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9TH Grade Mathematics [Algebra 1]

TABLE OF CONTENTS

Required Textbook: A Beka Algebra 1, Work-Text, Pensacola, Florida, current edition.
  • MAIN OBJECTIVES
  • INTRODUCTORY CONCEPTS
    --|Linear Expressions and Equations
    --|Slope, Intercepts, and Slope-Intercept Form
    --|Graphing Linear Equations
  • SYSTEMS OF EQUATIONS
    --|Simultaneous Equations
    --|Graphing Simultaneous Equations
    --|Inconsistent and Dependent Systems
  • POLYNOMIALS
    --|Monomials and Polynomials
    --|Simplifying Polynomials
    --|Adding and Subtracting Polynomials
    --|Multiplying Polynomials
    --|Dividing Polynomials
  • FACTORING
    --|Prime, Composite, and Square Numbers
    --|Factoring Monomials
    --|Factoring Polynomials
    --|Factoring the Difference of Two Squares
  • QUADRATIC EQUATIONS
    --|Quadratic Expressions, Equations, And Functions
    --|Solving Quadratic Equations by Factoring
    --|Solving Quadratic Equations by Graphing
    --|The Quadratic Formula
  • REFERENCES
    MAIN OBJECTIVES


    [a] Understand the basics of functions and know how to apply them to problem-solving situations.
    [i] Identify the mathematical domains and ranges and determine reasonable domain and range values for given situations.
    [ii] Know how to manipulate symbols in order to solve problems and use the necessary algebraic skills required to simplify algebraic expressions (factoring, properties of exponents etc.).
    [ii] Connect the function notation of y = and (x) =.
    [ii] Analyze situations and formulate systems of equations or inequalities in two or more unknowns to solve problems.
    [iii] Interpret and determine the reasonableness of solutions to systems of equations or inequalities.
    [iv] Identify and sketch graphs of parent functions, including linear (y = x), quadratic (y = x2), square root (y = √x), inverse (y = 1/x), exponential (y = ax), and logarithmic (y = logax) functions.

    [b] More advanced functions.
    [i] Describe a conic section as the intersection of a plane and a cone.
    [ii] Identify symmetries from graphs of conic sections and identify the conic section from a given equation.
    [iii] Understand that quadratic functions can be represented in different ways and learn to translate among their various representations.
    [iv] Determine the reasonable domain and range values of quadratic functions, interpret and determine the reasonableness of solutions to quadratic equations and inequalities.
    [v] Relate representations of quadratic functions, such as algebraic, tabular, graphical, and verbal descriptions.
    [iv] Determine a quadratic function from its roots or a graph.
    [v] Use characteristics of the quadratic parent function to sketch the related graphs and connect between the y = ax2 + bx + c and the y = a(x - h)2 + k symbolic representations.

    [c] Rational functions.
    [i] Formulate equations and inequalities based on rational functions.
    [ii] Use quotients to describe the graphs of rational functions, describe limitations on the domains and ranges, and examine asymptotic behavior.
    [iii] Determine the reasonable domain and range values of rational functions and determine the reasonableness of solutions rational equations and inequalities.
    [iv] Analyze a situation modeled by a rational function, formulate an equation or inequality.

    [d] Exponential and logarithmic functions.
    [i] Formulate equations and inequalities based on exponential and logarithmic functions.
    [ii] Develop the definition of logarithms by exploring and describing the relationship between exponential functions and their inverses.
    [iii] Analyze a situation modeled by an exponential function, formulate an equation or inequality.

    1 INTRODUCTORY CONCEPTS

    1.1 - Linear Expressions and Equations

    Sometimes, you find yourself having to write out your own algebraic expression based on the wording of a problem. 

    In that situation, you want to 

    1. read the problem carefully,
    2. pick out key words and phrases and determine their equivalent mathematical meaning,
    3. replace any unknowns with a variable, and
    4. put it all together in an algebraic expression.
    The following are some key words and phrases and their translations:
     
    Addition: sum, plus, add to, more than, increased by, total 

     
    Subtraction:  difference of, minus, subtracted from, less than, decreased by, less

     
    Multiplication:  product, times, multiply, twice, of 

     
    Division:  quotient divide, into, ratio 

    [1.1.a] Write the phrase as an algebraic expression.
    The sum of a number and 10

    It looks like the only reference to a mathematical operation is the word sum - so what operation will we have in this expression?   If you said addition, you are correct.

    The phrase 'a number' indicates that it is an unknown number - there was no specific value given to it.   So we will replace the phrase 'a number' with the variable x.  We want to let our variable represent any number that is unknown

    Putting everything together, we can translate the given english phrase with the following algebraic expression:
    *'sum' = +
    *'a number' = variable X
    *'The sum of a number and 10' = X + 10

    [1.1.b]

    The product of 5 and a number

    Again, we are wanting to rewrite this as an algebraic expression, not evaluate it.

    This time, the phrase that correlates with our operation is 'product' - so what operation will we be doing this time?   If you said multiplication, you are right on.

    Again, we have the phrase 'a number', which again is going to be replaced with a variable, since we do not know what the number is.

