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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
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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
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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...
2. 5, 10, 15, 20, 25...
3. 3, 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.
3.
4.
5.
2, 4, 6, 8, 10...12, 14, 16, 18, 20
5, 10, 15, 20, 25... 30, 35, 40, 45, 50
3, 6, 9, 12, 15…18, 21, 24, 27, 30
10, 20, 30, 40, 50…60, 70, 80, 90, 100
11, 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
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
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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. 2185 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 10th term
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5, 8, 11, 14, 17……
Start 5, next = now +3
3, -9, 27, -81, 243……
Start 3, next = now • -3
Recursive Patterns
• http://www.shodor.org/interactivate/activities
/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?
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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
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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 addons 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.