1.1 calculator viewing window find roots in your calculator 1.2 functions find domain and range (from a graph) may need to review interval notation

Transcription

1 1.1 calculator viewing window find roots in your calculator 1.2 functions find domain and range (from a graph) may need to review interval notation functions vertical line test function notation evaluate the value of the funct. for a given input (from eq. or graph) 1.3 find x-intercepts (same as zeros) and y-intercepts from an equation/graph 2.1 function properties state intervals of... (remember interval notation) continuity increasing/decreasing/constant 2.5 piecewise-defined functions Graph (when in doubt, graph all of the pieces in your calc., then go from there) Use a graph to write the function definition 3.1 imaginary/complex numbers simplify radicals with i simplify different powers of i multiply/divide radicals with negative radicands (insides) remember to simplify each with i first, THEN mult. or divide Add, subtract, multiply, divide complex #s Complex conjugates (use for division) 3.2 quadratic functions Degree of a function (quadratic degree = 2) Find vertex: (h, k) use then k f (h) (for ) Find the y-intercept vertex is either the min or max POINT h = WHERE/WHEN the min/max value occurs k = the min/max VALUE of the function 3.3 solving quadratic equations and inequalities if you have x2 and x -> get everything on one side (0 on the other) factor, set each factor = to 0, then solve OR use quadratic formula if you only have x2 or (... x... ) 2, then isolate that, take the sq. root of both sides (remember ±) Solve quadratic inequalities Solve associated eq. to find boundaries => test intervals (NOT the zeros) OR get everything on one side, graph in your calc., then look above/below the x-axis

2 3.5 nth degree polynomial functions domain is always ( ) always continuous Turning points # of turning points n - 1 Zeros (x-intercepts) # of zeros n find with your calculator End behavior (4 possibilities - if you get mixed up, think of x2, -x2, x3, -x3) find min/max points with your calculator be careful with your viewing window! 3.6 polynomial properties and factors Synthetic division shortcut k is a zero of a funct. if and only if (x - k) is a factor of that funct. (factor theorem) if a poly. function P(x) is divided by (x - k), the remainder = P(k) (remainder thrm.) => if remainder = 0, k is a zero => (x - k) is a factor given a zero, use division to get to the remaining factors 3.7 zeros of polynomials multiplicity: odd -> crosses x-axis, even -> touches x-axis use with zeros and end behavior to graph a rough sketch rational zeros theorem find possibilities use synthetic division to check 3.8 solving polynomial equations and inequalities use your calculator to graph and find zeros/check intervals (remember 3.3) 4.1 rational functions introduction find equations of asymptotes from a graph find domain and range from a graph 4.2 Graphing rational functions Find vertical asymptote(s) Find horizontal/oblique asymptote (or neither) Find intercepts (shortcut for x-int?) Does the function cross the horizontal/oblique asymptote? Find some other points => graph Use multiplicities of zeros (x-intercepts) and multiplicities of numerator factors (asymptotes) Remember to reduce your fraction to lowest terms (watch out for holes!) 4.3 Rational Equations and Inequalities find domain set each denominator = 0, and solve solve rational equations multiply both sides by least common denominator remember to throw out answers that aren t in domain solve rational inequalities get everything in one fraction on one side, 0 on the other side set numerator = 0, and solve set denominator = 0, and solve

