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8.1Exponential Growth
Learning Targets
Students should be able to…
Graph exponential growth functions.
Go over Chapter 7 test
Warm-up/Introduction
http://math.rice.edu/~lanius/pro/rich1.html
Exponential function-
involves the expression bx where b is a positive number other than 1.
Base
is the number b in an exponential function.
Asymptote-
is a line that a graph approaches as you move away from the origin. y = ab
Exponential growth function-
is an exponential function in the form where a > 0 and b > 1.
Graph exponential growth functions using y = abx - h + k
x y
0 3
1 6
Graph y = 3·2x
Make a table with x values 0 and 1
Plot the points and draw a curve that runs close to the x-axis and passes through the two points
D: all real
R: y > 0
Asymptote y = 0
Graph y = 3·4x -1
x y
0 3
1 12
x y
1 3
2 12
Make a table with x values 0 and 1
Begin by looking at the un shifted graph of y= 3·4x
Shift the two points right 1
D: all real R: y > 0
Asymptote y = 0
Graph y = 4·2 x - 3 + 1
x y
0 4
1 8
x y
3 5
4 9
Begin by looking at the un shifted graph of y= 4·2x
Make a table with x values 0 and 1
Shift the two points right 3 and up 1
D: all real
R: y > 1
Asymptote y = 1
2. Use exponential growth models in real life
When a real-life quantity increases by a fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by this equation:
y = a(1 + r)t
In this model, a is the initial amount and r is the percent increase expressed as a decimal. The quantity 1 + r is called the growth factor.
In 1980 wind turbines in Europe generated about 5 gigawatt-hours of energy. Over the next 15 years, the amount of energy increased by about 59% per year.
1. Write a model giving the amount E (gigawatt-hours) of energy t years after 1980. About how much wind energy was generated in 1984?
E=5(1.59)t
About 32 hours
2. Graph the model.
3. Estimate the year when 80 gigawatt-hours of energy were generated? About 5.98 near the end of 1985
You purchase a baseball card for $54 dollars. If it increases each year by 5%, write an exponential growth model.
V = 54(1.05)t
Compound Interest
Consider an initial principal P deposited in an account that pays interest at an annual rate r (expressed as a decimal) compounded n times per year. The amount A in the account after t years can be modeled by
You deposit $1500 in an account that pays 6% annual interest.
Find the balance after 1 year if the interest is compounded A. Annually
B. Semi-annually
C. Quarterly
nt
n
rPA )+1(=
1•1
)1
06.+1(1500=A =1590
1•2
)2
06.+1(1500=A =1591.35
=1592.051•4
)4
06.+1(1500=A
1. 2.
y = 2x