Warm Up Write down objective and homework in agenda Lay out homework (None) Homework (Recursive worksheet) Get a Calculator!!!

Unit 3 Functions -Common Core Standards 8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. 8.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. A-REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Note: At this level, focus on linear and exponential equations F-BF.1 Write a function that describes a relationship between two quantities. 1)Determine an explicit expression, a recursive process, or steps for calculation from a context. 2) Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Note: At this level, limit to addition or subtraction of constant to linear, exponential or quadratic functions or addition of linear functions to linear or quadratic functions. F-BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Note: At this level, limit to vertical and horizontal translations of linear and exponential functions. Even and odd functions are not addressed. F-IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Unit 3 Functions -Common Core Standards F-IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Note: At this level, the focus is linear and exponential functions. F-IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Note: At this level, focus on linear, exponential and quadratic functions; no end behavior or periodicity. F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Note: At this level, focus on linear and exponential functions F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Note: At this level, focus on linear functions and exponential functions whose domain is the subset of integers. N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

Warm Up Study the patterns below to determine the next five numbers in each sequence 1. 2, 4, 6, 8, , 10, 15, 20, , 6, 9, 12, 15… 4. 10, 20, 30, 40, 50… 5. 11, 22, 33, 44, 55…

Study the patterns below to determine the next five numbers in each sequence 1. 2, 4, 6, 8, , 14, 16, 18, , 10, 15, 20, , 35, 40, 45, , 6, 9, 12, 15…18, 21, 24, 27, , 20, 30, 40, 50…60, 70, 80, 90, , 22, 33, 44, 55…66, 77, 88, 99, 110

Vocabulary Domain: the set of all inputs or x-values explicit equation: an equation that tells how to calculate an output value based on a given input value Input: a value that is entered Iteration: a repetitive process mapping diagram: diagram that matches input values with their corresponding output values using arrows Output: a value that is calculated from the input value using a set rule Range: the set of all outputs or y-values recursive equation: an equation that shows how to calculate the value of the next term in a sequence from the value of the current term (or a combination of previous terms) Relation: a set of ordered pairs Sequence: a list of numbers where each term is based on the previous term or a combination of previous terms using a set pattern or rule

The Rabbit Problem In the year 1202, an Italian mathematician named Leonardo Fibonacci became interested in the reproduction of rabbits. He created an imaginary set of ideal conditions under which rabbits could breed, and posed the question "How many pairs of rabbits will there be a year from now?" The ideal set of conditions was a follows:

The Rabbit Problem 1. You begin with one male rabbit and one female rabbit. These rabbits have just been born. 2. A rabbit will reach sexual maturity after one month. 3. The gestation period of a rabbit is one month. 4. Once it has reached sexual maturity, a female rabbit will give birth every month. 5. A female rabbit will always give birth to one male rabbit and one female rabbit. 6. Rabbits never die.

The Rabbit Problem Newborn: Mature: So how many male/female rabbit pairs are there after one year (12 months)?

The Rabbit Problem Month Pairs of Rabbits Fill in the table What pattern do you notice? Write a rule to find out how many rabbits are in any given month, based on the number of rabbits from previous months. What is the START value?

The Rabbit Problem Answers

Now – Next Equations Sequences are lists of numbers where each term is based on the previous term or a combination of previous terms using a set pattern or rule.

Writing NOW-NEXT Rules for Sequences Given the sequence of numbers: 3, 6, 9, 12, 15… Step 1. The start is the number we begin with. In this case, the start number is 3 Step 2. Write the rule in NOW-NEXT form. In the pattern above, each number increases by 3 This means that the next number will be NOW + 3 Therefore, the rule should be written NEXT = NOW + 3. Start = 3

Simple Sequences  5, 10, 15, 20,...  START = 5, NEXT = NOW + 5  2, 4, 8, 16, 32,...  START = 2, NEXT = NOW  2

Now-Next Equation Examples We are only looking at the value column!

Now-Next Examples Start = 3, Now=Next + 1

Now-Next Examples For P – Start = 4, Now=Next + 4 For A – Start = 1, Now=Next + 2

You Try! Write a Now-Next Equation 3, 7, 11, 15, 19,... 1, 2, 4, 8, 16,... 5, 8, 11, 14, 17,... 1, 1, 2, 3, 5, 8,... 1, 10, 100, 1000,...

You Try! Write a Now-Next Equation 3, 7, 11, 15, 19,... – Start = 3, Next = Now + 4 1, 2, 4, 8, 16,... – Start = 1, Next = Now *2 5, 8, 11, 14, 17,... – Start = 5, Next = Now + 3 1, 1, 2, 3, 5, 8,... – Start = 1, Next = Now + number behind it 1, 10, 100, 1000,... – Start = 1, Next = Now *10

Vocabulary Iteration: a repetitive process recursive equation: an equation that shows how to calculate the value of the next term in a sequence from the value of the current term (or a combination of previous terms)

Iterating Functions on the Graphing Calculator Calculators can quickly iterate functions. Start by entering your initial value, then take advantage of the ANS key to create a function to iterate simply by pressing ENTER repeatedly. For example, to iterate NEXT = 5  NOW + 10, starting at an initial value of 1: Hit 1, ENTER. Enter 5Ans+10 and then hit ENTER repeatedly.

Iterating Functions on the Graphing Calculator Interpret the calculator: initial value is 1; the next is 15, and so on is the fourth iteration

Practice List the first 5 values generated by the recursive routine below and write a now-next equation 8 ENTER Ans – 4, ENTER, ENTER 8, 4, 0, -4, -8

Practice Write a now-next equation for each sequence. Then use your equation to find the 10 th term 5, 8, 11, 14, 17…… Start 5, next = now +3 3, -9, 27, -81, 243…… Start 3, next = now -3

Recursive Patterns /Sequencer/ /Sequencer/

Starting Numbers: What effect does a negative starting number have on the sequence? What effect does a positive starting number have on the sequence? Add-ons: What effect does a negative add-on have on the sequence? What effect does a positive add-on have on the sequence? Multipliers: What effect does a positive multiplier have on the sequence? What effect does a negative multiplier have on the sequence? What effect does a decimal multiplier have on the sequence? What effect does a large multiplier have on the sequence? What combinations of starting numbers, add-ons, and/or multipliers made the sequence increasing? What combinations of starting numbers, add-ons, and/or multipliers made the sequence decreasing? What combinations produced a linear graph? What combinations produced a non-linear graph?

Sample answers Students should associate the starting number with the location of the y-intercept on the graph. They should also note that when the multiplier is 1, positive add- ons give increasing sequences and negative add-ons give decreasing sequences. With multipliers, students should look at the different cases that tie back to the different outcomes of multiplying positive and negative numbers. Depending on their level of sophistication, they may describe things like the following: when the multiplier is negative, the sequence will bounce back and forth between positive and negative values, regardless of the sign of the starting number if the multiplier is a negative number between -1 and 0, the “bounces” will get closer and closer to zero if the multiplier is a negative number less than -1, then the “bounces” will get larger and large when the starting number is positive and the multiplier is larger than 1, the sequence is increasing and curved when the starting number is negative and the multiplier is positive, the sequence is decreasing and curved, etc.

Graph each ordered pair on a coordinate plane. 1.(2, –4) 2.(0, 3) 3. (–1, –2) 4.(–3, 0)

Give coordinates for each ordered pair on the graph. 5.A 6.B 7.C 8.D C D B A

Warm UP Evaluate. 7x + 3 if x = 2 8 – 2y if y = 12 Define the word set and give an example. Define the word relation and give an example.