1. 8.1 Exponential Growth

2. Objectives/Assignment
Graph exponential growth functions
Use exponential growth functions
Assignment: odd

3. Use a calculator to graph the following:

4. Exponential Function
f(x) = bx where the base b is a positive number other than one.
Graph f(x) = 2x
Note the end behavior
x→∞ f(x)→∞
x→-∞ f(x)→0
y=0 is horizontal asymptote

5. Asymptote
A horizontal line that a graph approaches as you move away from the origin
The graph gets closer
and closer to the line
y = 0 ……. As values
Of x become larger
And larger negative
Negative numbers…
But NEVER reaches it
2 raised to any power
Will NEVER be zero!!
y = 0

6. Look at the activity on p. 465
Graph y= a * 2x (first with a = 1/3 then a= 3)
Now let a = -1/5 then let a = -5….compare these graphs with those graphed above.
What is the effect of a on the graph of y?
Passes thru the point (0,a) (the y intercept is a)
The x-axis is the asymptote of the graph
D is all reals (the Domain)
R is y>0 if a>0 and y<0 if a<0
(the Range)

7. Consider:
y = abx
If a>0 & b>1 ………
The function is an Exponential Growth Function

8. Example 1 Graph y = (½) 3x Plot (0, ½) and (1, 3/2)
Then, from left to right, draw a curve that begins just above the x-axis, passes thru the 2 points, and moves up to the right

9. D= all reals
R= all reals>0
y = 0
Always mark asymptote!!

10. Example 2 Graph y = - (3/2)x Plot (0, -1) and (1, -3/2)
Connect with a curve
Mark asymptote
D=??
All reals
R=???
All reals < 0
y = 0

11. To graph a general Exponential Function:
y = a bx-h + k
Sketch y = a bx
h= ??? k= ???
Move your 2 points h units left or right …and k units up or down
Then sketch the graph with the 2 new points.

12. Example 3 Graph y = 3·2x-1-4 Lightly sketch y=3·2x
Passes thru (0,3) & (1,6)
h=1, k=-4
Move your 2 points to the right 1 and down 4
AND your asymptote k units (4 units down in this case)

13. D= all reals
R= all reals
>-4
y = -4

14. Now…you try one! Graph y= 2·3x-2 +1 State the Domain and Range!
D= all reals
R= all reals >1
y=1

15. Exponential Growth Models
When a real life quantity increases by fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by:
y = a(1+r)t
Where a is the initial amount and r is the percent increase expressed as a decimal.
The quantity 1+r is called the growth factor

16. A=P(1+r/n)nt Compound Interest P - Initial principal
r – annual rate expressed as a decimal
n – compounded n times a year
t – number of years
A – amount in account after t years

17. Compound interest example
A=P(1+r/n)nt
You deposit $1000 in an account that pays 8% annual interest.
Find the balance after I year if the interest is compounded with the given frequency.
a) annually b) quarterly c) daily
A=1000(1+ .08/1)1x1
= 1000(1.08)1
≈ $1080
A=1000(1+.08/4)4x1
=1000(1.02)4
≈ $
A=1000(1+.08/365)365x1
≈1000( )365
≈ $