MATH-40 Review Concepts (Haugen) Unit : Equations, Inequalities, Functions, and Graphs Rational Expressions Determine the domain of a rational expression Simplify rational expressions -factor and then cancel common factors Arithmetic operations with rational expressions -add, subtract, multiply, and divide rational expressions Complex rational expressions Equations Polynomial, radical, and absolute value equations Equations with rational exponents Equations that are quadratic in form Linear Inequalities and Absolute Value Inequalities Ways we can express the solution set of an inequality:. Simple or compound inequalities. Set-builder notation 3. Geometrically (using a number line) 4. Interval notation Basics of Functions Relation (a correspondence between two sets) Domain and range of a relation Function (a special type of relation) Vertical Line Test More on Functions Identify the open intervals on which a function is increasing, decreasing, or constant Determine the relative extrema of a function Symmetry tests Piecewise-defined functions Linear Functions and Slope Three ways to express the equation of a line:. Slope-Intercept Form y mx b. Point-Slope Form y y mx x 3. General Form Ax By C Parallel Lines (equal slopes) Perpendicular Lines (opposite reciprocal slopes) Average Rate of Change of a Function from x to x : f x f x x x Transformations of Functions Rigid Transformations: horizontal, vertical, and reflection Non-rigid Transformations: horizontal and vertical stretching/shrinking
Combinations of Functions; Composite Functions Sum, Difference, Product, and Quotient Functions Function Composition Inverse Functions One-to-One Functions and the Horizontal Line Test Finding the inverse of a function (Switch-and-Solve Approach) Unit : Polynomial and Rational Functions Quadratic Functions Standard Form: f x ax h k General Form: f x ax bx c Identify the vertex, axis of symmetry, x-intercept(s), and the y-intercept of a parabola Polynomial Functions p x a x a x a x... a x a x a n n n General Form: n n n 0 End behavior of a polynomial function Locating the zeros of a polynomial function Determine the behavior of a polynomial function at its zeros Intermediate Value Theorem Dividing Polynomials Polynomial long division Synthetic division Zeros of Polynomial Functions Rational Zero Theorem Conjugate Pairs Theorem Descartes Rule of Signs Graphs of Rational Functions Asymptotic behavior of rational functions Vertical, Horizontal, and Slant (or Oblique) Asymptotes Polynomial and Rational Inequalities Boundary points, test intervals, test values, endpoint analysis, and conclusion
Unit 3: Exponential and Logarithmic Functions Exponential Functions Standard Form: f x b x, b 0, b Properties of exponential functions Compound Interest Formulas: r Finite number of compounding periods: A P n Infinite number of compounding periods (continuous compounding): nt A Pe rt Logarithmic Functions f x log x, b 0, b Standard Form: b Evaluating logarithms Converting from logarithmic form to exponential form and vice versa Properties of logarithms Product Rule (the log of the product = the sum of the logs) Quotient Rule (the log of the quotient = the difference of the logs) Power Rule (special case of the Product Rule) loga M Change of Base Formula: logb M log b a Exponential and Logarithmic Equations Type exponential equations Express each side using the same base and then use - property of exponential functions Type exponential equations Take the logarithm of each side and then apply the Power Rule Type logarithmic equations Convert to exponential form and then solve Type logarithmic equations Take advantage of the Product, Quotient, and/or Power Rule Watch for extraneous solutions Exponential Growth and Decay; Modeling Data kt Equation: A A0e We have exponential growth when k 0 ; decay when k 0 A is the original amount or size of the growing/decaying entity 0 Logistic Growth Growth under restricted conditions c Equation: A bt ae
Unit 4: Conic Sections Distance and Midpoint Formulas Distance between the points Circles x y and,, d x x y y Midpoint of the line segment joining Midpoint x x, y y x y : x y and,, x y : x h y k r Standard form of the equation of a circle: General form of the equation of a circle: x y Dx Ey F 0 Convert from general form to standard form by completing the square Ellipses Standard form of an ellipse centered at the origin (assume ): x y horizontal major axis x y vertical major axis b a Standard form of an ellipse centered at the point hk, : x h y k a horizontal major axis b x h y k vertical major axis b a Identify center, vertices, foci, and the endpoints of the minor axis Use c a b to help locate the foci Ax By Dx Ey F 0, A and B 0 General form of the equation of an ellipse: Convert to standard form by completing the square
Hyperbolas Standard form of a hyperbola centered at the origin: x y horizontal transverse axis y x vertical transverse axis Standard form of a hyperbola centered at the point hk, : x h y k a horizontal transverse axis b y k x h vertical transverse axis Identify center, vertices, foci, and the endpoints of the conjugate axis Use c a b to help locate the foci Fundamental Rectangle helps us sketch each branch Equations of the asymptotes Ax By Dx Ey F 0, A and B 0 General form of the equation of a hyperbola: Convert from general form to standard form by completing the square Parabolas Standard form of a parabola whose vertex is located at the origin: x 4py opens up or down y 4px opens left or right Focal length, p, is the directed distance from the vertex to the focus of the parabola Standard form of a parabola with vertex hk, : x h 4p y k y k 4px h opens up or down opens left or right Identify the vertex, focus, and directrix of a parabola Latus rectum helps us sketch parabolas Length of a parabola s latus rectum = 4 p General form of the equation of a parabola: Ax By Dx Ey F 0, A or B 0 Convert to standard form by completing the square
Unit 5: Systems of Equations/Inequalities and Matrices Systems of Linear Equations in Two Variables A solution to a system must satisfy all equations simultaneously Solve by graphing, substitution, or elimination Recognize when a system has an infinite number of solutions Recognize when a system has no solutions Systems of Linear Equations in Three Variables To solve a system of three equations with three unknowns:. Pick two equations and eliminate an unknown. Pick another two equations and eliminate the same unknown from step 3. Solve the x system formed using the equations from steps and Recognize when a system has an infinite number of solutions Recognize when a system has no solutions Systems of Nonlinear Equations in Two Variables We can use the elimination method is some cases; otherwise, substitution must be used Watch for extraneous solutions It helps to sketch each equation in the system Systems of Inequalities For each inequality in the system:. Sketch the boundary line (dashed or solid depending on the inequality). Pick a test point not on the boundary line 3. Shade on the appropriate side of the boundary line The solution to the system is given by the overlapping shaded regions (assuming such an overlap exists) Matrix Solutions to Linear Systems Augmented Matrix Elementary Row Operations Row-Echelon Form We convert the augmented matrix to row-echelon form using Gaussian elimination Reduced Row-Echelon Form Gauss-Jordan elimination Recognize when a system has an infinite number of solutions Recognize when a system has no solutions Matrix Operations Matrix addition and subtraction Scalar multiplication Matrix multiplication