8.1 Exponential Growth p. 465
Dec 31, 2015
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