Multiplicity of zeros- Even multiplicity, then touches the x-axis and turns around. Odd multiplicity then crosses the x-axis.

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1 Useful Fact Sheet Final Exam Polynomial function of degree n: with leading coefficient,, with maximum number of turning points is given by (n-1) Leading Term Behavior: If n is even, and If n is even, and If n is odd, and If n is odd, and Multiplicity of zeros- Even multiplicity, then touches the x-axis and turns around. Odd multiplicity then crosses the x-axis. Intermediate Value Theorem For any polynomial function P(x) with real coefficients, suppose that for a b, P(a) and P(b) are opposite signs. Then the function has a real zero between a and b. The Division Algorithm Remainder Theorem, then The Rational Zero Theorem If has integer coefficients and (where is reduced to lowest terms) is a rational zero of f, then p is a factor of the constant term and q is a factor of the leading coefficient,. The Linear Factorization Theorem If where and, then where are complex numbers. Descartes s Rule of Signs P(x), written in decreasing exponential order or ascending exponential order then The number of positive real zeros of P(x) is either: The number of negative real zeros of P(x) is either: 1. the same as the number of variations of sign in P(x) 1. the same as the number of variations of sign in P(-x) 2. Less than the number of sign variations by a positive 2. Less than the number of sign variations by a positive even integer even integer Determining a Vertical Asymptotes If is a zero of the denominator, then the line is a vertical asymptote. Determining a Horizontal Asymptotes For rational function the degree of the numerator is n and the degree of the denominator is m. 1. if, the x-axis, or, 2. if, the line, or, is is the horizontal asymptote. the horizontal asymptote. 3. if, the graph has no horizontal asymptote., Slant Asymptotes division will take the form, simplified, and the degree of p is one degree higher than the degree of q, then divide. The. The slant asymptote is obtained by dropping the remainder. Modeling using Variation: English Statement Equation English Statement Equation y varies directly as x, y is directly proportional to x y varies inversely as y is inversely proportional to y varies directly as, y is Y varies jointly as x and z directly proportional to y varies inversely as y is inversely proportional to Compound Interest:, where A is the balance, P is the principal, r is the interest rate (in decimal form), and t is time in years. Interest is paid more than once a year:, where n is number of compound periods in a year. Continuous compounding, 1 P a g e

2 Converting Between Exponential and Logarithmic Equation The Product Rule The Power Rule Change of Base Property: The Quotient Rule Base Exponent Property: For any b>0 and b 1 Factorial Notation: Summation Notation: One to One Property of Logarithms: For any M>0, N>0, b>0, and b 1 Expressing in base e: is equivalent to Exponential Growth and Decay Models: If k>0, the function models the amount, or size, of a growing entity. If k<0, the function models the amount, or size, of a decaying entity, is the original amount, or size, of the growing/decaying entity at time t=0, A is the amount at time t, and k is a constant representing the growth/decay time. Logistic Growth Models :, a, b, and c are constants with c>0, b>0. Newton s Law of Cooling, temperature T, time t, C is the constant temperature of the surrounding medium, is the initial temperature of object, and k is a negative constant. General Term of a Geometric Sequence: The Sum of the First n terms of a Geometric Sequence: where r is the common ratio The Sum of an Infinite Geometric Series: if then the infinite geometric series is given by, if the infinite series does not have a sum. Value of an Annuity: Interest Company n Times per Year:, P is deposit made at end of each compound period, r is percent annual interest, t is time in years, n is compound periods per year, A is value of annuity. General Term of an Arithmetic Sequence (the nth term): where d is the common difference The Sum of the First n Terms of an Arithmetic Sequence: The Law of Sines: The Law of Cosines: or or Plotting points in the polar coordinate system: is located r units from the pole. If r>0, the point lies on the terminal side of. If r<0, the point lies along the ray opposite the terminal side of. If r=0, the point lies at the pole, regardless of the value of. Equation Conversion from Polar to Rectangular Coordinates: Heron s Formula for the area of a triangle: Multiple Representation of Points: 2 P a g e

3 Area of an Oblique Triangle: Converting a point from Rectangular to Polar Coordinates : 1. Plot the point (x,y) 2. Find r by computing the distance from the origin to (x,y): 3. Find using with the terminal side of passing through (x,y) Representing Vectors in Rectangular Coordinates: Vector v, from (0,0) to (a,b) is represented as a is the horixontal component of vector v, and v is the vertical component of vector v. Linear Combination: ai+bj Magnitude of Vector v: The magnitude of is Vector Components of v: Vector v can be expressed as the sum of two orthogonal vectors, where is parrelle to w and is orthogonal to w. Writing a vector in terms of its Magnitude and Direction: Representing Vectors in Rectangular Coordinates: Vector v with initial point and terminal point is equal to the position vector Adding and Subtracting Vectors in terms of i and j: Let and let then Absolute Value of a Complex Number: Polar Form of a Complex Number: Let this in polar form is where and and and Product of Two Complex Number in Polar Form: Let and then [ [ ]] Quotient of Two Complex Numbers in Polar Form: Let and then [ [ ]] DeMoivre s Theorem: Let then [ ] DeMoivre s Theorem for Finding Complex Roots: ( ) in radians ( ) in degrees. Where k=0, 1, 2,, n-1 Representing Vectors in Rectangular Coordinates: Vector v, with intial point and terminal point is equal to the position vector Finding a Unit Vector w/ Same Direction as given Vector: 3 P a g e

4 PARABOLA ELLIPSE: Standard form of Equations of Ellipses Centered at h,k Center Major Axis Vertices Graph the x-axis (h a, k) (h + a, k) the y-axis (h, k a) (h, k + a) 4 P a g e

5 HYPERBOLA: Standard form of Equations of Hyperbola Centered at h,k Center Major Axis Vertices Graph the x-axis (h a, k) (h + a, k) the y-axis (h, k a) (h, k + a) Standard form of the Equation of a Circle with center and radius r is General Form of the Equation of Circle:, where D,E and F are real numbers Plane Curves and Parametric Equations: Suppose that t is a number in an interval I. A plane curve is the set of ordered pairs (x,y) where The variable t is called a parameter and the equations are called parametric equations for the curve. Eliminating the Parameter: Begin with the parametric equations, solve for t in one of the equations, substitute the expression for t in the other parametric equation. Plane curves can be sketched by eliminating the parameter, t, and graphing the resulting rectangular equation. It is sometimes necessary to change the domain of the rectangular equation to be consistent with the domain for the parametric equation in x. Parametric Equations for the function : A set of parametric equations for the plane curve defined by is and 5 P a g e