• INTERMEDIATE ALGEBRA
    Notes Semester 1
     
     IAUnit5 Polynomials
     
    IAUnit1 Ch1 Numbers & Ch2 Equations

     

     
    Day 1  Number Sets and Properties
     IntAlg1Notes2 Solving Equations
    IntAlg1Notes3  Absolute Value Equations
     
    NZQRC  Number Sets
    Sets Open Closed  absolute value def
     
    A scene in the movie Smilla's Sense of Snow compares human life to the number system:
     

    Properties of Equality
     
    Addition  Subtraction  Multiplication  Division
    add equals  subtract equals  multiply equals  dividing nonzero equals
     
    Identity  additive and multiplicative IDs  Inverse Properties  Additive and Multiplicative Inverses
     
    Reflexive  Symmetric Property  Transitive Property
    a = a  a=b iff b=a  transitive equations
     
    Associative Properties   of addition and multiplication
    Commutative Properties  of addition and multiplication
     
    Distributive Property of multiplication over addition
     

    Notes 2
    Solving Equations
     
    equation wordle
    What are the order of operations PEMDAS?
    PEMDAS  
     There are two kinds of equations--arithmetic and algebraic.
    The first is easy to solve.  Just do what it says:  Add 5 to 3:
    equation arithmetic algebra
     
    The second is more difficult.  What number plus 2 is 6? 
    You can guess, or use the Properties of Equality to isolate the variable, m.
    Do the opposite of adding 2.  Subtract 2.  m = 6 - 2 so m is 4.
    This is called a one-step equation.  Here's another:
     
    one step add 3  
    Do the opposite of subtracting 3 to isolate x. 
    Add 3 to both sides using Addition Property.
    Use the Additive Identity to get rid of 0.
     
     
     
     
     
    solving one step  
    Here's a one step equation using the Division Property.
     
    one step divide  
     
    This example is a two-step equation.
     
    Solve x on left  
     
    Here's another equation that requires two steps to solve.
    solve two step  
    Since the equation says multiply x by 4 first, then add 8 to get 24.
    Do the inverse operations in reverse order.  Subtract 8, then divide by 4.
     
    What happens to n?  Multiplied by 5 then subtract 11.
    Inverses in opposite order to 24:  Add 11 then divide by 5.
    solve two step add divide  
     
    Sometimes equations take many steps to solve.  These are multi-step equations.
     
    multi step combine like terms
    How did they ditch the 6?  Subract 6. 
    How did they ditch the 8?  Divide by 8.
     
    What if the distributive property is involved?
     
    solve distributive eq
    Now it's a multistep equation.  Isolate x on the right by subtracting 5x from both sides then adding 5.
     
    Try this example:
    solve dist like terms
     
     Here's one with a fraction:
    solve distributive fraction  
    Now there's a fraction. It would have been better to multiply by 4 first:
    3(x+3) = 9*4
    3x + 9 = 36
    3x = 36-9
    x = 27/3
    x = 9
     Here's a multi-step distributive equation with a fraction.

     Solve distributive fraction
     
     
     This one has a double distributive property:
     

     
    Day 3
    Absolute Value Equations
     
    These techniques are useful in absolute value equations:
    solve abs val eq  
    Note there are two solutions.  By definition of absolute value either x+2 = 7 or its opposite does: -(x+2)=7.
    First, solve x = 7 - 2. There are two ways to go about the second part:
    -x - 2 = 7 OR x + 2 = -7. 
    Both give the same answer:
    -x = 9 OR x = -9.
     
    How can you be sure to find both solutions?
    solve abs val multi steps
    The inside could be 7 or -7.  So we have two equations:
    x - 2 = 7 OR x - 2 = -7
    x = 7+2 OR x = -7+2
    x = 9 or x = -5.
     
    Review of absolute value definition:
     abs val def
    and as a fuction.
    abs val function  
    In other words, if x is negative, make it positive.
    If x is positive or zero, keep it the same.
     
     

     Unit 2 Chapter 3
    Graphing Linear Equations
     
    IntAlg2Notes1 Graphing Linear Equations
    IntAlg2Notes2 Function Notation
    IntAlg2Notes4  Slope Intercepts
    IntAlg2Notes5 Functions, Relations, Vertical Line Test
     IntAlg2Notes6 Vertical, Horizontal, Parallel, Perpendicular Lines
     IntAlg2Notes7  Linear Inequalities
     

     
     
    Graphing Linear Equations
     linear equation graphic
     
     A Cartesian grid (named after Rene DesCartes) is formed when two number lines intersect to form a right angle.
    cartesian grid  
    The intersection of the lines is called the origin.  
    The horizontal number line is the x-axis and the vertical line is the y-axis.
    Moving right on the x-axis corresponds to a positive number.  A negative number is to the left of the origin.
    On the y-axis, positive numbers mean moving up from the origin, and negative numbers mean moving down.
     
