 2 / 5 ) x 1) First draw an x/y box and find 2 points: –1 –5–4–3–2–112543 4 1 2 3 5 6 9 8 7 10 x y 0101 4 8/58/5

7. Example 3 Graph y = 23 (x-2) + 1 To graph: y = ab (x-h) + k, begin by graphing y = ab x. Then translate the graph horizontally by h units and vertically k units. 1) First graph y = 23 x x y 0101 2 6 –1 –5–4–3–2–112543 4 1 2 3 5 6 9 8 7 10 2) Then move h units horizontally and k units vertically h = 2 k = 1

8. THE NATURAL BASE e The natural base e is irrational. It is defined as follows: As n approaches + , (1 + 1 / n ) n approaches e  2.718281828459

9. The graph of is upward-sloping, and increases faster as x increases. The x-axis is a horizontal asymptote. The inverse function is the natural logarithm ln(x); The Natural Exponential Function

10. Pg. 62 7 – 16