8.1 Exponential Growth

p. 465

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 an asymptote

Asymptote• A line that a graph approaches as you move

away from the origin

The graph gets closer and closer to the line y = 0 …….But NEVER reaches it

y = 0

2 raised to any powerWill NEVER be zero!!

Lets look at the activity on p. 465

• This shows of y= a * 2x

• 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)

• These are true of:

• y = abx

• If a>0 & b>1 ………

• The function is an Exponential Growth Function

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-axsi, passes thru the 2 points, and moves up to the right

y = 0

Always mark asymptote!!

D+

D= all realsR= all reals>0

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

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.

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)

y = -4

D= all realsR= all reals >-4

Now…you try one!

• Graph y= 2·3x-2 +1• State the Domain

and Range!• D= all reals• R= all reals >1

y=1

Compound Interest

•A=P(1+r/n)nt

• 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

Compound interest example

• 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

≈ $1082.43

A=1000(1+.08/365)365x1

≈1000(1.000219)365

≈ $1083.28