Chapter 5 Concept Map

Chapter 5 Concept Map

1. Toolkit B.1/Section 6.1

   • Algebraic Expression:
     • Arithmetic expression that contains variables
       • Key feature: the number of terms it has
   • Monomial:
     • Algebraic expression with ONE term
       • 7x^2
         • Binomial:
           • Algebraic expression with TWO terms
             • 6x- 4
               • Trinomial:
                 • Algebraic expression with THREE terms
                   • x^2 - x +1
   • Special Product Formulas
     • If A and B are any real numbers or algebraic expressions, then
       • (A+B)(A-B)= A^2-B^2
         • *Inside and outside of FOIL cancel
           • ex:
             • (5t+1)(5t-1)=
               • *FOIL*
                 • 25t^2-1
   • Sum and Differences of Functions
     • The SUM of f and g:
       • is the function (f+g) defined by (f+g)(x)= f(x)+g(x)
     • The DIFFERENCE of f and g:
       • is the function (f-g) defined by (f-g)(x) = f(x) - g(x)
     • ex:
       • Let f(x)= 3x-4 and g(x)= x^2+1
         • (f+g)(x)=
           • (3x-4)(x^2+1)
             • x^2 + 3x -3

2. Toolkit C.2

   • Steps to solving Polynomial Equations by Factoring
     • 1.) Set the polynomial equal to zero
       • 2.) Factor the polynomial completely
         • 3.) Set each of the factors equal to zero and solve
           • 4.) Check your answers in the original equation
   • Steps to solving Quadratic Equations Using the Square Root Property
     • 1.) Isolate the squared variable expression
       • 2.) Use the square root property to undo the square. Use +/- on the side away from the variable
         • 3.) Rewrite as two equations and solve
           • 4.) Check answer(s) in the original equation. Be sure that answers are within the domain of the model and that they make sense in the context of the problem
   • Steps to solving Quadratic Equations by Completing the Square
     • 1.) Isolate the variable terms on one side of the equation
       • 2.) If the coefficient of x^2 is not 1, divide both sides of the equation by the coefficient of x^2
         • 3.) Take half of the coefficient of x, then square it. Add this number to both sides of the equation
           • 4.) Factor the quadratic into the square of the binomial
             • 5.) Solve using the square root property
               • 6.) Check your answers in the original equation
   • Steps to solving Equations using the Quadratic Formula
     • 1.) Set the quadratic equal to zero
       • 2.) Put the quadratic into standard form
         • 3.) Substitute the values of a, b and c into the quadratic formula
           • x= -b+/- √b^2 - 4ac / 2a
             • 4.) Simplify the quadratic formula
               • 5.) Check answers in the original equation

3. Section 5.2

   • General Form of a Quadratic Function
     • F(x)= ax^2 + bx + c
       • where a, b, and c are real numbers and a≠0
   • Expressing the function:
     • ex: f(x)= (3x-6)(x+5)
       • Foil the equation
         • 3x^2+15x-6x-30
           • f(x)= 3x^2+9x-30
             • in order of a, b, and c
   • Standard Form of a Quadratic Function
     • f(x)=a(x-h)^2 + k
       • can be found from General Form by Completing the Square
     • Vertex form of a parabola
       • y= a(x-h)^2 + k
         • h= horizontal shift
           • k= vertical shift
             • a=
               • skinnier if... a>1
               • wider if... 0<a<1
               • opens up if... a is positive/opens down if... a is negative
   • Graphing using standard form
     • Positive a value
     • Negative a value
     • find the vertex and sketch the graph of each quadratic function
       • make sure to label the vertex and at least one more point on the curve

