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ALGEBRA 2 Essential Learner Outcomes The Learner Will: Foundations for Functions • • • • • • • • • • • • • simplify algebraic expressions using order of operations evaluate algebraic expressions solve linear equations − one and two step − combining like terms − variables on both sides − distributive property simplify and solve linear equations containing − fractions (multiply by LCD) − decimals (multiply by power of 10) solve one-variable inequalities solve absolute value equations Identify domain and range of a function − From a graph − From a verbal situation − From a table Identify domain and range of a function − Set notation − Interval notation Determine reasonable domain and range values from a given situation − Continuous situations − Discrete situations − From a graph − From a verbal description − From a table predict the effects of parameter changes on the graphs of given functions (Move the Monster) − vertical shifts − horizontal shifts − vertical expansion(stretch) − vertical compression − reflection across the x- axis − reflection across the y-axis − horizontal expansion − horizontal compression identify changes in domain and range and compare to parameter change and ordered pair values record/describe parameter changes of absolute value function using function notation Use symbolic representation to describe transformations on the absolute value function. Linear Functions, Equations, and Inequalities • • • • • • write and solve linear equations and inequalities from problem situations Identify domain and range of a linear function − From a graph − From a verbal situation − From a table − From an equation graph two-variable inequalities − dotted line − solid line − test points transform equations from one form to another and graph identify and sketch the graph of the linear parent function describe the effects of parameter changes on the graph of the linear parent function − y = mx + b • • • • • • • • − changes in m − changes in b − connect changes in m and b to changes in a problem situation. make connections between the point-slope formula of a linear equation and the horizontal and vertical translations of the parent function write the equation of a line given − two points in a table (or as ordered pairs) − two points on a graph − a point and a slope − a slope and y-intercept − x- and y- intercepts − graph of a line translate equations from standard form to slope-intercept form write equations of parallel lines write equations of perpendicular lines write equation of a line parallel to a given line through a given point graph the equation of a line given − two points in a table (or as ordered pairs) − a point and a slope − a slope and y-intercept − from standard form using x- and y-intercepts write linear equations in various forms − slope-intercept form − point-lsope form − standard form Higher Order Systems • • • • • • • solve 2 X 2 systems − linear combination − substitution − graphing (with a graphing calculator) intersecting lines parallel lines lines that coincide − table (graphing calculator) − matrices define a matrix and use a matrix to represent data inverse matrices by hand (2X2 only) inverse matrices with calculator use operations of matrices to solve problems systems of inequalities − graphing calculator − graph by hand − Test a point Solve 3 X 3 systems − Matrices (calculator) Perform matrix multiplication to show multiplication is not commutative − By hand (demo only) − With technology determine what a solution to a system of equations/inequalities means in relationship to the problem determine if the solution to a system of equations/inequalities is reasonable for given contexts connect algebraic solutions to graphical and tabular solutions Quadratics • • • sketch the graph of the quadratic parent function identify the graph of the quadratic parent function recognize the attributes of the quadratic parent function − goes through origin • • • • • • • • • • • • • • • • • • • • • • • − symmetric about line x = 0 − table of values has a 2nd difference of 2 determine the axis of symmetry from the graph of a quadratic function graph a quadratic function using x-intercepts use transformations to sketch y = a( x − h )2 + k from the parent function − vertical shift − horizontal shift − vertical expansion (stretch) − vertical n across x- axis, y-axis or y = x − Identify the vertical compression Predict vertex of the graph from y = a( x − h )2 + k predict changes to the graph when a, h, or k are changed connect the effects of changing a, h, or k to a problem situation simplify polynomials − with concrete models (algebra tiles) − algebraically − verify on graphing calculator multiply polynomials − with concrete models − algebraically − verify on graphing calculator factor polynomial expressions − GCF − Trinomial factoring (include “a” not equal to 1) − difference of two squares − factor by grouping − sums and differences of cubes (optional) − perfect square trinomials express the solution of a quadratic equation in terms of a complex number connect the solution of a quadratic equation to the graph of the function simplify complex numbers −4 = 2i calculate and use the discriminant in determining types of solutions of quadratic equations − real − imaginary − rational − irrational solve quadratic equations using the quadratic formula use completing the square to change a quadratic function from standard