    So (2, 5) means from the origin move right 2 units and up 5.  This location point lies in Quadrant I.
    Moving left 3 and up 6 would be (-3, 6) and places the point in the second quadrant, QII.
    (-1, -1) is in the third quadrant (QIII), and (2, -3) would be in the fourth (QIV).
     
     
    To graph a line, there are several forms of equations.
     
    linear equation forms
     For the slope-intercept form, y = mx + b, ordered pairs can be generated in a table.
     linear equation table
    To graph a line, say y = 3x+1, only two points are needed.
    So evaluate for x = 0 and x = 1.
    3(0) +1 = 1 and 3(1) + 1 = 4.  Graph (0,1) and (3, 4) then draw the line.
     
    3xplus1 graph  
     
    To graph an equation in standard form, Ax + By = C, plug 0 into each variable and solve for the other.
    Let x = 0 and solve 3(0) + 2y = 6 for y.  Then substitute y = 0 and solve 3x + 2(0) = 6 for x.
     
     
    standard form 3x + 2y = 6  
    By graphing the points (2,0) and (0,3) we can graph the line and also locate both intercepts.
     
    Graph x - 2y = 2.
    standard form x - 2y 2=2
     
    Here are steps for graphing the slope intercept form.
    step by step slope intercept  
     Here is another form called point-slope.
    point slope form  
     
    Here's a formula map for all linear equations.
     
     linear EQ formulas
     

    Function Notation
     
     DEFINITION:  A "function" is a rule or correspondence that maps one number (or input element) to exactly one output value.  If there are two possible outputs for the same input, the correspondence is not a function and is called a "relation" instead.
     
     function notation
    domain rule range  pictuer of a function
     
    Find the rule that maps the Domain numbers into the Range bubble:
     
    function square map  function squared map
    The rule is to square the input number.  f(-4) = 16 because -4*-4 is 16.  f(x) = x^2.  
    We say that as "f of x equals x squared."  
     
    You can think of a function as a box or a room where something happens to the number going in.
    function +1 room  function x plus 1
     
     In this case, 1 is added to the input number, and the output is the sum.  f plus 1 equation
    function room
    In general, 
    function notation def  
    The input is called the Domain and the output is the Range.
    What are the domain and range values in the mapping below.
     input output map
    Domain:  {4, 5, 8}   Range:  {2, 5, 7, 9}
    Note this example is not a function as the input 5 has two different output values, 2 and 9. 
    Still, each relation can be written as an ordered pair: (4, 7); (5, 2); (5, 9); (8, 5)
     
    function ordered pair  
     
    If the relation is a function, the rule can be written as an equation.
    Function notation can also be written as an equation starting with "y =".
     
    function notation y
     
    function y equals 2x
     
    Now the equation can be described as a rule of correspondence.  
    For example, this rule is to multiply by 3 and then add 5.
     
     
    rule of correspondence
     
    function recap  
    function example  
     

     

     
     
     
     SLOPES and LINEAR EQUATIONS
    Linear EQ graphic
     
     
    Slope triangles
     
    Notes reviewing linear equations (need not copy systems examples.)
     
    linear EQ review  
     

     
     

     
    There's a typo in #5.  It should read "Up 1, Left 4" as it is a negative slope.
    Graphing Slopes/Slope Intercepts
     Graphing Slopes
     
    Graphing Slope Intercepts

    Functions and Relations
     
    Write your name in the first input bubble.  Then count the number of vowels and draw an arrow to that number in the second bubble.
     
    Name vowels  
    This is a mapping diagram for a function.  Each name has exactly one number of vowels in it. 
    As each input value corresponds to one output value, this relation is a function.
    However, each output value can come from more than one input. 
    For example, two different names can have the same number of vowels.
    But no name can have two different number of vowels in it.
     
    Is every relation mapping one set of values to another a function?  Not necessarily.
     
     Function Rule:  Map your birthday month to the sum of digits in your birthday.
    For example, August 19 would map August -> 10 because 1+9 = 10.
    Birthday digit sum
     
    Convert the month to their number (Jan = 1, Feb = 2, ... ) and convert the birth data into ordered pairs.
    For example, March 14 becomes (3, 5) because March is the 3rd month and 1+4 = 5.
     
    birthdate ordered pairs
     
    Now graph the ordered pairs.  Note that the repeated months prevent this rule from being a function.
    For example, there are two birthdays in December so the input x = 12 has two outputs, y = 4 and y = 6.
     
    birth digit sum graph  
     
    Graphing repeated x-values with different y-output values has points vertically above each other.
    This is an example of the relation failing the vertical line test.  So this rule is NOT a function.
     