    Lets see what we get for this answer:

    *'product' = multiplication
    *'a number' = variable X
    The product of 5 and a number = 5X

    [1.1.c]

    3 less than twice a number

    First of all, we have the phrase 'less than' which mathematically translates as subtraction.  You need to be careful with this phrase,  it is very tempting to start off with 3 and put your subtraction sign after the 3.  However, think about it, if you want 3 less than something, you are 3 below it. In order to be 3 below something, you would have to subtract the 3.  So you would not have 3 minus, but minus 3 as PART of your expression.

    The other part of the expression involves the phrase 'twice a number'.  'Twice' translates as two times a number and, as above, we will replace the phrase 'a number' with our variable x.
     

    Putting this together we get:

    *'less than' = -
    *'twice' = 2 times
    *'a number' = variable X
    *'3 less than twice a number' = 3 - 2X


    [1.1.d]

    The quotient of 3 and the difference of a number and 2

    First of all, the term 'quotient' is going to be replaced with what mathematical operation?  If you said division, you are right on the mark.

    Note how 3 immediately follows the phrase 'the quotient of', this means that 3 is going to be in the numerator.  The phrase that immediately follows the word 'quotient' is going to be in the numerator of it.

    After the word and', you have the phrase 'the difference of a number and 2'.  That is the second part of your quotient which means it will go in the denominator.  And what operation will we have when we do write that difference down below?  I hope you said subtraction.
     

    Lets see what we get when we put all of this together:
    *'quotient' = division
    *'difference' = -
    *'a number' = variable X
    *'The quotient of 3 and the difference of a number and 2' = 3 / [X - 2]


    1.2 - Slope, Intercepts, and Slope-Intercept Form

    Straight-line equations, or linear equations, graph as straight lines, and have simple variables with no exponents on them. For example, the linear function, f(x) = mx + b, may be graphed on the x, y plane as the equation,

    y = mx + b

    In summary
  • This equation is called the slope-intercept form for a line.
  • The graph of this equation is a straight line.
  • The slope of the line is m.
  • The line crosses the y-axis at b.
  • The point where the line crosses the y-axis is called the y-intercept.
  • The x, y coordinates for the y-intercept are (0, b)
    Given some information about a line, you can figure out the equation of that line.
  • [1.2.a]Find the equation of the straight line that has slope m = 4 and passes through the point (1, 6).
    First you write the standard equation of a line,
    y = mx + b

    Then plug in the numbers you are given. You've been given the slope m. The point (-1, -6), gives me an x-value and a y-value: x = 1 and y = 6. The slope-intercept form of a straight line has y, m, x, and b. So the only missing component is is b (the y-intercept).
           y = mx + b
           y = 4x + b       <-- replace m with 4
           y = 4(-1) + b    <-- replace x with -1
          -6 = 4(-1) + b    <-- replace y with -6
          -6 = -4 + b
          -6-(-4) = b
          -2 = b
    
    The line equation is y = 4x 2



    1.3 - Graphing Linear Equations

    Follow the example y = 3x + 2 through this explanation.
  • The number in front of x is the slope. (If necessary, place this number over 1 to form a fraction.
    This fraction is your rise/run.) slope = 3/1
  • The "b" value is where the line crosses the y-axis. Be sure to check the sign of this number. b = 2
  • Plot the b value on the y-axis. see graph below
  • Standing at that point use your rise and run values to plot your second point.
    (If rise is positive, move up. If rise is negative, move down.) (If run is positive, move right. If run is negative, move left.)
  • Connect the two points to form the line.

    ...
    2 SYSTEMS OF EQUATIONS

    2.1 - Simultaneous Equations

    When in a given problem, more than one algebraic equation is true at a time, it is said there is a system of simultaneous equations which are all true together at once. Such sets of multiple equations may help solve for more than one unknown variable in a problem, since having more than one unknown in one equation is typically not enough information to solve any of the unknowns.
    ...
    3 POLYNOMIALS

    3.1 - Monomials and Polynomials

    An algebraic expression consisting of one term that is a constant, a variable, or the product of constants and variables is called a monomial. Examples of monomials are,
                  4           x           4y           7y2x3z  
    
    Two or more terms containing the same variables, with correspomnding variables having the same exponents are called like terms. Examples of like terms are,
                  4x    and    x            5y    and     -7y
                  2x2   and  3x2            5z3   and  3z3    
    
    The degree of a monomial in one variable is the exponent of the variable. For example,
                  The degree of 4x is 1    
                  The degree of 2x2 is 2
                  The degree of 5z3 is 3    
    
    The degree of a nonzero constant is zero. For example, 5 is the same as 5x0.
    A monomial such as 7x3 may be considered a polynomial of one term. A polynomial is on its simplest form when it contains no like terms. For example,
                  5z3 + 2x2 + 3z3 + 4
    The terms 5z3 and 3z3 are like terms and can be added up.
    The simplified form of the polinomial is,
                  8z3 + 2x2 + 4
                
    
    ...

    4 FACTORING

    4.1 - Prime, Composite, and Square Numbers

    ...

    5 QUADRATIC EQUATIONS

    5.1 - Quadratic Expressions, Equations, And Functions

    ...

    REFERENCES

    [1] A Beka Algebra 1, Work-Text, Pensacola, Florida.