Useful Fact Sheet Final Exam Interval, Set Builder Notation (a,b) = {x a<x<b} [a,b) = {x a x<b} (a,b] = {x a<x b} [a,b] = {x a x b} (-,a) = {x x<a} [a, ) = {x x a} Rational Exponents Symmetric w/ Respect to y-axis: for every point (x,y) on the graph, the point (-x,y) is also on the graph. Symmetric w/ respect to x-axis: for every point (x,y) on the graph, the point (x,-y) is also on the graph. Symmetric w/ respect to origin: if for every point (x,y) on the graph, the point (-x,-y) is also on the graph Even If the graph is symmetric with respect to the y-axis. Odd If the graph is symmetric with respect to the origin. Difference quotient or average rate of change Algebra of Functions: ; provided g(x) 0 The composition function: ( ), x is in the domain of g and g(x) is in the domain of f. Definition of the Inverse function: ( ) and for every x in the domain of and ( ) for every x in the domain of Obtaining an inverse relation: we interchange the first and second coordinates the relation obtained is an inverse of the relation Obtaining a formula for an inverse: Step 1: Replace f(x) with y Step 2: Interchange x and y Step 3: Solve for y Step 4: Replace y with f(x) Horizontal Line Test If it is possible for a horizontal line to intersect the graph of a function more than once, then the function is not 1-1 and its inverse is not a function. Vertical line test which states that if any vertical line intersects the graph of a relation in more than one point, then the relation graphed is not a function. 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, is the horizontal asymptote of the graph of f. 2. if, the line, or, is the horizontal asymptote of the graph of f. 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. 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, Converting Between Exponential and Logarithmic Equation The Product Rule The Power Rule The Quotient Rule The Change of Base Property: Base Exponent Property: For any b>0 and b 1 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. 2 P a g e
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. Matrix Addition and Subtraction: matrices of the same order Matrix Multiplication: [ ] [ ] [ ] [ ] [ ] Determinate of a 2x2 Matrix : (second order determinate) Determinate of a 3x3 Matrix : (third order determinate) 3 P a g e
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
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 Factorial Notation: Summation Notation: General Term of a Geometric Sequence: where r is the common ratio The Sum of the First n terms of a Geometric Sequence: 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. 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. 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: 5 P a g e