Exponential Functions4.2

Explorations of growth and decay

SAT #1

Quick Poll!

SAT #2

Quick Poll!

SAT #3

Quick Poll!

A half-life POD

If a substance has a half-life of 25 hours, how much of a 100-gram sample is left after 75 hours? 115 hours?

A POD to start with

If a substance has a half-life of 25 hours, how much of a 100-gram sample is left after 75 hours? 115 hours?

At 75 hours, there will have been 3 half-lives, decreasing from 100 to 50 to 25 to 12.5 g.

At 115 hours, there will have been some fraction of a half-life. We need to use an exponential equation.

Exponential Basics

Form:f(x)=abx

a is the initial amount (why?)

b is the base (it is also the rate of change-- why?)

x is the exponent (which is why they are called exponential functions)– it is often the variable for time.

Graph y=3x and y=(1/3)x.

How do the graphs compare?

What are the asymptotes? Intercepts?

What do the phrases “exponential decay” and “exponential growth” have to do with these graphs?

Could we find inverses for these graphs?

The Graphs

Reflected over y-axis

The y-intercept for both is (0,1).

Both are one-on-one functions.

HA for both is the x-axis.

No VA for either– no restriction on the domain.

One is growth, one is decay.

Inverses involve logs.

The Graphs

Graph y=3x and y = log3x.

Then add y = x to the graph.

What do you see?

The Graphs

Exponential Graphs

Now, graph y=3x and y=3-x on the same set of axises on your calculator.

How do these graphs compare? What is the transformation from the first to the second?

Why does the second one do what it does?

Exponential Graphs

Now, graph y=3x and y=2x on the same set of axises on your calculator.

How do they compare?

How do the graphs of y=3-x and y=2-x compare?

In general, if 1 < c < d,

then cx < dx for positive values of x

and cx > dx for negative values of x.

Exponential Graphs

What would the graph of y=3*2x look like?

Growth or decay?

How does it compare to the graph of y=2x? What is the transformation?

Asymptote?

Intercepts? What does the intercept mean?

Exponential Graphs

Graph the following and compare:

Change y-int asymptoteh(x)=2x

f(x)=2x - 2

g(x)=2x-2

What sorts of transformations are we looking at?

Exponential Applications

Let’s go back to the POD.

a = 100b = .5x = 115/25

y = 100(.5)115/25 = 4.12 gr.

Does this fit with our other data?

Exponential Applications

Find the total amount of dollars for $1000 invested at 6% compounded annually and monthly, after 4 years. (Remember the formulas?)

Exponential Applications

Find the total amount of dollars for $1000 invested at 6% compounded annually and monthly, after 4 years. (Remember the formulas?)

49.1270$

12

06.11000

48.1262$

06.11000

124

4

A

A

Exponential Equations

Solve for x:

67-x = 62x+1

2x+1 = 4x

Exponential Equations

Solve for x:

67-x = 62x+1 Same base– set exponents equal to each other7 – x = 2x + 1 x = 2

2x+1 = 4x Different bases– get to common base before setting exponents equal

2x+1 = 22x x + 1 = 2x

x = 1

Exponential Equations

Find an exponential function of the form y=abx with a y-intercept (0, 6) and that contains the point (2, 3/32).

Exponential Equations

Find an exponential function of the form y=abx with a y-intercept (0, 6) and that contains the point (2, 3/32).

a = 6x = 2y = 3/32

Find b and the final equation.

8

1

64

1632

3

632

3

2

2

2

b

b

b

b

x

y

8

16