ALGEBRA II WITH TRIGONOMETRY COURSE OUTLINE SPRING 2009. MR. YATES Vocabulary Unit 1: Polynomials Scientific Notation Exponent Base Polynomial Degree (of a polynomial) Constant Linear Quadratic Cubic Monomial Binomial Trinomial Term Leading Term Leading Coefficient Monic Standard Form Factored Form Zero Factor Property End Behavior Zeros Distributive Property Like Terms Unit 2: Quadratics Factors Distributive Property FOIL Factoring Quadratic Monic Binomial Trinomial Parabola Vertex Zero Root x-intercept Axis of Symmetry Standard Form Vertex Form Factored Form Quadratic Formula Like Terms Velocity Acceleration Regression Pendulum Period Unit 3: Multidimensional Linearity Matrix Inverse Scalar Identity Dimensions (of a Cramer s Rule matrix) Matrix Equation System of Linear Linear Regression Equations Point of intersection Determinant Linear Programming Objective Function Feasible Area Vertex Optimize Maximize Minimize
Unit 4: Trigonometry Right Triangle Pythagorean Theorem Sine Cosine Tangent Ratio Trig Inverses Law of Sines Law of Cosines Unit Circle Radius Angle Standard Position Initial Side Terminal Side Degrees Radians Cartesian Coordinates (x,y) Slope Periodic Function Cycle Domain Range Maximum Minimum Amplitude Period Zeros Sine Function Cosine Function Tangent Function Asymptote Unit 5: Exponentials Exponent Base Result Coefficient Exponential Function Exponential Growth Exponential Decay Percent Increase / Decrease Initial Value Growth Rate Population Radioactivity Half-life Compound Interest Principal Interest Rate e Logarithm Common Log (base 10) Natural Log (base e) Change of Base Rule Inverse Function
Concepts and Skills Unit 1: Polynomials I. Scientific Notation A. Write a number in scientific notation B. Convert a number from scientific notation back to standard notation II. Exponents A. Apply rule about multiplying numbers (add exponents) B. Apply rule about dividing numbers (subtract exponents) C. Apply rule about raising to a power (multiply exponents) D. Only if the bases are the same III. Polynomial Arithmetic A. Distinguish between a monomial and a polynomial B. Find the degree of a monomial or a polynomial C. Add/Subtract Like Terms (depending on sign) D. To subtract a polynomial, add its opposite (reverse all its signs) E. Multiply by distributing every piece to every other piece IV. Polynomial Analysis A. Graph a polynomial in the coordinate plane B. Identify the zeros of a polynomial in factored form using the Zero Factor Property C. Identify the zeros of a polynomial from its graph D. Evaluate a polynomial at a given value E. Identify end behavior of a polynomial from its equation or graph
Unit 2: Quadratics V. FOIL & Factoring A. When multiplying a binomial times a binomial, you distribute every piece to be multiplied by every other piece B. To simplify, you multiply the Firsts, then the Outsides, then the Insides, then the Lasts C. Factoring is reverse distribution: you can always check your work by using FOIL or the distributive property D. To factor x 2 + bx + c, find numbers that multiply to give c but add to give b VI. Graphing Parabolas A. Identify if a quadratic function s parabola points up or down B. Identify if it is wide, normal, or narrow C. Identify its vertex from a graph D. Identify its vertex if the equation is in vertex form E. Identify its zeros from the graph F. Identify its zeros no matter what form the equation is in (see next category) G. Substitute values for x into a quadratic function H. Fill out a table of values and plot the points in a graph VII. Solving Quadratic Equations A. Identify if an equation is in standard form (ax 2 + bx + c = 0) or vertex form (a(x-h) 2 + k = 0) or factored form (a(x-r1) (x-r2) = 0) B. In factored form, set each piece equal to zero, then solve one piece at a time C. In vertex form, solve by getting the x alone, step-by-step reversing every operation by doing its opposite 2 b ± b 4 a c D. In standard form, use the Quadratic Formula: 2 a Negative b Plus or minus the Square root of b squared Minus four times a times c All over two a, hey! E. Combine like terms to get an equation in standard form VIII. Application A. Various word problems may ask you to evaluate a quadratic function at a value, find its vertex, or find its zeros 1 2 B. Physics applications with gravity use the formula h = gt + v0t + h0, with 2 g = 32 ft/s/s = 9.8m/s/s, v 0 = starting velocity, h 0 = starting height C. Physics applications with pendula use the formula L g 2 T= 2π or L= T, 2 g 4π with L the pendulum s length, T the period in seconds, and the gravitational constant g = 32 ft/s/s = 9.8m/s/s
