Dec 5, 2013 ... Unit 2a: Functions and Graphing equations. Teacher's Page. Objectives:
Introduction of functions through sequences and linear graphing with ...
Unit 2a sequence Notes.notebook
December 05, 2013
Unit 2a: Functions and Graphing equations Teacher's Page Objectives: Introduction of functions through sequences and linear graphing with two variables. Extends to Inequalities of two variables. (No writing equation topics in this section, on hold for next section.) Textbook sections: 1.7, 2.2 Benchmarks: Part 1: A.CED.21: Create linear...equations in two or more variables to represent a relationship between quantities and graph equations on coordinate axes with labels and scales. A.REI.10: Understand a graph of an equation in two varibles is the set of all solutions plotted in the cooridinate plane (any shape) F.IF.1: Understand the function from one set (called domain) to another set (called range) assigns to each element of the domian exactly one elementof the range. y =f(x). F.IF.2: Use function notation, evaluate functions for their inputs in their domains and interpret statements that use function notation in terms of context. F.IF.3: Recognize that sequences are functions, sometimes recursively, whose domainis a subset of integers. F.IF.51: Relate the domain of a linear ...function to it's graph, and to the quantity it describes. F.IF.61: Calculate and interpret the average rate of change of a linear...over a specific interval. Estimate the rate of change from a graph pf a function over a specified interval. F.BF.31: Identify the effect of a graph for replacing f(x) by f(x) + k, kf(x), f(xk), and f(x+k) for specified valued of K(Positive and negative). Experiment with cases and illustrate an explanation of the effects on the graph using technology. F.LE.1b Recognize situations in which one quantity changes at a constant rate per unit interval related to another. All in parts not in their entirety. Time Frame: 34 weeks 930 to 1023 Part 1: Day 1: Introduction to basic sequences...leading to functions. (HW: 2a1) Day 2: Relations and Functions in ordered pair form...high vocabulary day includes vertical line test. (1.7) (HW: Function Vocabulary WS) Day 3: Function notation and using ordered pairs (ttables) to create a function. (HW: 2a2) Day 4: Introduction to graphing function including nonlinear. Ttable method. (HW: 2a3) Day 5: Quiz on Basics (Unit 2a1 Quiz) Day 6: Discuss topics on QuizFunction Notation and tying it all together. (CW: Function Notation Review WS) Day 7: Introduction to slope(m). Understanding slope (positive, negative, zero, N/A) (HW: 2a4) Slope formula and calculating slope whether on graphs or ordered pairs Day 8: Graphing using Slopeintercept form.(HW: 2a5) Day 9: Review (maybe used as a second day to graph) Day 10: Quiz on graphing and slope. Freshmen Fundamental Algebra students Day 11: Intro to calculator graphing and checking slopeintercept form Day 12: Calculators Continue Day 13: (Makeup day)PSATS...all sophomores will be testing.
Sophomores Study for PSAT Study for PSAT
Day 14: Introduction to graphing 2variable inequalities (HW: 2a6) Day 15: Contining graphing in 2 variables Day 16: Review for Unit 2a Test Day 17: Review for Unit 2a Test Day 18: Unit 2a Test (End of the Quarter) Part 2: Unit 2b
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Unit 2a sequence Notes.notebook
December 05, 2013
101413 Algebra Warmup: Create a pattern of your choice.
EQ: Create a sequence and state it's rule. HW: Finish Sequence WS Daily Proficiency:
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Unit 2a sequence Notes.notebook
93013 HW: Finish Sequence WS
December 05, 2013
EQ: Create a sequence and state it's rule. Unit 2a Linear Equations Day 1: Sequences in Math
Sequence: a list of numbers that are created from a common difference
Model: 1, 3, 5, 7, 9, 11, 13, ...
Term: an individual number in a sequence.
Some sequences are created from a known definition and some are created from using operations.
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Unit 2a sequence Notes.notebook
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Popular definition sequences: a.) 1, 4, 9, 16, 25, 36, 49, 64 ___, 81 ___,100 ___, ...
b.) 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 ___, 31 ___, ___, 37 ...
c.) 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 ___,144 ___,233 ___, ...
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Unit 2a sequence Notes.notebook
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Mathematical sequences: Algebraic sequences: Use addition, and/or subtraction. Geometric Sequences: Use multiplication, and/or division. Any pattern is possible. You may even see exponents to create patterns. d.) 1, 2, 4, 8, 16, 32, 64, ...
e.) 81, 27, 9, 3, 1, 1/3, 1/9, ...
f.) 5, 10, 15, 20, 25, 30, 35, ...
