Seu SlideShare está sendo baixado. ×

# Exponential_Functions.ppt

Anúncio
Anúncio
Anúncio
Anúncio
Anúncio
Anúncio
Anúncio
Anúncio
Anúncio
Anúncio
Anúncio
Próximos SlideShares
Exponential_Functions.ppt
Carregando em…3
×

1 de 25 Anúncio

# Exponential_Functions.ppt

sample lecture

sample lecture

Anúncio
Anúncio

Anúncio

### Exponential_Functions.ppt

1. 1. Exponential Functions
2. 2. Objectives To use the properties of exponents to:  Simplify exponential expressions.  Solve exponential equations. To sketch graphs of exponential functions.
3. 3. Exponential Functions A polynomial function has the basic form: f (x) = x3 An exponential function has the basic form: f (x) = 3x An exponential function has the variable in the exponent, not in the base. General Form of an Exponential Function: f (x) = Nx, N > 0
4. 4. Properties of Exponents X Y A A   X Y A  XY A X Y A    Y X A  X Y A A    X AB  X X A B X A B        X X A B X A  1 X A 1 X A  X A X Y A  Y X A   X Y A 
5. 5. Properties of Exponents 2 3 2 2   5 2 32  2 6 2 2   4 2 4 1 2  1 16    2 3 2  6 2 64  Simplify:
6. 6. Properties of Exponents 3 2 3         3 3 2 3   3 3 3 2  7 9 3 3  2 3 2 1 3  1 9     1 1 2 2 2 8  1 2 16 16  27 8  4    1 2 2 8   Simplify:
7. 7. Exponential Equations Solve: Solve: 5 125 x  3 5 5 x  3 x  1 2 ( 1) 7 7 x    1 2 1 x    1 2 x 
8. 8. Exponential Equations 8 4 x    3 2 2 2 x  3 2 x  2 3 x  3 2 2 2 x  8 2 x    3 1 2 2 x  3 1 x  1 3 x  3 1 2 2 x  Solve: Solve:
9. 9. Exponential Equations 1 3 27 x    1 3 3 3 27 x  19,683 x    1 3 27 x    1 3 27 x   3 x   3 x   3 3 3 x   Not considered an exponential equation, because the variable is now in the base. Solve: Solve:
10. 10. Exponential Equations 3 4 8 x    4 3 4 3 3 4 8 x    4 3 8 x    4 2 x  16 x  Not considered an exponential equation, because the variable is in the base. Solve:
11. 11. Exponential Functions General Form of an Exponential Function: f (x) = Nx, N > 0 g(x) = 2x x 2x g(3) = g(2) = g(1) = g(0) = g(–1) = g(–2) = 8 4 2 1 1 2 1 2  2 2 2 1 2  1 4 
12. 12. Exponential Functions g(x) = 2x 2 0 x 3 2 1 0 -1 -2 g(x) 8 4 2 1 1/2 1/4
13. 13. Exponential Functions g(x) = 2x
14. 14. Exponential Functions h(x) = 3x x 2 1 0 9 3 1   h x –1 –2 1 3 1 9
15. 15. Exponential Functions h(x) = 3x
16. 16. Exponential Functions Exponential functions with positive bases greater than 1 have graphs that are increasing. The function never crosses the x-axis because there is nothing we can plug in for x that will yield a zero answer. The x-axis is a horizontal asymptote. h(x) = 3x (red) g(x) = 2x (blue)
17. 17. Exponential Functions A smaller base means the graph rises more gradually. A larger base means the graph rises more quickly. Exponential functions will not have negative bases. h(x) = 3x (red) g(x) = 2x (blue)
18. 18. Exponential Functions     1 2 x j x  Exponential functions with positive bases less than 1 have graphs that are decreasing.
19. 19. Properties of graphs of exponential functions Function and 1 to 1 y intercept is (0,1) and no x intercept(s) Domain is all real numbers Range is {y|y>0} Graph approaches but does not touch x axis – x axis is asymptote Growth or decay determined by base
20. 20. The Number e e  2.71828169 A base often associated with exponential functions is:
21. 21. The Number e Compute:   1 0 lim 1 x x x   x –.1 –.01 –.001 2.868 2.732 2.7196   1 1 x x    1 0 lim 1 x x x    x .1 .01 .001 2.5937 2.7048 2.7169   1 1 x x    1 0 lim 1 x x x    2.71828169 
22. 22. The Number e Euler’s number Leonhard Euler (pronounced “oiler”) Swiss mathematician and physicist
23. 23. The Exponential Function f (x) = ex
24. 24. Exponential Functions x 2 1 0 4 2 1   j x –1 –2 1 2 1 4     1 2 x j x 
25. 25. Why study exponential functions? Exponential functions are used in our real world to measure growth, interest, and decay. Growth obeys exponential functions. Ex: rumors, human population, bacteria, computer technology, nuclear chain reactions, compound interest Decay obeys exponential functions. Ex: Carbon-14 dating, half-life, Newton’s Law of Cooling