    The set of values for the input is called the Domain.  The set of output values is called the Range.
    What are the Domain and Range for this data set?
     
    birth month digit sum
    DOMAIN:  {Feb, Mar, May, July, Sept, Oct, Nov Dec}
     RANGE:  {1, 2, 3, 4, 5, 6, 7, 9, 11}
    Mathematically, what could the smallest number for the range be? The largest number?
     
    Convert the data above into ordered pairs (month, digit sum).
    ordered pair birth digits  
     The circled pairs indicate repeated input values mapping to different output values. 
    So this is not a function.
    birth digit sum graph
     The repeated input values when graphed fail the vertical line test.
     

     
    SPECIAL LINEAR EQUATIONS
     Vertical and Horizontal Lines, Parallel and Perpendicular Lines
     
    Special Lines Notes
     
    Vertical Lines x = k
     
    Horizintal line y = b
     
    parallel lines (= slopes)
    parallel lines graph
     
    perpendicular line slopes
     
    perpendicular lines graphed
     
    perpendicular right angle  
     
    parallel perpendicular formulas
    parallel rise over run
     
    parallel m = 2 parallel same slopes  
     
    slope two thirds  
     
    parallel 1/2 find equations
    slope formula parallel perpendicular
    perpendicular right slopes
    opposite reciprocals  
    perpendicular slopes 2 perpendicular slopes 3 halfs

     perpendicular slopes negative half
    slope 3/4*(-4/3) = -1
     
    formula recap parallel perpendicular  
     
     
     

     
     
     
    IntAlg2Notes 7
    Graphing Inequalities
     
    Vertical lines are of the form x = k. 
    What do the graphs of x < k or x > k look like?
     
    x < - 1 x  /> 2
     
    The solution of ordered pairs (x,y) that satisfy the inequality is called a half-plane.
     
    Which graph is the true graph of  y > 1 ?
    horizontal graphs  
    Graph 2 is y > 1.  The others are y < 1, y < 1, and y > 1.
     
    For linear equations in slope-intercept form form y = mx + b, which side do we shade?
    shade half
     
    If the inequality is > or >, shade the upper half-plane.
    (y > mx+b, y > mx+b)
     
    Shade the lower half-plane if the inequality reads
    y < mx+b or y < mx + b.
    y < mx + b
     
    Here are the steps to graph y < (3/2)x + 3. 
    Use a solid line for "or equals to" symbols.
    graph line, then shade
    Shade below, or check the point (0, 0).  Is 0 < (2/3)0 + 3? 
    Yes, 0 < 3.  Shade on side of origin.
     
     Graph y > (-1/2)x -3.  Use a dashed line when graphing y = (-1/2)x - 3 since it is a strict inequality.
    y  /> (-1/2)x - 3
     
    Graph y > 2x -5.
    y  />= 2x - 5
    Graph 2x + 5y > -5.  First put inequality in slope intercept form.
    shade above with a solid line  
     
    Graphing standard form inequality Ax + By < C
     
    half plane solutions
     
    Test points for graphing x + y < 3.
    which side?
    So shade below on the same side as point (-1, 2).
    x + y < = 3
     
    Why can't (0, 0) be used as a good testing point for graphing this inequality?
    test points
    inequality instructions  
    inequality symbols  

     
     
    Unit 3 Chapter 4
    Systems of Equations
     systems wordle
    IntAlg4Notes1 Solve by Graphing
    IntAlg3Notes2 Solve by Substitution
    IntAlg3Notes3 Solve by Elimination
    IntAlg2Notes4  Solve with Technology
    IntAlg4Notes5 Solving with Matrices
    Solving with Cramer's Rule
     
    Lesson 1 - Solving Systems by Graphing

     
     
    three kinds of systems  
     
    inconsistent and consisten
    number of solutions
    solve a system by graphing  
    solve and check  
    consistent, independent
    inconsistent  
    (2,0) (2,1)  
    (2,2)  
    x = 2, y = 3
    y = 3x - 2 AND y = -x + 2  

     
    solve each system
     

    SOLVE Systems by Substitution
    (p226 evens 16-22 done as class notes)
     
    p 226 substitution steps
     
    p226 #20 #22  
     
    System Elimination
     elimination method
    elimination steps  
    solve system
    solve this system
    eliminate x
    back substitute y = 6
    eliminate y
    back substitute x = -1
    and x = 4
    eliminate x and y
     
    Use this online software to check answers and see solutions:
     
    elimination exercises  

    Solving Systems with Technology
    GRAPHING CALCULATORS:
     
    To graph the system of equations, follow these steps:
     
    Press Y= .
    Enter first equation into Y1. Press ENTER.
    Enter 2nd equation into Y2.
    Press ZOOM and then press  6 : ZStandard. 
    Or arrow to 6 and press ENTER.
    (This sets the xy-axes on -10 to 10.)
    A graphing window should appear.
     