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Unit 2a sequence Notes.notebook
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Creating patterns using rules: Make a pattern... up to 5 terms. g.) ... starting with 7 and adding 4 to each term.
h.) ... starting with 1, multiplying 3, and subtract 1 to each term.
i.) ... starting with 50 and subtract 8 to each term.
j.) ...starting with 0 and add nine to each term.
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Unit 2a sequence Notes.notebook
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Writing patterns to the nth term... When a pattern has a rule... it can be written as an expression using n.
n= the order of the term . So, if n = 1, that's the first term of the sequence. If n = 2, that's the second term of the sequence.
(Coefficient)n + (a number) = the term Using example g, h, and i g.) 7, 11, 15, 19, 23, ...
1st: Find the difference between the numbers That value is your coefficient.
h.) 1, 4, 7, 10, 13, ...
i.)
50, 42, 34, 26, 18, ...
2nd: Decide what must be added to get the 1st value of the sequence. It may add or subtract.
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Unit 2a sequence Notes.notebook
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Sequence WS Name: ________________________________ Per: ______ Date: _________________ 1.) From the pattern created by the images below, how many triangles will be in the 5th image?
2.) Describe the pattern of the images from question 1.
3.) What is the nrule for the following sequence: 12, 24, 36, 48, 60, 72, 84, 96, 108, ...
4.) If the sequence followed the rule 2n 5, what are the first 5 terms of the sequence?
5.) Predict the next term in the sequence: 48, 42, 36, 30, ___. Explain your reasoning to get that term.
6.) Find the next three terms of the sequence: 1, 5, 4, 20, ______, ______, ______ 7.) Explain how you found the next three terms in problem 6.
8.) Create your own number pattern. It must have at least 7 terms, and you must state the rule. Extra credit if you create the rule using n correctly.
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Unit 2a sequence Notes.notebook
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100113 Algebra
Warmup: What number is missing from the pattern: 2, 8, 18, ____, 50, 72,...
Daily Proficiency:
EQ: Explain visually and orally what a function is. HW: WS 2a2
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Unit 2a sequence Notes.notebook
101413 HW: 2a2
December 05, 2013
EQ: Explain visually and orally what a function is. Unit 2a Linear Equations Day 2: Relations and Functions
Review: Create a pattern using 2n + 3...write as a table
n result
This table is also known as an inputoutput table.
Input= the numbers you plug into a rule or expression, x Output = the numbers that result from the expression, y
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Unit 2a sequence Notes.notebook
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When you have a group of input values, it's called DOMAIN !!! When you have a group of output values, it's called RANGE !!! State the domain and range of:
Use { Ex. 1)
} when lists numbers of domain and range. input output
7 6 5 4
2 4 1 3
Ex. 2)
input output 0 2 3 4 3 4 9 8
Domain and Range combine to create a RELATION.
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Unit 2a sequence Notes.notebook
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Function = a relation that has 1 input for each output. Which of the previous examples is a function? Example 1 or Example 2 Are the following a function? Ex. 3.)
Ex. 4.) {(5, 9), (4, 8), (3, 7), (2, 6)}
Ex.5)
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Unit 2a sequence Notes.notebook
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Vertical line Test: A quick test you can do to see if a graph is a function. The Test: Draw a vertical line on the graph. If it hits the picture once, then its considered a function. If it hits the picture more than once, it's not a function. Ex. 5 Again
Model:
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Unit 2a sequence Notes.notebook
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Try: 1.) 2.) 3.)
4.) 5.) 6.)
State the domain and range for the following:
7.)
1 2 -6 -2
3 2
4 5 6 10
8.) 8
4
2 -3 10 0 1 -1.5 5 0
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Unit 2a sequence Notes.notebook
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Vocabulary Check WS Name: _____________________ Period: __ Date: ________ Directions: Define the math vocabulary in your own words. Create an example of each.
1.) Relation
2.) Algebraic sequences
3.) Geometric Sequences
4.) Input
5.) Output
6.) Function
7.) Sequence
8.) Vertical line Test
9.) Term
10.) Range
11.) Domain 15
Unit 2a sequence Notes.notebook
December 05, 2013
How do you read function notation?
