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5.3 Exponential Functions

5.3 Exponential Functions. Anything in the form of f(x)= ab x where a>0 and b>0 and b≠1 is called an EXPONENTIAL FUNCTION WITH BASE B. (a is simply a scalar multiplier, the most general form would have a=1). The variable of the function is found in the exponent. (0,a). (0,a).

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5.3 Exponential Functions

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  1. 5.3 Exponential Functions

  2. Anything in the form of f(x)=abx where a>0 and b>0 and b≠1 is called an EXPONENTIAL FUNCTION WITH BASE B. (a is simply a scalar multiplier, the most general form would have a=1). • The variable of the function is found in the exponent. (0,a) (0,a)

  3. If f is an exponential function, and f(0)=4 and f(2)=36, find f(x) and f(-2). • Since f(0)=4, then ab0=4, thus a=4. • Then since f(2)=36, then 4b2=36, so then b=3. • Thus f(x)=4*3x • And f(-2)=??

  4. The graph of y=abx has y-intercept of 7. Find the value of a. • If h(x)=abx, h(0)=5, and h(1)=15, find values for a and b.

  5. For what values of b is the function f(x)=abx increasing? And for which values of b is it decreasing? • What is the domain for a function like f(x)=4*3x • What about the range?

  6. Transformations • Exponential graphs, like other functions we have studied, can be dilated, • reflected and translated. • It is important to maintain the same base as you analyze the transformations. Reflect @ x-axis Vertical stretch 3 Vertical shift down 1 Vertical shift up 3

  7. More Transformations Reflect about the x-axis. Vertical shrink ½ . Horizontal shift left 2. Horizontal shift right 1. Vertical shift up 1. Vertical shift down 3. Domain: Domain: Range: Range: Horizontal Asymptote: Horizontal Asymptote: Y-intercept: Y-intercept: Inc/dec? decreasing Inc/dec? increasing Concavity? down Concavity? up

  8. 5-4 The number e and the Function ex e, is the first letter of the last name of the man who first used the notation Leonhard Euler (pronounced oiler). e=2.71828182845904523536028747135266249775724709369995… Obviously this is an irrational number and its uses are immense, We often use e to help simplify more complex equations. e is called the natural number, therefore f(x)=ex is called the natural exponential function.

  9. How do we know that e is that number? Euler recognized that the expression of as n got very large and approached ∞ the value of the expression converged towards 2.71828… In advanced mathematics e is defined as the following limit

  10. The graph of the natural exponential function should come as no surprise if we understand that e is simply a number and in this case it is our base.

  11. How does f(x)=ex, compare to something like f(x)=2x or 3x? Hopefully this makes sense because e, is simply a number just like 2 and 3 are, so because of its value the exponential function behaves more like 3x than 2x.

  12. HWK. Pg. 183 3-6, • Pg. 190 13-16

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