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Exponential Functions

L. Waihman 2002

- A function that can be expressed in the form
- and is positive,

is called an Exponential Function. - Exponential Functions with positive values of x

are increasing, one-to-one functions. - The parent form of the graph has a y-intercept at

(0,1) and passes through (1,b). - The value of b determines the steepness of the

curve. - The function is neither even nor odd. There is

no symmetry. - There is no local extrema.

More Characteristics of

- The domain is
- The range is
- End Behavior
- As
- As
- The y-intercept is
- The horizontal asymptote is

- There is no x-intercept.
- There are no vertical asymptotes.
- This is a continuous function.
- It is concave up.

- How would you graph

Domain Range Y-intercept

Horizontal Asymptote

Inc/dec?

increasing

Concavity?

up

- How would you graph

Domain Range Y-intercept

Horizontal Asymptote

Inc/dec?

increasing

up

Concavity?

- Recall that if then the graph

of is a reflection of about the

y-axis. - Thus, if then

Domain Range Y-intercept

Horizontal Asymptote

Concavity?

up

- How would you graph

Is this graph increasing or decreasing?

Decreasing.

- Notice that the reflection is decreasing, so the

end behavior is

How does b affect the function?

- If bgt1, then
- f is an increasing function,
- and

- If 0ltblt1, then
- f is a decreasing function,
- and

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 _at_ x-axis Vertical stretch 3 Vertical

shift down 1

Vertical shift up 3

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?

Inc/dec?

decreasing

increasing

Concavity?

Concavity?

down

up

The number e

- The letter e is the initial of the last name of

Leonhard Euler (1701-1783) - who introduced the notation.
- Since has special calculus

properties that simplify many - calculations, it is the natural base of

exponential functions. - The value of e is defined as the number that the

expression - approaches as n approaches infinity.
- The value of e to 16 decimal places is

2.7182818284590452. - The function is called the Natural

Exponential Function

Domain Range Y-intercept H.A.

Continuous Increasing No vertical asymptotes

and

Transformations

Vertical stretch 3.

Horizontal shift left 2.

Reflect _at_ x-axis.

Vertical shift up 2

Vertical shift up 2.

Vertical shift down 1.

Domain Range Y-intercept H.A.

Domain Range Y-intercept H.A.

Domain Range Y-intercept H.A.

Inc/dec?

increasing

Inc/dec?

increasing

Inc/dec?

decreasing

Concavity?

up

Concavity?

up

Concavity?

down

Exponential Equations

- Equations that contain one or more exponential

expressions are called exponential equations. - Steps to solving some exponential equations
- Express both sides in terms of same base.
- When bases are the same, exponents are equal.
- i.e.

Exponential Equations

- Sometimes it may be helpful to factor the

equation to solve

or

There is no value of x for which is equal

to 0.

Exponential Equations

- Try
- 1) 2)

or