101613 Algebra Warmup: State the Range of y = x + 3 when x = { 3, 7, 1, 4}
EQ:
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Unit 2a sequence Notes.notebook
101613 HW:
December 05, 2013
EQ: How do you read function notation? Unit 2a Linear Equations Day 3 T-tables and Function notation
Ordered Pairs: (x, y) each individual input with it's output. TTables: a place to write 2 or more ordered pairs. Model: Pick 4 input values (x), and find their output (y) given a formal sequence (function equation)
Note: Sometimes the question gives you the inputs/ domain
y = 3x 2 x
y
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Unit 2a sequence Notes.notebook
December 05, 2013
Create an inputoutput table for the following problems: Use 0, 1, 2, and 3 for your input values. 1. x 5 = y 2.) 5x 1 = y 3.) 2(x + 7) = y
x
y
x
y
x
y
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Unit 2a sequence Notes.notebook
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Function Notation: The notation given to an equation when the graph IS a function. NO checking using the ___vertical line test. _________________ required.
Models: f(x) = 3x + 2, g(x) = 3x, h(x) = 3, j(x) = x2 1 Treat f(x) as the output, y. Make an input output table using the following examples. Use x= {1, 2, 5, 7}
4.) f(x) = 3x + 2 5.) g(x) = 4x 6.) h(x) = 7
x
f(x)
x
g(x)
x
h(x)
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Unit 2a sequence Notes.notebook
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On your own: Make an input output table using the following examples.
Use your own values for x.
a.) f(x) = x 2 b.) g(x) = 3(x+5) c.) h(x) = (1/2)x
x
f(x)
x
g(x)
x
h(x)
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Unit 2a sequence Notes.notebook
December 05, 2013
101712 Algebra Warmup: Create a ttable for: j(x) = x
EQ: How do you graph function notation?
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Unit 2a sequence Notes.notebook
100313 HW: WS 2a3
December 05, 2013
EQ: How do you graph function notation? Unit 2a Linear Equations Day 4 Graphing using t-tables
Ordered Pairs: ______________________________________ Graphing definitions: yaxis: ______________________
Coordinate Plane Ordered Pairs are your guide to show how the function grows.
x axis: _____________ ___________________ ___________________
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Unit 2a sequence Notes.notebook
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Model: Graph f(x)= x + 1 x
f(x)
1 0 1 2 3
0 1 2 3 4
y
x
-x
Once the ordered pairs are plotted, connect the dots with a continuous line.
-y
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Unit 2a sequence Notes.notebook
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y
Ex. 1 Graph f(x)= 2x x
f(x)
x
-x
What is different about this graph compared to the model? -y
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Unit 2a sequence Notes.notebook
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Ex. 2 Graph f(x)= 2x - 3 x
y
f(x)
x
-x
What is different about this graph compared to f(x) = 2x?
-y
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Unit 2a sequence Notes.notebook
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Unit 2a sequence Notes.notebook
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Ex. 3 Graph f(x)= (1/2)x x
y
f(x)
x
-x
What is different about this graph compared to f(x) = 2x?
-y
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Unit 2a sequence Notes.notebook
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Ex. 4 Graph f(x)= -5x + 3 x
y
f(x)
x
-x
What is different about this graph compared to the others?
-y
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Unit 2a sequence Notes.notebook
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y
Ex. 5 Graph f(x)= x2 x
f(x)
x
-x
What is different about this graph compared to the others?
-y
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Unit 2a sequence Notes.notebook
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y
Try: Graph f(x)= -x + 10 x
f(x)
x
-x
Any noticible trends the graphs do?
-y
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Unit 2a sequence Notes.notebook
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y
Try: Graph f(x)= 3x - 5 x
f(x)
x
-x
Any noticible trends the graphs do?
-y
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Unit 2a sequence Notes.notebook
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102113 Algebra Warmup:
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Unit 2a sequence Notes.notebook
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Unit 2a sequence Notes.notebook
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102213 Algebra Warmup: In your own words, describe function notation.
EQ: How does function notation really work? HW: Finish Classwork.
Daily Proficiency:
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Unit 2a sequence Notes.notebook
10/7/13 HW: none
December 05, 2013
EQ:How does function notation really work?
Linear Function Review
Sequences: a list of numbers created by a rule or pattern 2, 7, 12, 17, 22, ...
State the pattern: State the Nthrule: What are the next two terms?
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Unit 2a sequence Notes.notebook
December 05, 2013
Your turn: 20, 17, 14, 11, 8,...
State the pattern: State the Nthrule:
What are the next two terms?
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Unit 2a sequence Notes.notebook
Creating a sequence from an nthrule.
December 05, 2013
2n 8
The first five terms. 1st term: 2nd term: 3rd term: 4th term: 5th term: n represents the location of the number in the list. When the sequence is part of a function the n turns to x.
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Unit 2a sequence Notes.notebook
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Your turn:
Creating a sequence from an nthrule.
