Exponential Functions Functions, Study notes of Business Taxation and Tax Management

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Created by: M. Signore & G. Garcia

Lesson #38: Zero & Negative Exponents

Do Now:

Zero Exponents

• 1. 2 -^4 2. 4 -^2 3. x-
• 4. 3z-^2 5. - 3 - - 6.
• 7. 2 -^5 · 23 8. x^3 · x-^7 9.
• 10. x
  • x-
    • 11. x^0 12. 1001 -

Homework #38: Zero & Negative Exponents

Directions: Rewrite each item as an equivalent expression in exponential notation.

Answers should only have positive exponents.

1) = ⋅

⋅ ⋅ ⋅ 5 5

= −

− 2

2

3) (^0.^12 ( 0 .) 12 (^0 .)(^120 .) 12 (^0 .)^12 )= 4) 7 ⋅ 77 ⋅⋅ 77 ⋅^ ⋅ 77 ⋅ 7 =

5) (^) 9 =

6 15

= −

− 3

5 7

7

7) =  

 

  



4

3

4

3 5 8) (^) 4 =

4 3

12

9) (^) 9 =

9 6

(^6) 10) ⋅ = 9

4 5 7

7 7

11) (^ )^5 ⋅ =

30 2 5

9 5

Write the following algebraic problems in exponential notation.

12) (^) 3 =

7 x

x (^) 13) = 6 2

2 a b

a b

Lesson #39: Intro to Exponential Functions

Do Now:

Simplify the Following:

”Negative Exponents are Bad Manners in Math!” 1] (-4)^2 2] - 42 3] 32 • (-2)^3

4] a^2 • a^3 5] (d^4 )(d^6 ) 6] (x^4 )^3

7] (2x^2 y^3 )^4 8] z 0 • z^2 9] x-^3

10] 5 -^2 11] (-2x^2 )(6x^3 )(x^2 ) (^) 12] 4

12a −^2

3] f(x) = 0.5x

x y

4] h(x) = - 2 x

x y

PRACTICE:

Graph the functions

1. f^ (^^ x )=^4 x

x y

• 3
• 2
• 1 0 1 2 3

2. f^ (^^ x )^ =^1.^25 x

x y

• 3
• 2
• 1 0 1 2 3

Determine the equation of equation of each exponential function below

Lesson #40: Exponential Growth and Decay

Do Now:

Jump Start your Prior Knowledge

1] Given y = 3x, evaluate y when x = 3. __________

2] Given y = 3x, evaluate y when x = -2. __________

3] Which ordered pair represents the y-intercept for the function y = 2 x?

a) (0,0) b) (0, 1) c) (0, 2)

4] The graph of y = 2x^ lies in which Quadrants? a) I, II b) I, III c) I, IV

5] The graph of y = 2x^ contains which of these points?

a) (0,0) b) (0, 1) c) (0, 2)

Exponential Growth vs. Decay:

Example: Would the graph of y = 0.5x^ show exponential growth or exponential decay?

Example: Would the graph of y = 1.5x^ show exponential growth or exponential decay?

Exponentials in the Real World? Many real world phenomena can be modeled by functions that describe how things grow or decay as time passes. Examples of such phenomena include the studies of populations, bacteria, the AIDS virus, radioactive substances, electricity, temperatures and credit payments.

Growth Example:

Exponential Growth: Exponential Decay:

y = a(1 + r)x^ y = a(1 - r)x

Ingredients:

a = initial amount before measuring growth/decay r = growth/decay rate (often a percent)

IMPORTANT



convert to a decimal x = number of time intervals that have passed (years)

A bank account balance, b, for an account starting with s dollars,

earning an annual interest rate, r, and left untouched for n years can

be calculated as b = s(1 + r)n^ (an exponential growth formula). Find a

bank account balance to the nearest dollar, if the account starts with

$100, has an annual rate of 4%, and the money left in the account for 12 years.

Decay Example: Daniel’s Print Shop purchased a new printer for $35,000. Each year it depreciates (loses value) at a rate of 5%. What will its approximate value be at the end of the fourth year?

A) $33,250.00 B) $30,008.13 C) $28,507.72 D) $27,082.

U-Try:

1] Cassandra bought an antique dresser for $500. If the value of her dresser increases 6% annually, what will be the value of Cassandra's dresser at the end of 3 years to the nearest dollar?

Homework #40: Exponential Growth and Decay

For each word problem, write the exponential equation to model the situation.

4. A zombie infection in Yonkers Public Schools grows by 15% per hour. The initial group of zombies was a group of 4 freshmen. How many zombies are there after 6 hours.

5. Ryan is saving for his college tuition. He has $2,550 in a savings account that pays 6.25% annual interest.

6. Cars depreciate in value over time. A used car was purchased for $12,329 this year. Each year the car’s value decreases 8.5%.

7. Jeremiah owns a side business detailing cars. His first year he made $10, and each of the following years his profit increased 9%.

8. There are 128 teams entered in a basketball tournament. Half of the teams are eliminated each round. How many teams are left after 4 rounds?

9. Bacteria in a dirty glass triple every year. If there are 25 bacteria to start, how many are in the glass after 1 day?

10. The population of a city with 750,000 is devastated by an unknown virus that kills 20% of the population per day. How many people are left after a week?