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

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Functions, Graphs, and Tables Investigation 5C Advanced Integrated Math I

You-Trys Determine whether each table below represents a linear function.

You-Trys Determine whether each table below represents a linear function.

You-Try (with partner) Use the tables above to make predictions. Will there be enough grain to feed the world in 2020? In 2030? What kind of factors could change your predictions?

Assignment Page 429 #11-13

Page 429

Warm-Up: January 8, 2015 Describe how to find each output using the previous output.

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Section 5.13 Advanced Integrated Math I Constant Differences Section 5.13 Advanced Integrated Math I

Example Copy and complete the table Input Output Δ 5 1 12 2 19 3 26 4 33 40

Definitions 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.

You-Try Copy and complete the table Input Output Δ 4 3 1 2 -4 7 5

You-Try (with partner) In the table, variables a and b represent arbitrary numbers. Copy and complete the table in terms of a and b. Find a linear function that matches the table. Input Output Δ b a 1 2 3 4 5

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

Section 5.14 Advanced Integrated Math I Recursive Rules Section 5.14 Advanced Integrated Math I

Recursive Rules Recursive rules tell how to get from one output to the next output. An initial output, called the base case, must also be given.

Fibonacci Sequence Base Case: Recursive Rule: What are the first 10 numbers in the Fibonacci Sequence?

You-Try Describe a recursive formula that agrees with the table. Input Output 3 1 7 2 11 15 4 19 5 23

You-Try (with partner) Does the recursive rule define the function ? Test your answer with an input of ½

You-Try (with partner) 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. Which investment is better in the short run? Which investment is better in the long run?

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

Warm-Up: January 12, 2015 You may use a calculator (graphing or otherwise) to complete this warm-up.

Input, x Output, y 1 2 3 4 5

Homework Questions?

Investigation 5A Quiz Questions?

Section 5.15 Advanced Integrated Math I Constant Ratios Section 5.15 Advanced Integrated Math I

Definitions 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.

Explicit  Recursive

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

Page 452 See textbook for #5, 6, 11

Warm-Up: January 13, 2015 Tony invests $600 in a savings account that earns 3% interest at the end of each year. How much interest will he have earned after 1 year? How much interest will he have earned after 2 years?

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Section 5.16 Advanced Integrated Math I Compound Interest Section 5.16 Advanced Integrated Math I

You-Try (with partner) Tony finds a new investment for his $600 that earns 6% interest at the end of each year. How much interest will he earn after 2 years? Is it double the amount he would earn with the 3% investment?

Interest 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.

Example 1 Tony finds another investment that offers 5.9% interest compounded monthly. Why is this the best option so far?

Interest Formula 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

Example 2 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?

Assignment Read Section 5.16 (pages 456-459)

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

Warm-Up: January 14, 2015 On Monday we looked at functions of the form For what values of b will the outputs be increasing (for increasing inputs)? For what values of b will the outputs be decreasing (for increasing inputs)?

Homework Questions?

Graphs of Exponential Functions Section 5.17 Advanced Integrated Math I

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

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)

Assignments Read Section 5.17 (pages 462-464) Page 465 #5-8 Graphs must be on graph paper Page 467 #1-5

Page 465 See textbook for #7