3 these are the boundaries for your intervals => check any # BUT those or just use calculator get every term on 1 side (no need for just 1 fraction), 0 on other graph in calculator when do you look above the x-axis? When do you look below? 4.4 Power and Root Functions find domain of root functions (even root? odd root?) simplify expressions with roots/fractional exponents use exponents to rewrite expressions with roots/radicals 4.5 Root Equations and Inequalities solve root equations isolate one root (or one quantity with a rational exponent) on one side of the equation, then raise both sides to a power that will eliminate the root; (repeat if you have another root, then) solve the equation, and CHECK all answers solve equations with different rational or negative exponents (does it look quadratic?) -> u substitution solve root inequalities (same idea as in 3.3, except you need to check the domain) solve the associated equation -> boundary/boundaries for your intervals OR get everything on one side, and graph in your calculator when do you look above the x-axis? When do you look below? 5.1 Inverse Functions one-to-one functions check f(a)=f(b) a=b OR check graph with horizontal line test check if two functions are inverses of each other use function composition given a function, find its inverse 5.2 Exponential Functions graph an exponential function (remember basic shape -> shift and/or reflect) y-intercept solve Type 1 equations try to get the same base on both sides compound interest/continually compounded interest 5.3 Logarithms write exponential equation in log form write log equation in exponential form solve log equations switch to exp. form, then try to solve (but remember, base > 0) common log (no base indicated) -> base = 10 natural log (ln) -> base = e product, quotient, and power rules => combine or break down logs with the same base change-of-base rule (to evaluate a log in calculator) 5.4 Log Functions graph log functions (remember basic shape -> shift and/or reflect) find domain (argument > 0)

4 5.5 Exponential and Log Equations and Inequalities use the fact:. solve Type 2 equations (can t get same base on both sides) - > take log or ln of both sides, use the power rule, then solve solve more complicated log equations - use power/product/quot. rule to get one log on each side with the same base, then solve 5.6 Modeling with Exponential and Log Functions Applications exponential growth/decay, sound intensity, 6.1 Circles and Parabolas Draw the graph of a circle given the center-radius form of its equation. Draw the graph of a parabola given its equation. (What is the vertex? How do you find c?) c = the distance between the vertex and the focus AND between the vertex and the directrix 6.2 Ellipses and Hyperbolas Draw the graph of an ellipse given its equation. (What is the center? a > b => Which is the major axis? What are the endpoints?) Draw the graph of a hyperbola given its equation. (a 2 under the positive variable => Which is the major axis? Draw the box to find your asymptotes.) 7.1 Systems of Equations Solve a system using substitution, elimination, or graphing (any intersection point is a solution). dependent / consistent / inconsistent 7.3 Solving Linear Systems with Matrices Write a system of equations as an augmented matrix. Write an augmented matrix as a system of equations. Perform row transformations on a matrix. row echelon form

5 Solve a system (of 3 equations with 3 variables) with an augmented matrix. In TI83/84: Enter the matrix into your calculator, then use rref. Analytically: Use row transformations to convert the matrix to row echelon form, then use back substitution to solve for all of the variables. REMEMBER when you solve a system this way, what does an inconsistent system s matrix look like? What does a dependent system s matrix look like? 7.4 Matrix Properties and Operations dimension square / row / column matrices equality of matrices Add or subtract two matrices. (They need to have the same dimension.) Multiply one matrix by a scalar (a constant). Multiply two matrices together. (What dimensions do you need?) 7.5 Determinants and Cramer s Rule shortcut for determinant of a 2x2 matrix Find the minor of an element. Find the cofactor of an element. Use cofactors to find the determinant of an nxn matrix. Use Cramer s Rule to solve a system of equations sequences and series Given an equation, write the first few terms of that sequence Finite vs. infinite Recursive definitions Summation notation Evaluate a series (add up the terms in the sequence)

6 11.2 Arithmetic Sequences and Series How to recognize Find the common difference Find a specific term, given a 1 and d Find a 1, given some other term and d (or just 2 other terms) Write a general formula for a sequence (a n = ) Use sum formulas to evaluate series OR to find specific terms Remember (find a 1, and go from there using a sum formula) 11.3 Geometric Sequences and Series How to recognize Find the common ratio Use formulas to find a sum, a 1, or any other term Convergent (-1 < r < 1) vs. divergent sums Remember (use sum formula) Find the future value of an annuity (remember what each variable represents) 11.4 Binomial Expansion Pascal s Triangle For the exponent on x decreases from n to 0; the exponent on y increases from 0 to n Factorials Use combinations to find coefficients (n C r in the calculator) Memorize binomial theorem: Any r th term: YOU ARE RESPONSIBLE FOR KNOWING ANY FORMULA THAT YOU MAY NEED. AREN T YOU GLAD YOU CAN BRING A CHEAT SHEET? DON T FORGET THAT CHEAT SHEET.