2n + 4
The first five terms. 1st term: 2nd term: 3rd term: 4th term: 5th term:
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Unit 2a sequence Notes.notebook
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Function Notation places the nth rule in an equation and now it becomes a trend. We must graph the trend. f(x) = x + 4 ttable is a tool to organize your numbers.
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Unit 2a sequence Notes.notebook
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When we only want to find one specific term: The problem will look like: We want to find the value when x is 3. Substitute 3 everywhere there is an x. (TTables are not necessary)
1.) f(x) = 9 + 3x; f(2)
Try: 3. f(x)= x + 7; f(8)
f(x) = 5x +2; f(3)
2.) f(x) = 3x 7x + 9 ; f(1)
4.) f(x) = x 5 ; f(6) 2
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Unit 2a sequence Notes.notebook
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102113 Algebra Warmup: Draw a steep line graph, Draw a flat line graph.
EQ: What is Slope? HW: WS 2a4
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Unit 2a sequence Notes.notebook
December 05, 2013
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Unit 2a sequence Notes.notebook
102113 HW: WS 2a4
December 05, 2013
EQ: What is Slope? Unit 2a Linear Equations Day 5 Finding Slopes
Slope: (m)The pattern or trend a line travels Types of slopes: Slopes create four directions.
Positive Slope 1.) ___________________________ 2.) ___________________________ Negative Slope 3.) ___________________________ Zero Slope 4.) ___________________________ No Slope
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Unit 2a sequence Notes.notebook
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Finding Slope off a graph:
Pick an ordered pair to start, Pick a second ordered pair to stop. Count the number of grid lines up OR down FIRST Count the number of grid lines right OR left SECOND. Depending on the direction of the line will depend on if it's up, down, right, or left. If you counted down, make sure to include a negative. This is the same if you count left.
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Unit 2a sequence Notes.notebook
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Model: Find the slope of the line.
First ordered pair
Second ordered pair
The line travels Down 5 units and Left 2 units. So m = 5/2 (Positive Slope) ... translation for every 5 units up the line travels 2 units to the right.
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Ex. 1)
So m = (____________ Slope) ... translation for every ______________, the line travels _________________.
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Unit 2a sequence Notes.notebook
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Ex. 2)
So m = (____________ Slope) ... translation for every ______________, the line travels _________________.
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Unit 2a sequence Notes.notebook
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Ex. 3)
So m = (____________ Slope) ... translation for every ______________, the line travels _________________.
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Unit 2a sequence Notes.notebook
Slope = Rise also known as Run
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Δy Δx
Δ= the greek letter "delta" that means difference of... So when given two ordered pairs (either from a graph or a list) find the difference of the y's in the ordered pairs and divide that by the difference of the x's in the ordered pairs. Slope Formula:
y2 - y1 x2 - x1
where the given ordered pairs are: (x 1, y1) and (x2, y2)
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Unit 2a sequence Notes.notebook
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Finding Slope using the formula: 1.) (4, 6) and (0, 0)
2.) (2, 5) and (2, 3)
3.) (10, 10) and (10, 10) 4.) (6, 2) and (6, 4)
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Unit 2a sequence Notes.notebook
Try:
December 05, 2013
Find the slope and direction
a.) (9, 4) and (7, 4)
b.) (3, 3) and (2, 2)
c.) (5, 0) and (5, 5)
d.) ( 0, 6) and (3, 3)
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Unit 2a sequence Notes.notebook
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102313 Algebra Warmup: Find the slope of (9,9) and (6,3)
EQ: Describe the slopeintercept form and how its formed. HW: WS 2a5
Daily Proficiency:
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Unit 2a sequence Notes.notebook
10/09/13 Hw: WS 2a5
December 05, 2013
EQ: Describe the slopeintercept form and how it is formed. Unit 2a Linear Equations Day 6 Slope and Slope Intercept Form
Recap: We've found slope from graphs....
And from ordered pairs....
(1, 4) and (7, 2)
What we haven't done is find slopes from equations...
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Unit 2a sequence Notes.notebook
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Two main equations you will see... Standard Form: Ax + By = C Slope Intercept Form: y = mx + b m = _slope __ b = (0, b)_yintercept_ The number in front of x When starting with Standard Form, use opposite operations to SOLVE FOR Y. Ex. 1) x + y = 4 Ex. 2) 6x + 3y = 9
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Unit 2a sequence Notes.notebook
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What do you do once the equation is in slope- intercept form? Find the Slope, of course.
What is the slope and yintercept of the following equations?
