Exponential Functions Honors Precalculus

Mr. Velazquez

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Exponential Functions Defined

2

Exponential Functions

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The exponential equation 13.49 .967 1 predicts the number of O-rings

that are expected to fail at the temperature x F on the space shuttles. The

O-rings were used to seal the connections between d

x

o

f x

ifferent sections of the shuttle

engines. Use a calculator to find the number expected to fail at the temperature of

40 degrees.

We are looking for 𝑓(40):

𝑓 40 = 13.49 0.967 40 − 1 𝑓 40 = 13.49 0.261 − 1

𝑓 40 = 2.524

So, realistically, at least 2 O-rings are expected to fail.

Graphing Exponential Functions

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Graphing Exponential Functions

5

Graph the following two functions:

𝑓 𝑥 =1

4

𝑥

𝑔 𝑥 = 4𝑥

Transformations of Exponential Fucntions

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Transformations of Exponential Fucntions

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Use the graph of f(x)=4 to obtain the graph of g(x)=4 3.

What is the domain and range of each function?

x x

Transformations of Exponential Fucntions

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2Use the graph of f(x)=4 to obtain the graph of g(x)=4

Find the domain and range for the g(x) function.

x x

Transformations of Exponential Fucntions

9

Use the graph of f(x)=4 to obtain the graph of g(x)=2 4

Find the domain and range for the g(x) function.

x x

The Natural Base 𝑒

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The values of 1 +1

𝑛

𝑛 for

increasingly large values of 𝑛. Notice that as 𝑛 → ∞, these values approach a specific number. This irrational and transcendental number is called the natural base 𝑒 ≈ 2.718281828. The function 𝑓 𝑥 = 𝑒𝑥 is called the natural exponential function.

The Natural Base 𝑒

The population of Shagnasty Island (real place, seriously) can be modeled by the equation 𝑃 𝑥 = 8𝑒0.8𝑥, where 𝑥 is the number of years since the people originally settled. (Round off answers to the nearest integer)

a) How many people originally settled in Shagnasty? (𝑥 = 0)

b) Assuming no new immigration, what will be the population of Shagnasty 10 years after settlement? What about after 15 years?

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Compound Interest Compound interest is the amount of interest computed on your original investment as well as on any accumulated interest. The original amount of your investment is called the principal 𝑃. This amount is invested at an annual interest rate 𝑟 for a number of years 𝑡.

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Annually Compounded Interest: 𝑨 = 𝑷 𝟏 + 𝒓 𝒕

Compound Interest Sometimes interest is compounded semiannually, monthly, quarterly, etc. This introduces a new variable 𝑛, which simply represents the number of times the interest will be compounded in one year. If there are 𝑛 compounding periods in a year, then the interest rate per time period would be 𝑖 =

𝑟

𝑛 and

there are 𝑛𝑡 time periods in 𝑡 years. This results in the formula below for the balance after 𝑡 years.

13 Annually Compounded Interest:

𝑨 = 𝑷 𝟏 + 𝒓 𝒕

Interest Compounded 𝑛 Times Per Year:

𝑨 = 𝑷 𝟏 +𝒓

𝒏

𝒏𝒕

Continuous Compounding of Interest

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Compound Interest

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Interest Compounded 𝒏 Times Per Year:

𝑨 = 𝑷 𝟏 +𝒓

𝒏

𝒏𝒕

Continuously Compounded Interest:

𝑨 = 𝑷𝒆𝒓𝒕

Find the accumulated value of an investment of $15,000

for 2 years at an interest rate of 3.2% if the money is

a. compounded semiannually

b. compounded quarterly

c. compounded monthly

d. compounded continuously

Classwork/Exit Ticket

Calculate which of the following back accounts will have a higher balance after 15 years.

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Account 1: Principal Investment = $5,000 Annual Interest Rate = 12% Compounded Quarterly (every 3 mo.)

Account 2: Principal Investment = $9,000 Annual Interest Rate = 8% Compounded Monthly

Account 3: Principal Investment = $7,500 Annual Interest Rate = 10% Compounded Continuously

Interest Compounded 𝒏 Times Per Year:

𝑨 = 𝑷 𝟏 +𝒓

𝒏

𝒏𝒕

Continuously Compounded Interest:

𝑨 = 𝑷𝒆𝒓𝒕

Homework Due 12/15

Pg. 397, #53-56