Aim: What is the exponential function?

Do Now: Given y = 2x, fill in the table

x -3 -2 -1 0 1 2 3

1/8 ¼ ½ 1 2 4 8

y

HW: Worksheet

Math Composer 1. 1. 5http: / / www. mathcomposer. com

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y = 2x

Graph the curves y = 2x, y = 3x, y = 4x onthe same coordinate system

x

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y = 2x

y = 3xy = 4x

1

x 2x 3x 4x

-3 1/8 1/27 1/64-2 1/4 1/9 1/1

6-1 1/2 1/3 1/40 1 1 11 2 3 42 3 9 163 4 27 64

Exponential FunctionsExponential Functions

(1/2)x

8

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1/2

1/4

1/8

y=(1/2)x

What about y=(1/2)x ?

All base exponential functions are similar because they all go through the point (0,1), regardless of the size of their base number

Exponential Functions are written in the form:

f(x) = abx a = constant b = base x = variable

Properties of the Properties of the basicbasic ff((xx) = ab) = abxx function function

• The domain consist of all real numbers x• The range consists of all positive all real

numbers y • The function is increasing when b >1 and

decreasing when 0 < b < 1• It is one to one function • The x –axis is the horizontal asymptote to

curve, toward the left when b>0 and toward the right for 0 < b <1

Exponential Function: Any equation in the form f(x) = Cbx. if 0 < b < 1 , the graph represents

exponential decay – the y-value is going down as x increases

if b > 1, the graph represents exponential growth – the y-value is going up as x increases

Examples: f(x) = (1/2)x f(x) = 2x

Exponential Decay Exponential Growth

We will take a look at how these graphs “shift” according to changes in their equation...

Take a look at how the following graphs compare to the original graph of f(x) = (1/2)x :

f(x) = (1/2)x f(x) = (1/2)x + 1 f(x) = (1/2)x – 3

Vertical Shift: The graphs of f(x) = Cbx + k are shifted vertically by k units.

Take a look at how the following graphs compare to the original graph of f(x) = (2)x :

f(x) = (2)x f(x) = –(2)x f(x) = –(2)x + 2 – 3

Notice that f(0) = 1

(0,1)

This graphis a reflection of f(x) = (2)x . The graph isreflected over the x-axis.

(0,-1)

Shift the graph of f(x) = (2)x ,2 units to the left. Reflect the graph over the x-axis. Then, shift the graph 3 units down

(-2,-4)

e = 2.71828….

This number is called the natural base.

It is called this because the value seems to occur naturally in many situations. e is an irrational number that is similar to the property of .e is a very important in calculus since the derivative of e is itself.

The function f(x) = ex is the natural exponential function.

Math Composer 1. 1. 5http: / / www. mathcomposer. com

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f(x) = ex

The population of the United States can be modeled by the function p(x) = 80.12e0.131x where x is the number of decades since 1900 and p(x) is the population in millions

a. graph p(x) over the interval 0 x 15

b. If the population of United States continues to grow at this rate, predict the population in the years 2010 and 2020.

Sketch the graph of y = 3x over -3 x 3Math Composer 1. 1. 5http: / / www. mathcomposer. com

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Sketch the graph of over -2 x 2 x

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Math Composer 1. 1. 5http: / / www. mathcomposer. com

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