     
    To find the point of intersection:
    Press 2nd TRACE to access the CALC menu.
    Select or press 6  : Intersect. 
    The calculator asks "First curve?" and if the cursor is on Y1 press ENTER.
    Repeat to answer "Second curve?" and ENTER.
    To answer "Guess?" arrow the cursor to the point of intersection and press ENTER.
    The bottom left of the screen says "X = " and the bottom right "Y = " with the solution.
     
     ALSO, on the internet check out this website online:
     
    It lets you enter equations in any form (Ax+By=C, y=mx+b, y-y1=m(x-x1), etc.).  
    Separate each equation by a semi-colon ";" and it SOLVES and GRAPHS it for you.
     

    SYSTEM WORD PROBLEMS
     
     coin problems 1-9
     

     
    How many pennies are there in circulation?

     
     
     Matrix Row Operations

     
     Coin #6
     
    5 nickels and 19 dimes
     
    Solve these with Matrix Row Operations:
     
     elimination exercises
     
    Let's do the two with fractions with graphing calculators.
    Write system #27 as a matrix:
    equation matrix
    Now enter it into a TI graphing calculator.
    Press 2nd MATRIX, and arrow to "EDIT".  Press ENTER or 1 to edit Matrix A.
    Press 2 ENTER and 3 ENTER to set the size of matrix A to 2 rows and 3 columns (2x3).
    Enter each fraction and press ENTER between each to fill in the matrix.
    Then  press 2nd QUIT, 2nd MATRIX, ENTER, MATH 1: Frac ENTER.
     
    [A] MathFrac
     Math Frac
     
    Now to solve it.  Press 2nd MATRIX, move to "MATH", move down to A:ref(, and press ENTER.
    Then press 2nd MATRIX, ENTER or 1, press MATH and ENTER or 1:>Frac to display in fractions.
     
    Ans → Frac
    Matrix 27b
     
    The is called the "reduced echelon form" and it tells us that 0x + 1y = 72, or y = 72.
    To find x we can back substitute, or repeat the process above this time with B:rref( from the MATRIX MATH menu.
    Press 2nd MATRIX, move to "MATH", move down to B:rref(, and press ENTER.
    Then press 2nd MATRIX, ENTER or 1, and ENTER again.
     
     
     
    x=34        or        x=34
     
    This form is called "reduced row echelon form" and tells us that 1x + 0y = 34, or x = 34, and that to y = 2.
     
    Now let's do #28 with the rref command.  First, enter the system into matrix A.
    matrix 28  
     
    Now Press 2nd MATRIX, move to EDIT, select B: rref(, 2nd MATRIX ENTER.
     
    rref([A]
    decimals
     
    It displayed it in decimals so let's frac it!
    MATH 1: Frac ENTER
     
    rref(Ans
    fractions  
    x = -1/7 and y = 75/14 or 5 5/14.

    3-by-3 Systems
    Solving 3 by 3
    p233 #10 1st
    Continue until you get a 3-by-3 diagonal of 1s with the other entries 0s:
     
    p233 #10 2nd
    If you end up with an entire row of 0s, then the system has infinitely many solutions and is dependent.

    Cramer's Rule 
     determinant = 1*7 -5(-3)
    solving for x  
    Solve cramer 2 by 2
     
    cramer solves 2 by 2  
     
    3 by 3 systems
    cramer 3 by 3  
    cramer solutions
    cramer colors  

     
     
    3 by 3 Determinants
     
    add down diagonals sum diagonals
     
    subtract up diagonals  
    subtract downward diagonals  
     
    determinant example
    Copy the first two columns to the right:
     
    copy first two columns  
    Now draw diagonals through the numbers:
     
    add diagonal products subtract diagonals products
    Add the downward diagonal products, and subtract the upward diagonal products.
     
    downward - upward  
     Evaluate the products, sum, and difference:
    determinant =  
     
    If a determinant is 0, then the system is either inconsistent or dependent,
    indicating its matrix is not invertible.
     
     
    3 by 3 determinants
     
     
     POLYNOMIALS
    naming polynomials  
    Polynomial Addition  
     
    foiling box
    foil distribute  
    FIRST OUT IN LAST  
    diff of squares
     
    Mr. FOIL  
     
    foil  
    FOIL
    FOIDL fold  

     

     box method
    polynomial cartoon
     

    PZ91
     
    PZ92
     
    PZ101
     

     
    PZ 101 1,5  
    pz 101 2, 6
    pz 101 #3
     
    PZ 101 4  
     
    PZ 101 8-9  
     
    PZ101 10-13
    pz101 14  

     
    PZ102  
     
     
     
Last Modified on January 6, 2016