determine whether each table below represents a linear function

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Determine whether each table below represents a linear function. Warm-Up: January 7, 2015

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Determine whether each table below represents a linear function.

Warm-Up: January 7, 2015

Homework Questions?

Functions, Graphs, and Tables

Investigation 5CAdvanced Integrated Math I

Determine whether each table below represents a linear function.

You-Trys

Determine whether each table below represents a linear function.

You-Trys

a) Use the tables above to make predictions. Will there be enough grain to feed the world in 2020? In 2030?

b) What kind of factors could change your predictions?

You-Try (with partner)

Page 429 #11-13

Assignment

Page 429

Describe how to find each output using the previous output.

Warm-Up: January 8, 2015

Homework Questions?

Constant DifferencesSection 5.13

Advanced Integrated Math I

Copy and complete the table

Example

Input Output

Δ

0 51 122 193 264 335 40

Consecutive outputs are the outputs for consecutive inputs.

The differences between consecutive outputs are recorded in the Δ column.

If these differences are the same, that value is called the constant difference or common difference.◦ This also means the function is linear.

Definitions

Copy and complete the table

You-Try

Input Output Δ0 4 31 22 -43 04 75

In the table, variables a and b represent arbitrary numbers.

a) Copy and complete the table in terms of a and b.

b) Find a linear function that matches the table.

You-Try (with partner)

Input Output Δ0 b a1 a2 a3 a4 a5

Read Section 5.13 (pages 431-434) Page 437 #6, 7, 8, 10, 12

Assignment

Recursive RulesSection 5.14

Advanced Integrated Math I

Recursive rules tell how to get from one output to the next output.

An initial output, called the base case, must also be given.

Recursive Rules

Base Case:

Recursive Rule:

What are the first 10 numbers in the Fibonacci Sequence?

Fibonacci Sequence 11

00

F

F

21 nFnFnF

Describe a recursive formula that agrees with the table.

You-Try

Input Output0 31 72 113 154 195 23

Does the recursive rule

define the function ?

Test your answer with an input of ½

You-Try (with partner)

0if51

0if3

nnf

nnf

xx 53

James saves $85 and wants to invest it. Investment L will add $5 at the end of every

year to James’s account. Investment E will add 5% of the current

amount at the end of every year.a) Which investment is better in the short

run?b) Which investment is better in the long

run?

You-Try (with partner)

Read Section 5.14 (pages 440-442) Page 445 #8-11, 13, 14

Assignment

Warm-Up: January 12, 2015 You may use a calculator (graphing or

otherwise) to complete this warm-up.

Input, x

Output, y

012345

Homework Questions?

Investigation 5A Quiz Questions?

Constant RatiosSection 5.15

Advanced Integrated Math I

An exponential function is one where x is in the exponent.

The ratio of outputs is one output divided by the previous output.

Exponential functions have a constant ratio.

Definitions

Explicit Recursive

0if1

0if

xbxf

xaxf

baxf x

Read Section 5.15 (page 447-449) Page 452 #5, 6, 8, 11

Assignment

See textbook for #5, 6, 11

Page 452

Tony invests $600 in a savings account that earns 3% interest at the end of each year.

a) How much interest will he have earned after 1 year?

b) How much interest will he have earned after 2 years?

Warm-Up: January 13, 2015

Homework Questions?

Compound InterestSection 5.16

Advanced Integrated Math I

Tony finds a new investment for his $600 that earns 6% interest at the end of each year.

a) How much interest will he earn after 2 years?

b) Is it double the amount he would earn with the 3% investment?

You-Try (with partner)

Most bank accounts calculate interest based on the current balance and add the amount to the account.

When the interest on the current balance includes interest on previous interest, it is called compound interest.

Compound interest is usually calculated and added to the account more often than once per year.

Interest

Tony finds another investment that offers 5.9% interest compounded monthly. Why is this the best option so far?

Example 1

A = current amount P = principal (the original investment) r = Interest rate (APR), expressed as a

decimal, not as a percent n = Number of times per year interest is

calculated t = Time, measured in years

Interest Formulatn

n

rPA

1

Tony decides to invest his money in a CD (certificate of deposit) that earns 6% APR, compounded annually. How many years will it take for his investment to double?

Example 2

Read Section 5.16 (pages 456-459) Page 460 #5, 6, 10, 11, 12

Assignment

Page 460 #5, 6, 10, 11, 12

On Monday we looked at functions of the form

a) For what values of b will the outputs be increasing (for increasing inputs)?

b) For what values of b will the outputs be decreasing (for increasing inputs)?

Warm-Up: January 14, 2015

xbaxf

Homework Questions?

Graphs of Exponential

FunctionsSection 5.17

Advanced Integrated Math I

Exponential growth occurs when a>0 and b>1.

Exponential decay occurs when a>0 and 0<b<1.

Exponential Growth and Decay xbaxf

Real World Applications

Exponential Growth Exponential Decay

Population growth◦ People◦ Bacteria

Continuous compounding of interest

Nuclear reactions Processing power of

computers (Moore’s Law)

Half-lives of radioactive isotopes◦ Carbon dating◦ Other dating to determine

ages of dinosaurs, etc. Rate of cooling (temp.) First order chemical

reaction rates Atmospheric pressure

(as a function of height)

Read Section 5.17 (pages 462-464) Page 465 #5-8

◦ Graphs must be on graph paper

Page 467 #1-5

Assignments

Page 465

See textbook for #7