4. Section 5.3

   • Quadratic Functions
     • General Form:
       • f(x)= ax^2 + bx + c
         • where a, b, and c are real number and a ≠ 0
     • Standard Form:
       • f(x)= a(x-h)^2 + k
         • where a, h, and k are real numbers and a ≠ 0
     • By completing the square, the general form f(x)= ax^2 + bx +c
       • of a quadratic function can be converted to the standard form:
         • f(x)= a(x-h)^2 + k
           • where h= - b/2a
   • Maximum or Minimum value of a Quadratic Function
     • if f(x)= ax^2 + bx + c
       • then the maximum or minimum values of f(x) occurs at the vertex, x= - b/2a
     • If a > 0,
       • then the minimum value is f(- b/2a)
     • If a < 0,
       • then the maximum value is f(- b/2a)
   • ex: f(x)= x^2 +10x +20
     • x^2= a
       • 10x= b
         • 20= c
     • Find vertex
       • x= - b/2a
         • x= -10/2(1)
           • = -5
             • Plug it back into the equation
               • -5^2 - 20 +20
                 • = -5
                   • Vertex= (-5, -5)
   • Does the graph have a maximum or minimum value?
     • Maximum
     • Minimum

5. Section 5.4

   • Finding x- and y-intercepts of the Graph of a Quadratic Function
     • For a quadratic function f(x)= ax^2 + bx + c,
       • To find the x-intercept(s), solve the equation ax^2 +bx + c =0
         • To find the y-intercept, set x=0

6. Continuation of Chapter 5

   • Rational Functions
     • A rational function is the quotient of two polynomials
       • R(x) = p(x) / q(x)
         • The domain of a rational function includes all real numbers except those that make the denominator equal to 0.
       • ex:
         • f(x)= x/x-1
           • x-1 ≠ 0
             • add 1 to both sides
               • x ≠ 1
                 • Domain: ℝ ≠ 1
     • The line:
       • x=a is a vertical asymptote of the graph of f(x) if f(x)= ∞ OR f(x)= -∞ as 'x' approaches 'a' from left or right
       • y=b is a horizontal asymptote of the function y=f(x) if the output approaches 'b' as 'x' approaches ∞ or -∞
     • A graph cannot cross a vertical asymptote, but it CAN cross a horizontal asymptote
   • Simplifying Rational Functions
     • A rational expression
       • a fractional expression in which the numerator and denominator
         • To simplify is a rational expression, factor the numerator and the denominator completely and divide out any common factors using the following property of fractions
           • AC/BC= A/B
     • ex:
       • 5x/10
         • 5/10 = 2
           • =x/2
   • Adding/Subtracting Rational Functions
     • To ADD or SUBTRACT rational expressions:
       • First, find a common denominator and then use these properties
         • A/C + B/C = A+B/C
         • A/C - B/C = A-B/C
         • Adding and Subtracting Rational Expressions
           • 1.) Factor the denominators and find the LCD (if necessary)
             • 2.) Rewrite each rational expression with the common denominator
               • 3.) Add or subtract, and then simplify
                 • ex:
                   • 5x/x+3 + 15/x+3
                     • Common denominators
                     • 5x+15/x+3
                       • check to see if anything simplifies
                         • 5(x+3)/x+3
                           • x+3's cancel out
                             • =5
   • Multiplying/Dividing Rational Expressions
     • To multiply rational expressions,
       • 1.) Factor the numerators and denominators completely
         • 2.) Simplify by dividing out common factors in the numerator and the denominator
           • 3.) Add or subtract, and then simplify
     • ex: (multiplying)
       • a+3/a+7 * a+7/a-5
         • a+7's cancel out
           • = a+3/a-5
     • ex: (dividing)
       • t^2v/w^3 / t/w
         • tv/w^2
   • Direct and Inverse Variation
     • Direct Variation:
       • Y varies directly with x
         • (y is directly proportional to x)
           • if y=kx for some real number k ≠ 0
             • k is called the variation constant or constant of proportionality
       • In Direct Variation, both variables either increase together or decrease together
     • Inverse Variation:
       • The variable y varies inversely with x if y= k/x
         • OR
           • y varies indirectly with x or y is inversely proportional to x if y= k/x
             • k is called the variation constant or constant of proportionality
               • Two quantities vary inversely if when one gets larger, the other one gets smaller

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