form to vertex form identify y-intercept from standard form of a quadratic function choose the appropriate form of a quadratic function based on the situation (standard or vertex form) determine domain and range of a quadratic function when − given a graph − given a table − given an equation − given a situation that can be modeled with a quadratic function interpret the solution of a quadratic equation or inequality in context of the situation determine if the solution to a quadratic equation or inequality is reasonable in context of the situation compare the domain and range of a quadratic function and the domain and range of a situation that can be modeled by the same quadratic function solve quadratic equations in problem situations and in purely mathematical situations by − factoring − quadratic formula − graph − table solve quadratic inequalities in problem situations and in purely mathematical situations from a − graph (graphing calculator) − table (graphing calculator) • • • • • • • • create the other representations of quadratic functions when given one of the following: − verbal description − equation − graph − table determine which form of a quadratic function is appropriate when solving problems. write a quadratic equation or inequality to solve application problems use a quadratic function to answer questions and make predictions in a given situation determine if a situation can be modeled by a quadratic function Interpret the meaning of the maximum/minimum values from a graph or table to the situation write a quadratic function when given − two roots o write in factored form o use sum and product of roots − graph − three points connect the solution to a quadratic equation to − x-intercepts of the function from a table and a graph − roots of the equation − zeroes of the function Relations and Functions • • • • • • • • • • • • • Illustrate a function as − A table − A set of ordered pairs − A mapping diagram − An equation − A graph Determine whether relations are functions utilizing various methods such as − Investigating the table − Definition of function − Vertical line test Given a situation, choose a reasonable graph Given a graph, choose a situation Given a graph, create a situation connect equation notation with function notation recognize that all functions can be denoted in many forms, such as y = d = y1 = f(x) = determine which form is the most appropriate for a given situation recognize and apply function notation find the value of a function (f(3) = 5) interpret a specific function value from function notation as an ordered pair [ie: f(3) = 5 represents the ordered pair (3 , 5)] identify the graph of a parent function Use function notation to represent equations involving parent functions Square Roots and Inverses • • • • identify the graph of the square root parent function sketch the graph of the square root parent function predict parameter changes on the graph of square root functions and verify on graphing calculator − vertical shift − horizontal shift − vertical expansion (stretch) − vertical compression − reflection across x- axis, y-axis or y = x sketch the graph of square root functions with the following parameter changes − vertical shift − horizontal shift • • • • • • • • • • • • • • • • • − vertical expansion (stretch) − vertical compression − reflection across x- axis, y-axis or y = x determine the equation of square root functions from a graph express limitations on the domain and range of square root functions with the following parameter changes − vertical shift − horizontal shift − vertical expansion (stretch) − vertical compression − reflection across x- axis, y-axis or y = x develop inverse relations and functions − from a situation − algebraically − graphically (calculator) − tables (calculator) compare and contrast the domain and range of a relation or function and its inverse graph a function and its inverse − using a table of values − on calculator − by hand use composition of functions to verify an inverse discover graph of square root functions are inverse of parabolas with restrictions Explore and formalize the relationship between the parent functions y = x2 and y = x . create the other representations of square root functions when given one of the following: − verbal description − equation − graph − table determine which form of a square root function is appropriate when solving problems. determine domain and range of a square root function when − given a graph − given a table − given a situation that can be modeled with a square root function interpret the solution of a radical equation or inequality in context of the situation determine if the solution to a radical equation or inequality is reasonable in context of the situation. solve radical equations − algebraically − from a graph (calculator) − from a table (calculator) − for a given situation and from a purely mathematical situation solve one-variable square root inequalities from a graph (calculator) and set appropriate windows interpret graph in relationship to solution from a table (calculator) interpret table values in relationship to solution graph solutions to a square root inequality on a number line use the graphing calculator to represent the solution to a square root inequality − y1 = ( x + 3) ≤ 3 − interpret solution from graph Exponents • • use laws of exponents to simplify expressions, including negative and rational exponents − product rule − quotient