Unit 3

Unit 3

1. Chapter 5: Quadratic Functions and Models

   • Section 5.2: Quadratic functions and their graphs
     • A quadratic functions is int he form f(x)=ax^2+bx+c
       • a,b, and c are real numbers
       • a does not equal 0 ever
     • Can be shows in the standard form f(x) = a(x-h)^2 + k by completing the square
     • Graphing using standard form
       • f(x)=ax^2+bx+c is a parabola with a vertex (h,k)
       • a>0 then it opens upwards
       • a<0 then it opens downwards
   • 5.4 Quadratic Equations: Getting info from a model
     • Finding x and y intercepts of the graph of a quadratic function for f(x)=ax^2+bx+c
       • to find x-intercept(s), solve ... ax^2+bx+c= 0
       • a does not = 0 ever
       • to find y-intercepts, set x=0 in f(x)=ax^2+bx+c
   • Toolkit B.2 Factoring algebraic expressions supplemental material
     • Factoring by common factors
       • 1. Find the largest common number. (GCF)
       • 2. Factor the GCF from each term, then simplify if possible
       • 3. Verify the factorization is correct
     • Factoring by grouping
       • 1. Factor ou the GCF
       • 2. Make two groups of terms
       • 3. For each of the groups find the GCF
       • 4. Remove the GCF
       • 5. If each group has a common polynomial factor, factor it out. if not do step 2 with different grouping
       • 6. Verify the Factorization is correct
     • Factoring Trinomials of the Form: x^2+bx+c
       • To factor x^2+bx+c = (x+m)(x+n) find the pair of integers (M and N) of which (mXn=c) and (m+n=b)
       • If c is positive, m/n have the same sign
       • If c is negative, m/n have opposite signs
       • A trinomial that does not factor is said to be a prime polynomial
     • Factoring by using AC method
       • 1. Factor ou the GCF
       • 2. Find AC
       • 3. Find the factors to rewrite bx as the sum of two terms
       • 4. Factor by grouping
       • 5. Verify the factorization is correct
     • Factoring using trial and error method
       • to factor ax^2+bx+c = (mx+p)(nx+q)
         • 1. List factors pairs m and n of a
         • 2. List factor pairs of p and q of c
         • 3. Find the combination of factor pairs form steps 2 and 3 such that mq+np=b
     • Factoring the sum/difference of two cubes
       • A^3 + B^3 = (A+B)(A^2-AB+B^2)
       • A^3 - B^3 = (A-B)(A^2+AB+B^2
     • Special Factoring Formulas
       • A^2+B^2 = PRIME
       • A^2+2AB+B^2 = (A+B)^2
       • A^2-2AB+B^2 = (A-B)^2
       • A^2-B^2= (A+B)(A-B)
   • Toolkit C.2 Solving Quadratic Equations Supplemental Material
     • Solving using the Square root property
       • If k is equal to or greater than zero, the solutions to x^2=k are x= +/- sqr rt of k
     • Zero product property
       • AB=0 if and only if A=0 or B=0
     • Solving a polynomial equation by factoring
       • 1. Write the equation in standard form
       • 2. factor the polynomial expression into 2 linear factors
       • 3. Set each factor equal to zero
       • solve each equation
     • Solving quadratic equations using the quadratic formula
       • x= -b +/- sqr rt of b^2 -4ac all divided by 2a
       • ax^2+bx+c=0
   • 5.3 Maxima and Minima: Getting info from a model
     • Minimum or maximum value of a quadratic function
       • if f(x)=ax^2+bx+c, then max/min values of f(x) occurs at the vertex , X = -b/2a
       • if a>0, then the min is f(-b/2a)
       • if a<0, then the max is f(-b/2a)

2. Chapter 6: Algebraic Operations

   • Toolkit B1: Combining Algebraic Expressions
     • An Algebraic expression is an arithmetic expression that contains variables, on being the amount of terms it has
       • Monomial= one term
       • Binomial = two terms
       • Trinomial= three term
     • To multiply, we use the distributive property
       • A(B+C)= AB+AC
       • (a+b)(x+y)= ax+ay+bx+by
       • (a+b)(x+y+z)= ax+ay+az+bx+by+bz
     • Special product formulas
       • (A+B)(A-B) = A^2-B^2
       • (A+B)^2 = A^2-2AB+B^2
       • (A-B)^2 = A^2-2AB+B^2
   • 6.1 Working with functions: Algebraic Operations
     • Let f and g be functions with domains A and B
       • Sum of f and g (f+g) is defined by (f+g)(x) = f(x) + g(x)
       • Difference of f and g (f-g) is defined by (f-g)(x) = f(x) + g(x)
     • As long as its a polynomial, the domain is all real number except when dividing
     • P(x) = R(x) - C(x)
       • P(x)= profit
       • R(x) =revenue from selling x units
       • C(x) = cost of producing x units of a product
       • Let f and q be functions with domains A and B Respectively. We define new functions as follows...
         • Product of f and g (fg) is defined by (fg)(x) = f(x) x g(x)
         • Quotient of f and g (f/g) is defined by (f/g)(x) = f(x)/g(x)

3. Rational Functions and Radical Functions

   • Rational expressions
     • 1. Multiply by Least common denominator (LDC)
     • 2. Solve the resulting equations
     • 3. Check the solutions in the originalequation. If a value makes any of the denominators zero, it is not a solution
   • Radical Functions and Equations
   • Inverse functions
     • Inverse variation (Proportionality) : Q and R are related by the equation Q=k/R
       • k= constant of proportionality
       • Q is inversely proportional to R
     • Direct variation (proportionality): both variables either increase/decrease together.
   • Toolkit B.3 Part II: Adding and Subtracting Rational Expressions
   • Toolkit B.3 Rational expressions part 1
     • Rational expression: A fractional expression in which the numerator and denominator are both polynomials
       • To simplify factor the numerator and the denominator and divide out any common factors using the following property of fractions
         • AC/BC = A/B
       • Multiplying
         • 1. Factor Numerators and denominators completely
           • 2. Simplify by dividing out common factors
             • 3. Multiply what you have left
       • Dividing
         • Flip the fraction that you dividing by and then multiply
           • (A/B)/ (C/D)..... (A/B)(D/C)
   • Rational Equations