Ex. 3) y = 6x 5
Ex. 4) y = 9x
Ex. 5) y = (1/2) x + 3
Ex. 6) 8x + 7 = y
Ex. 7) y = 5
Ex. 8) x = 2
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Unit 2a sequence Notes.notebook
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Graph: y = 2x + 7
1st: Identify the y-intercept,(b), by ignoring the x term. b = ____
2nd: Create an ordered pair by using x = 0 and the b. (0, b) (
,
)
3rd: Plot the ordered pair on a coordinate plane. It is your starting point. 4th: Find the slope and write as a fraction. m = rise = run
Last: (repeat step x2) At the starting point go ____________, then _________________
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Unit 2a sequence Notes.notebook
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Let's put it all together... Solve for y, state the slope, then graph it. 1.) x + y = 6
2.) 5x + y = 9
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Unit 2a sequence Notes.notebook
3.) x y = 7
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4.) y x + 9 = 0
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Try: Identify the slope, yintercept and graph for the equations below: a.) y = 4x 3 b.) x + 4y = 8
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Unit 2a sequence Notes.notebook
December 05, 2013
101013 Algebra Warmup: Graph: y = (1/4)x 7 Daily Proficiency:
EQ: On a scale of 110 how difficult is gr aphing. Explain your score. HW: none
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Unit 2a sequence Notes.notebook
December 05, 2013
101113 Algebra Warmup: Graph: y = (1/2)x +3 Daily Proficiency:
EQ: On a scale of 110 how difficult is gr aphing. Explain your score. HW: none
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Unit 2a Quiz 2 Slopes Name: _____________________________________Per: ____Date: _________________
For Problems 13, Identify the slope and yintercepts.
1.) y = 1 x + 7 2
m= ___ b = ___
2.) x + 3y = 9
m= ___ b = ___
3.) 4x = y 5
m= ___ b = ___
For Problems 45, Use the slope formula to find the slope of two ordered pairs. State the direction.
4.) (-3, 3) and (-7, 13)
5.) (3, 7) and (-1, 9)
For Problems 6, 7. Graph the equations.
6.) 4x - 2y = -6
7.) y = -3x + 5 4
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Quiz on Slopes and Yintercept form Name: _____________________________________Per: ____Date: _________________
For Problems 13, Identify the slope and yintercepts.
1.) y = 3x + 2 4
2.) 2x + y = - 3
m= ___ b = ___
m= ___ b = ___
3.) 2x = y 5
m= ___ b = ___
For Problems 45, Use the slope formula to find the slope of two ordered pairs. State the direction.
4.) (2, 4) and (-4, -8)
5.) (0, 0) and (-7, 7)
For Problems 6, 7. Graph the equations.
6.) y = 2x + 5 3
7.) 4x - 2y = -6
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Algebra 10/14/13 Warmup: Have you ever used a graphing calculator before? Daily Proficiency:
EQ: Are you ready for the second half of graphing? HW: None
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Unit 2a sequence Notes.notebook
10/14/13 Hw: None
December 05, 2013
EQ: Are you ready for the second half of graphing? Linear Equations: Day 7 Calculator training Day
Quick poll: On a scale of 1(bad)10(I can graph forever), how comfortable are you with graphing? Would you like to check your graphs?
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How to use a graphing calculator:
Graphing buttons ______________________
= most commonly used buttons
asks for functions in blue deletes small errors
deletes a full line or screen of errors turns the calculator on (Saves batteries)
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Model: Graph the following problems:
Find and press the y = button
Use to enter x The equation will graph when there is a black box around the equal sign.
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Graph the following problems on the calculator and on paper: 2.) y = (1/4)x 2 1.) y = 5x + 3
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3.) y = 3x
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4.) y = 7
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5.) x= y 3
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6.) x= 5
What makes these equations special? ____________________________________________ __________________________________________________________________________
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Extra Graphing Practice: Name:___________________________________________ Per: ______ Date: _____________ Graph the equations on the line. State the slope, the yintercept, and the direction each are going. 1) y = 2x
2) y = 5x 6
3) y = 4
4) y = 2x + 4 3
5) x = 10
6) 2x 3y = 9
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Algebra 10/17/13 Warmup: Graph: x > 9 Daily Proficiency:
EQ: What do 2d inequalities look like? HW: WS 2a7
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10/17/12 Hw: WS 2a7
December 05, 2013
EQ: Name two differences between lines of equations and lines of inequalities? Linear Equations: Day 8 Graphing Inequalities
Review: What do these symbols mean? > = __________________ < = __________________ ≥ = __________________ ≤ = __________________ Graph: a.) x > -7 10
9
8
7
6
5 4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
7
6
5 4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
b.) x ≤ 5
10
9
8
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c.) We need to graph: y > x - 7 Using slope-intercept form: Is the line dotted or solid?________
> or
