Polynomial and Rational Functions

Polynomial and Rational Functions

1. 3.1 Quadratic Functions

   • Exponents are ALWAYS positive
     • The highest exponent determines the degree
   • The proper form is f (x) = ax2 + bx + c
     • c is called the constant term and equals the y-intercept
   • Parabolas
     • general form f (x) = a( x – h )2 + k
       • (h,k) = Vertex
     • Vertex from proper form
       • ( (-b)/2a , f((-b)/2a))

2. 3.5 Rational Functions

   • x = vertical asymptote which can NEVER be crossed
     • vertical asymptote is determined by the value in which the denominator is zero
   • y = horizontal asymptote which might be crossed
     • finding horizontal asymptote of (a_n x^n)/(b_n y^m )
       • IF n < m than y = 0 , IF n = m than y = a_n/b_n , IF n > m than y = None

3. 3.3 Dividing Polynomials

   • If the remainder is zero the divisor is a factor of the quotient
   • Synthetic Division
   • Long Division

4. 3.2 Polynomial Functions and their Graphs

   • Lead Coefficient Test
     • Even : The ends point in same direction
       • Positive : both ends point up Negative : both ends point down
     • Odd : end point in oppisite directions
       • Positive : the line point down left and up right Negative : the line point up right down left
   • Factor VS Zero
     • IF ( x – a ) is a factor of f(x) THAN x = a is a zero of f (x)
     • multiplicity
       • When a function has multiple of the same zero

5. 3.4 Zeros of Polynomial Functions

   • Zeros are the location in which the graphed line crosses the x-axis
     • Rational Zero Test : Gives a list of all possible RATIONAL zero
       • Rational Zero Test = p/q = (factors of the constant)/(factors of the lead coefficient)
     • • The Degree of the polynomial is equal to the Max number of zeros

6. 3.6 Solving Polynomial and Rational Inequalities

   • • A polynomial inequality is any inequality that can be put into one of the forms f ( x ) < 0, f ( x ) > 0, f ( x ) ≤ 0, f ( x ) ≥ 0
     • Inequalities are graphed on a number line
     • • IF < or > the point is not included and ( ) are to be used • IF ≤ or ≥ the point is to be included and [ ] are to be used

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