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# Exponential functions

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### Exponential functions

1. 1. The exponential function is very important in math because it is used to model many real life situations.  ◦ For example: population growth and decay, compound interest, economics, and much more.
2. 2. f ( x) b x Question… How is this function different from functions that we have worked with previously?
3. 3. Answer: The exponent is the variable! f ( x) b x Question… b = The base b >0 and b ≠ 1 x = The exponent x=any real number Is f(x)=x3 an exponential function? NO
4. 4.  One of the most common exponential functions is x f ( x)  2 The graph looks like this:
5. 5. f ( x) 2 x The graph starts off slow but then increases very rapidly  The x-axis (y=0) is an asymptote.  X can be any real number, for example: f (x) 2 3  (0,1) is the y intercept  Models Exponential Growth
6. 6. Question… What would a graph look like if b is between 0 and 1? For example: x f ( x) 1 2
7. 7. f ( x) 1 2 x    The graph starts off very high but then decreases very rapidly The x-axis (y=0) is an asymptote. X can be any real number, for example: f (x)  1 2 (0,1) is the y intercept Models Exponential Decay
8. 8. Definition f ( x) b b>1 x b = The base b >0 and b ≠ 1 x = The exponent x=any real number 0<b<1
9. 9.  What is an exponential equation? ◦ An equation where the variable is the exponent. ◦ Example: 2 x 1 8
10. 10. 2 1 8 x Any ideas? What if we changed the right side to 2 x 2 3 2 3 Now What?
11. 11. x 2 ◦If 2 ◦Then x=? ◦Check: -3= 1 ◦2 8 3 3 Then we have solved 2 x 1 8
12. 12. 1.Express each side of the equation as a power of the same base. 2.Set the exponents equal and solve. 3.Check your answer Main idea If the bases are the same, set the exponents equal!
13. 13.  Solve for x: 7 x 2 7 3x Are the bases equal? YES Set the exponents equal. Solve for x. x 2 3x All we have here is a simple Algebra problem x 1
14. 14. x = -1 7 7 x 2 ( 1) 2 7 3 3x 7 3( 1) 7 3 7 It checks!
15. 15.  Solve for x: 5 4 t t 1 25 Are the bases equal? NO Change the right side to: Simplify: Solve! 5 4 t t 5 2 2t 2 2 t 1 (5 )
16. 16. t=2 4 t t 1 5 25 4 2 2 1 5 25 2 1 5 25 It checks!
17. 17.  Solve for x: 49 x 2 7 7 11 x 4
18. 18.  Solve for x: 49 x 2 7 7 Are the bases equal? NO 2 x Change both sides to: (7 ) Simplify: Solve! 7 2x 4 2 7*7 3 2 7 3 2x 4 2 11 x 4 1 2
19. 19. x = 11/4 x 2 49 11 49 4 2 3 4 3 2 4 73 7 7 23 2 3 2 49 7 (7 ) 7 It checks!
20. 20. REVIEW 1. Exponential function: f ( x) b Key Point: 2. Exponential Equations: Example: Rewrite as: x The exponent is the variable b= the base b >0 and b ≠1 X= the exponent X = any real number An equation where the exponent is the variable 4x 6 2 16 4x 6 4 2How to solve: 2 Set exponents 4x 6 4 If the bases are the equal: Check: same, set the10 exponents x Solve: equal! 4 2 4( 10 ) 6 4 10 6 2 2 4 16 16 16 It checks!
21. 21. Homework: Exponential Equations Worksheet