Let a and b real number constants. An exponential function in x is a function that can be written in the form: f(x) = a  b x
Where a is nonzero, b is positive and b ≠ 1.
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 a (the constant) is the initial value of f (the value at x = 0) b is the base

Identifying exponential functions y = x 8 y = 5 x y = x √x
Computing Exact Value f(x) = 3  5 x x = 0 f(x) = -2  3 x x = 1/3 f(x) = 2 x x = -3/2

Determine a formula for the exponential function: g(x) = h(x) =

For any exponential function f(x) = a  b x and any real number x, f(x +1) = b  f(x)
If a > 0 and b > 1 , the function f is increasing and is an exponential growth function. The base b is its growth factor
If a > 0 and b < 1 , the function f is decreasing and is an exponential decay function. The base b is its decay factor.

Exponential Functions f(x) = b x f(x) = b x for b > 1
 Domain:
 Range:
 Continuity:
 Symmetry:
 Bounded?
 Extrema:
 Horizontal Asymptote:
 Vertical Asymptote:
 End Behavior for function #1
 End Behavior for function #2 f(x) = b x for 0 < b < 1

g(x) = 2 x-1 h(x) = 2 -x k(x) = 3  2 x

f(x) = e x
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Domain:
Range:
Continuity:
Symmetry:
Bounded?
Extrema:
Horizontal Asymptote:
Vertical Asymptote:
End Behavior:
Definition:

Exponential Functions and the Base e
Any exponential function f(x) = a b x can be rewritten as f(x) = a e kx for an appropriately chosen real number constant k.
If a > 0 and k > 0 , then f(x) = a  e kx is an exponential growth function.
If a > 0 and k < 0 , then f(x) = a  e kx is an exponential decay function.

Exponential function transformations h(x) = e -x g(x) = e 2x k(x) = 3e x

Logistic Growth Functions or
If b < 1: growth
If b > 1: decay limit to growth
End
Behavior:
Horizontal Asymptotes: y= 0 and y = c

 Domain:
 Range:
 Continuity:
 Symmetry:
 Bounded?
 Extrema:
 Horizontal Asymptote:
 Vertical Asymptote:
 End Behavior

Graph the function and find the y-intercept and horizontal asymptotes .
f(x) = 16
1 + 3e -2x
Y-intercept:
Horizontal Asymptotes:

The population of North Carolina can be modeled by:
P(t) = 12.79
1 + 2.402e -0.0309x
P: Population in Millions t: Number of years since 1900
When was the population of North Carolina 10 million?

Population Models: Assuming the growth is exponential, when will the population of Austin surpass 800,000 people?

 Pg. 286-288 (2,6,8, 11, 12, 14,20, 24,32,34,42,56)
 Define Terms (Entire Sheet due Friday)