We’ve looked at linear and quadratic functions, polynomial functions and rational functions. We are now going to study a new function called exponential functions. They are different than any of the other types of functions we’ve studied because the independent variable is in the exponent.

xxf 2

Let’s look at the graph of this function by plotting some points. x 2x

3 8 2 4 1 2 0 1

-1 1/2 -2 1/4 -3 1/8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

7

123456

8

-2-3-4-5-6-7

2

121 1 f

Recall what a negative exponent means:

BASE

The asymptote

The asymptote is a line that the graph is heading towards but will never meet. At Studies these are usually horizontal, or vertical lines.

y 2x

The asymptote

y 0

xxf 2

xxf 3

Compare the graphs 2x, 3x , and 4x

Characteristics about the Graph of an Exponential Function where a > 1 xaxf

What is the domain of an exponential function?

1. Domain is all real numbers

xxf 4

What is the range of an exponential function?

2. Range is positive real numbers

What is the x intercept of these exponential functions?

3. There are no x intercepts because there is no x value that you can put in the function to make it = 0

What is the y intercept of these exponential functions?

4. The y intercept is always (0,1) because a 0 = 1

5. The graph is always increasing

Are these exponential functions increasing or decreasing?

6. The x-axis (where y = 0) is a horizontal asymptote for x -

Can you see the horizontal asymptote for these functions?

All of the transformations that you learned apply to all functions, so what would the graph of look like?

xy 232 xy

up 3

xy 21up 1

Reflected over x axis 12 2 xy

down 1right 2

xy 2

Reflected over y-axis This equation could be rewritten in a different form: x

xxy

2

1

2

12

So if the base of our exponential function is between 0 and 1 (which will be a fraction), the graph will be decreasing. It will have the same domain, range, intercepts, and asymptote.

There are many occurrences in nature that can be modeled with an exponential function (we’ll see some of these later this chapter). To model these we need to learn about a special base.

xxf 2

xxf 3

xexf

This says that if we have exponential functions in equations and we can write both sides of the equation using the same base, we know the exponents are equal.

If au = av, then u = v

82 43 x The left hand side is 2 to the something. Can we re-write the right hand side as 2 to the something?

343 22 xNow we use the property above. The bases are both 2 so the exponents must be equal.

343 x We did not cancel the 2’s, We just used the property and equated the exponents.

You could solve this for x now.

Let’s try one more:8

14 x The left hand side is 4

to the something but the right hand side can’t be written as 4 to the something (using integer exponents)

We could however re-write both the left and right hand sides as 2 to the something.

32 22 x

32 22 xSo now that each side is written with the same base we know the exponents must be equal.

32 x

2

3x

Check:

8

14 2

3

8

1

4

1

2

3 8

1

4

12 3

Solving equations with exponentials

a) Solve the equation 2x - 3 0

correct to 2 decimal places.

b) Solve the equation 2x - 3 10 correct to 2 decimal places.

Draw the graph of y 2x 3.

2x 3 0 is where the graph cuts

the x-axis (the root).

From the graph menu, with the

graph drawn:

G-SOLV ROOT x 1.58

Draw on the same graph, y 10

From the graph menu, with the

graph drawn:

G-SOLV ISCT x 3.70

y 2 x 3

y 10

Exponential questions

1. f (x) 3 x 5

a) Graph f (x).

b) Write down the domain

and range of f (x).

c) Write down the equation of

the horizontal asymptote of f (x).

d) Find f (x) 12.

e) Find f (3).

domain

range>5

y 5

x 1.77

f (x) 5.04