rule − power to a power rule − power of the product rule − power of the quotient rule use a graphing calculator to verify answers 1 • convert an exponential expression to a radical expression (ie: 3 2 = 3 ) and verify on graphing calculator − number bases − variable bases • convert a radical expression to an exponential expression calculator, simplify expressions containing complex numbers add, subtract, multiply, and divide complex numbers simplify powers of i 3 • • • (ie: 4 23 = 2 4 ) and verify on graphing Exponential and Logarithmic Functions • • • identify the graph of exponential and logarithmic parent function sketch the graph of the exponential and logarithmic parent function define the inverse of y = 2x as a logarithmic function y = log2 x • • • • compare the domain, range, and asymptotes of y = 2x to y = log2 x connect exponential notation and logarithmic notation define the inverse of y = 2 x as a logarithmic function y = log2 x . predict and describe the effect of parameter changes in the graph of the parent exponential function use graphing calculators to check predictions sketch the graph determine if the graph is increasing or decreasing discover changes to the domain, range, and asymptote predict and describe the effect of parameter changes in the graph of natural and common logarithmic functions − use graphing calculators to check predictions − sketch the graph − discover changes to the domain, range, and asymptote write equations of asymptotes determine domain and range of a exponential and logarithmic function when − given a graph − given a table − given a situation that can be modeled with an exponential or logarithmic function − write in set or interval notation write exponential equations in logarithmic form write logarithmic equations in exponential form solve exponential equations − by reducing both sides to a common base − using logarithms − from a graph (calculator) − from a table (calculator) − from a given situation simplify logarithmic expressions using logarithmic properties find common and natural logarithms and antilogarithms on the graphing calculator solve logarithmic equations − with like bases (by rewriting them as exponential equations and finding a common base) − from a graph (calculator) − from a table (calculator) − from a given situation write and solve an exponential equation from a given situation, such as bacterial growth and decay, population growth and decay, and finances − determine the independent and dependent variables of the situation − select an appropriate method for solving the equation (algebraically, graphically, tabular) − solve the equation and relate the solution to the situation write and solve an exponential inequality from a given situation such as bacterial growth and decay, population growth and decay, and finances − select an appropriate method for solving the inequality (graphical or tabular) − solve the inequality and relate the solution to the situation • • • • • • • • • • • Rational Functions • sketch the graphs of rational functions using transformations and long division. • Use tables and graphs to compare the functions y = • predict and describe the effect of parameter changes in the graph of parent rational function, using graphing calculators to check Use tables and graphs to identify discontinuities Using tables and graphs, describe the domain, range, horizontal asymptote, and vertical asymptote of the graphs of rational functions and how they change given certain parameter changes. Write equations of horizontal and vertical asymptotes. Using tables and graphs, describe end behavior and behavior near asymptotes Write the equation of a rational function from the graph. • • • • • • • • • • • • a a and y = 2 . x x simplify rational expressions involving addition, subtraction, multiplication, and division solve rational equations − algebraically − from a graph (calculator) represent a real-world situation using a table, appropriate symbolic representation, and a graph. Use transformations to fit a rational function to model a data set. from a table (calculator) solve rational inequalities − from a graph (calculator) − from a table (calculator) y list attributes of proportional relationships, including constant ratio of , graph passing through the origin, x and equation of the form y = kx make predictions in problem situations involving direct variation and indirect variation Conic Sections • • • • • describe each conic section as the set of points resulting from a double napped cone being intersected by a plane at different angles to the edge of the cone. graph each conic section given the equation in graphing form (from a transformations perspective). define and model a conic from its geometric description. correctly identify a conic section from a given equation − from the standard form of the equation − from the graphing form of the equation complete the square to convert a conic equation from standard form to graphing form. Trigonometry • Define sine, cosine and tangent in terms of x, y and r • Know the sine, cosine and tangent values of special angles (multiples • • • • • π π π π , , , ) 6 4 3 2 Find the values of cotangent, secant, and cosecant from sine, cosine, and tangent Graph all six trig functions and identify their period, amplitude, domain, range and zeros Model real-life data using sine and cosine functions Solve right triangles using sin, cos, and tan Use Law of Sines and Law of Cosines
