8.1 Exponential Growth. Learning Targets Students should be able to… Graph exponential growth...
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Transcript of 8.1 Exponential Growth. Learning Targets Students should be able to… Graph exponential growth...
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