Math II Unit 2(Part 2)

ExponentsEQ: How do you use properties of exponents to

simplify algebraic expressions?

Properties of Exponents - GO

EXPONENTS

Ax x is called the __________ A is called the __________

PROPERTIES OF EXPONENTS

Multiplication Properties Zero and Negative Exponents

When multiplying two bases that are the same,

you __________ the exponents.

Ex: x9 x6 x2 = _____

When raising a power to another power you

__________ the exponents.

Ex: (x2 )5 = _______

To find the power of a product you

____________ the exponents.

Ex: (2x2y5)3 = __________

Any nonzero number raised to the “0” power is equal to

__________ Ex: (2x)0 = ______

How do negative exponents become

positive? ______________________ ______________________ ______________________ ______________________ ______________________ ______________________

Division Properties

When dividing two bases that are the same,

you _________ the exponents.

Ex:

When raising a quotient to a power, you

__________ the exponents.

Ex.

Apply What You Have Learned…

Mr. Higgins told his wife, the mathematics professor that he would make her breakfast. She handed him this message:

What should Mr. Higgins fix his wife for breakfast?

I want y

EggterEas 01 )()(

EQ:How do you use the properties of exponents to solve exponential equations?

Solving Exponential Equations

•Identify the bases on each side of the equal sign.

•What is a commonbase for both?

Next

•Rewrite one (or both) to have common bases.

•Simplify exponents using the Power to a Power Rule.

Now.. •Set both exponents equal to each other.

•Solve for x.•CHECK YOUR ANSWER!

Done!

Solving Exponential Equations Using Properties of Exponents

Examples:

Example: Solve: 3x+1 = 9x

*Rewrite 9 as a base w/ 3 3x+1 = (32)x

*Power to a Power* 3x+1 = 32x

*Bases are equal so set exponents equal and solve. x + 1 = 2x

x = 1

Examples:12515 32x 82x 93x7x 168

3x 22

3x 3

32x515

332x 55

332x 62x

3x

93x47x3 )(2)(2

9)4(3x3(7x) 22 3612x21x 22

3612x21x

369x4x

Exponential Functions

EQ: What is an exponential function?

EQ: How do exponential functions relate to real-world phenomena?

now It… eed to Know It… earned It…

Word Bank: Exponential function Domain Exponential GROWTH function Range Exponential DECAY function Zeros End Behavior Intercept Geometric Sequence Integer Exponents Constant Ratio Inequalities Natural Base e Asymptotes

Exponential Functions

An exponential function with a base b is written f(x)=abx , where b is a positive number other than 1.

This is easy to identify because the variable is in the place of (or part of) the exponent.

Exponential Growth

An exponential growth function can be written in the form of y = abx where a > 0 (positive) and b > 1.

The function increases from (-∞, +∞) When looking at real-world data, we are often given

the percent rate of growth (r). We can then make our own function by using the formula

a is the starting value

r is the percent growth rate (changed to a decimal)

x is time

xr)a(1y

Exponential Decay An exponential decay function can be written in the form of

y = abx where a > 0 (positive) and 0 < b < 1. The graph decreases from (-∞, +∞). When looking at real-world data, we are often

given the percent rate that it decreases (r). We can then make our own function by using the formula

a is the starting value

r is the percent decrease/decay rate

(changed to a decimal)

x is time

xr)a(1y

Example 1:You have purchased a car for

$19,550. This car will depreciate at a rate of 12% each year.

Is this an example of a growth or decay? 

Write a formula to represent the amount the car is worth after x number of years.

What is the value of the car after 2 years?

Example 2: In the year 2005, a small town had a

population of 15,000 people. Since, then it is growing at a rate of 3% each year.

Is this an example of a growth or decay? Write a formula that represents the

population after x years. What is the population after 7 years?

Example 3: The week of February 14, storeowner J.C. Nickels ordered hundreds of heart shaped red vacuum cleaners. The next week, he still had hundreds of heart-shaped red vacuum cleaners, so he told his manager to discount the price 25% each week until they were all sold. The original cost was $80.

Write an exponential equation that you can use to find the price of the vacuum cleaners in successive weeks.

What was the price in the second week? What was the price of the vacuums in the fourth week?

Compound Interestnt

nr )P(1A

A = Ending Amount

P = Starting (principle) Amount

r = interest rate

n = Number of times the interest is compounded in a year.