Unit 3: Multidimensional Linearity IX. Matrices A. State the dimensions of a matrix (e.g. 3x2) B. Add and subtract matrices (if their dimensions are the same) C. Multiply a matrix by a scalar D. Multiply matrices (rows by columns, if inside dimensions match) E. Find the determinant of a 2x2 matrix (ad bc) F. Find the inverse of a 2x2 matrix (by calculator or by hand: 1 1 d b A = ( ) ) det A c a X. Systems of Linear Equations A. Solve by Cramer s Rule (C, Cx, Cy, determinants) B. Solve by Matrix Inversion (A -1 *B) with calculator or by hand C. Solve by researched method (substitution or elimination) D. Graph and show solution as intersection of lines E. Interpret solution in real-world application F. Rearrange linear equations into standard form (Ax + By = C) or into slopeintercept form (y = mx + b) XI. Systems of Linear Inequalities (Linear Programming) A. Solve and graph real inequalities in one variable B. Solve and graph linear inequalities in two variables C. Graph a system of constraints (linear inequalities) D. Find coordinates of vertices of shaded regions E. Evaluate the objective function at vertices F. Pick optimal solution G. Write a real-world situation as constraints and objective function H. Interpret optimal solution in real-world context
Unit 4: Trigonometry XII. Trig of the Right Triangle A. Know the definitions of sine, cosine, and tangent (SOHCAHTOA) B. Find the sine, cosine, or tangent of a given angle, using definitions & a picture C. Find the sine, cosine, or tangent of a given angle, using calculator or trig table D. Set up a proportion to solve for a missing side E. Use inverses to find a missing angle 2 2 2 F. Use the Pythagorean Theorem c = a + b G. Draw a diagram to represent a real-world situation XIII. Trig of Other Triangles sin A sin B sin C A. Law of Sines: = = a b c 2 2 2 B. Law of Cosines: c = a + b 2abcos( C) C. Use the Law of Sines if you have two angles and a side opposite one of them D. Use the Law of Sines if you have two sides and an angle opposite one of them E. Use the Law of Cosines if you have all sides and wish to know an angle (opposite side c) F. Use the Law of Cosines if your have two sides and the angle opposite the side you wish to know XIV. Trig of the Unit Circle A. Measure angles in degrees B. Convert between degrees and radians (conversion factor is π/180 or 180/π) C. Draw an angle in standard position D. Find the coordinates of the point where the terminal side meets the unit circle x = cos θ y = sin θ E. Find the slope of the terminal side (tan θ) F. Use the unit circle to help you with solving trig equations XV. Periodic Functions A. Sketch the graph of a periodic function B. Identify domain (possible x-values) & range (possible y-values) of a function C. Identify the zeros, maxima, and minima of a function D. Identify the period (length, in the x-direction, of one cycle) E. Identify the amplitude (half the total height, in the y-direction) F. Use the equation of a sine, cosine or tangent function transformed to determine period and amplitude (1) y = a sin (bθ): a = amplitude, 2π/b = period (2) y = a cos (bθ): a = amplitude, 2π/b = period (3) y = a tan (bθ): amplitude is infinite, π/b = period XVI. Solving Trig Equations A. Solve by doing the opposite of each operation B. Solve by graphing the function and the value it s supposed to equal on the calculator, and finding points of intersection
Unit 5: Exponentials XVII. Basic Exponents A. Identify base, exponent, result in an exponential equation B. Apply rule about multiplying numbers (add exponents) C. Apply rule about dividing numbers (subtract exponents) D. Apply rule about raising to a power (multiply exponents) E. Only if the bases are the same XVIII. Exponential Functions A. y = a b x B. Find values of points on an exponential function C. Identify the domain and range of an exponential function D. Identify the y-intercept (initial value) of an exponential function E. Identify as growth (b>1) or decay (0<b<1) F. Calculate percent increase or decrease G. Use initial value and growth rate to write the equation for an exponential function to model a population, then evaluate to predict population numbers H. Use the exponential decay function to model percent of a radioactive element left after x half-lives XIX. Logarithms A. Identify base, exponent, result in a logarithmic equation B. Convert between log equations and exponential equations C. Use this conversion to evaluate simple logarithmic expressions D. Use rules of logarithms (compare to Section II above) to simplify or expand logarithmic expressions E. Apply change of base formula to evaluate complicated logarithmic expressions F. Use logarithms (as inverse exponentiation) to solve exponential equations