# exponential functions graphs of exponential functions the natural base

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```					  Section 3-1

Exponential and Logistic Functions

Section 3-1

exponential functions
graphs of exponential functions
the natural base e
logistic functions
graphs of logistic functions
population models

1
Exponential Functions

standard form for an exponential function
f ( x) = a ⋅ bx
a is the initial value, b is the base
b must be positive and not equal to 1
if b > 1 then its exp. growth
if 0 < b < 1 then its exp. decay

Find the exp. function that passes through (0 , 4)
and (1 , 12)

2
Graphs
the basic exponential function graph passes
through (0 , a) and has a horizontal
asymptote at y = 0
growth (b > 1)

decay (0 < b < 1)

Graphs

the basic exponential      Try these on your
function can be             calculator:
stretched, reflected, and   y=2x
translated as we
previously learned.         y=-2x
y=2-x

x−        y=2x-2
f ( x) = a ⋅ b        +
y=2x-2
Y=3(2)x

3
Sketch the graph : f ( x ) = 3 ⋅ 2 x −2

The Natural Base e

one of the basic twelve functions introduced
in Chapter 1 was f ( x) = e x
e is known as the natural base e ≈ 2 . 72
the definition of e:
x
   1
e = lim  x + 
x→ ∞
   x

4
Sketch the graph : f ( x ) = 3 ⋅ e − x

Sketch the graph : f ( x ) = −4 ⋅ (0.3) x + 2

5
Solve graphically: (2.5) x ≥ (4) x

6

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