*annually = 1 *quarterly = 4*semi-annually = 2 *monthly = 12

Compound Interest Example:

You deposit $5000 into a savings account that earns 3% annual interest. If no other money is deposited, what is your balance after 4 years if it is compounded…

Semi-annually: Quarterly: Monthly: Daily: Continuously?

4)(22

.03)5000(1A 4)(4

4.03)5000(1A

4)(1212.03)5000(1A

4)(365365.03)5000(1A

Compounding Interest

ContinuouslyA = Pert A is ending balance

P is starting balance

e is the NUMBER…2.7182r is the interest ratet is the time in years

The Natural Base of e…e is an irrational number approximately equal

to 2.71828…

e is the base amount of growth shared by all continually growing processes. e lets you take a simple growth rate (where all change happens at the end of the year) and find the impact of compound, continuous growth, where every nanosecond (or faster) you are growing just a little bit.

e shows up whenever systems grow exponentially and continuously: population, radioactive decay, interest calculations, and more. Even jagged systems that don’t grow smoothly can be approximated by e.

e

Four Corners…

Graphing Exponential Functions

EQ:How do transformations of exponential equations affect the function analytically and graphically?

Troy and Gabriella are students at East High School, which has a

population of 2374 students. They met at registration and immediately

fell in love. However, their relationship was doomed to fail. By

the first day of school they were both dating someone new. By day

two both of those relationships had failed and each person involved had

found someone new. This unfortunate breakup pattern

continued each day until everyone at the school was in a relationship.

The breakups were so traumatic that the students involved only

paired up with those who had not yet been in a relationship.

x f(x)

-1

0

1

2

1/3139

x f(x)

-3

-2

-1

0

2x3y

1/3139

x f(x)

0

1

2

3

1x3y x3y

1/3139

x f(x)

-1

0

1

2

1/3139

x f(x)

-1

0

1

2

3510

x f(x)

-1

0

1

2

x3y

-2/3028

23y x 13y x

312

Asymptotes… What is happening to the graph of y = 3x

as it gets closer to the x-axis? Does it eventually cross and continue below the

axis? Let’s look at the table…

What happens to the y-value as

the x-value decreases? When will we reach zero? This is why there is an asymptote at y = 0. Asymptote: A straight line that a curve

approaches more and more closely but never touches as the curve goes off to infinity in one direction.

We draw the asymptote with a dotted/dashed line.

x y2 91 30 1-1 1/3-2 1/9-3 1/27-4 1/81

Step 1: Identify left/right

translations.

y = ab(x+h) left h units y = ab(x-h) right h units

Step 2: Identify up/down

translations.

Y = abx + k up k units Y = abx – k down k units

Step 3: Identify stretch/shrink/

reflections.

Step 4: Take common point (0, 1) and translations from steps 1 – 3 and

locate & plot the point in its new location.

Step 5: Identify and draw in

asymptote.

Examples:

-6 -4 -2 0 2 4 6

6 4

2 -2 -4 -6

-6 -4 -2 0 2 4 6

6 4

2 -2 -4 -6

-6 -4 -2 0 2 4 6

6 4

2 -2 -4 -6

Step 6: Identify and draw in the growth/decay exponential curve.

Characteristics of Graph

Domain: Range: Intercept(s): End

behaviors: Asymptote:

(-∞, +∞)

(3, +∞)

(0, 3.5)

Rises to leftFalls to the righty = 3

Geometric SequencesEQ:How are geometric sequences like exponential functions?

How many shapes are next?

Geometric Sequences

Explicit Formula/Rule for Geometric SequenceAn = a1(r)n-1 OR An = a0(r)n

a1 is first term a0 is starting value r is common ratio r is common ratio n is term number n is term number

**Look at 2nd formula. Looks a lot like… Exponential Function!

Example: Write the rule for the geometric sequence

below.6, 24, 96, 384, …

Answer: an = 6(4)n-1 OR an = 6/4(4)n

MM2A2. Students will explore exponential functions. a. Extend properties of exponents to include all integer exponents. b. Investigate and explain characteristics of exponential functions, including domain and range, asymptotes, zeros, intercepts, intervals of increase and decrease, rates of change, and end behavior. c. Graph functions as transformations of f(x) = ax d. Solve simple exponential equations and inequalities analytically, graphically, and by using appropriate technology. e. Understand and use basic exponential functions as models of real phenomena. f. Understand and recognize geometric sequences as exponential functions with domains that are whole numbers. g. Interpret the constant ratio in a geometric sequence as the base of the associated exponential function.

Standards Covered: