MATH 110: College Algebra Instructor: Reyes Semester Review Packet Remarks: This semester we have made a very detailed study of four classes of functions: Polynomial functions Linear Quadratic Higher degree polynomial functions Rational functions Exponential functions Logarithms This packet contains a list of the important skills and ideas you should possess as a result of our study of these classes of functions. 1. Functions (In General) You should be able to apply the following skills to all types of functions, including: polynomials, rational functions, exponential functions, and logarithms. Graph equations by plotting points. Identify x and y intercepts symbolically, numerically, and graphically. Identify a function symbolically, numerically, and graphically. Graph functions. Solve linear and non-linear inequalities. Use function notation (i.e. f(x)) to evaluate functions AND/OR solve equations involving functions symbolically, numerically, and graphically. Recognize and distinguish between the different types of functions (linear, quadratic, higher degree polynomials, rational, exponential, and logarithmic) symbolically, numerically, and/or graphically. State the domain AND range of a function represented symbolically, numerically, and/or graphically using interval and/or set builder notation. State the intervals over which a function is positive or negative AND increasing, decreasing, or constant. Identify concavity from a graph. Identify local extrema. 1
Express the end behavior of a function using the appropriate notation (i.e., y?). as x Perform addition, subtraction, multiplication, division, and composition of functions symbolically, numerically, and graphically. Perform AND recognize horizontal and vertical shifts, stretches and compressions, and reflections symbolically, numerically, and graphically. Identify when a function f(x) is one-to-one symbolically, numerically, and/or graphically. Expressing the inverse of a function symbolically, numerically, and graphically. State the domain and range of an inverse function and recognize the relationship between the domain and range of f(x). Verifying symbolically if two functions are inverses of each other. Determine where functions are continuous/discontinuous. 2. Linear Functions Slope-Intercept Form : f(x) = mx + b Point-Slope Form : y y 1 = m(x x 1 ) slope : m y-intercept : (0,b) point : (x 1, y 1 ) Recognize linear functions symbolically, numerically, and graphically. Solve linear equations symbolically, numerically, and graphically. Write linear functions in slope-intercept form AND point slope form. Express linear functions symbolically from a table AND/OR graph. Find the slope of a linear function and if applicable give a practical interpretation. Determine when lines are parallel and perpendicular. Find equations for parallel and perpendicular lines. Determine the domain, range, and all intercepts of a linear function from a table, graph, and equation. Recognize when linear functions are invertible AND find their inverses. 3. Quadratic Functions Standard Form : f(x) = ax 2 + bx + c Factored Form : f(x) = a(x p)(x q) Vertex Form : f(x) = a(x h) 2 + k 2
leading coefficient : a, (a 0) y-intercept : (0, c) zeros : p, q vertex : (h, k) Identify quadratics from equations and graphs. Graph quadratic functions from equations and tables. Factor quadratic functions. Solve quadratic equations by factoring, completing the square, and/or using the quadratic formula. Switch between writing quadratic functions in factored form, vertex, and/or standard form. Determine the number and types of zeros of a quadratic function using the discriminant. Determine a symbolic representation for a quadratic function from a table AND/OR graph. Identify the vertex and intercepts of a quadratic function symbolically, numerically, and graphically. State the domain and range of a quadratic function expressed symbolically and/or graphically. Write the factored form of a quadratic function given its zeros and a point. 4. Polynomial Functions of Higher Degree Standard Form : f(x) = a n x n + a n 1 x n 1 + + a 1 x + a 0 Factored Form : f(x) = a n (x c 1 )(x c 2 ) (x c n ) Recognize polynomials from equations. Graph polynomial equations. Leading Coefficient : a n, where a n 0 Degree : n y-intercept : (0, a 0 ) zeros : c 1, c 2,..., c n Write polynomial functions in standard form. Write polynomial functions in factored form. Add, subtract, multiply, and divide polynomials (using long division and/or synthetic division). 3
Solve polynomial equations by factoring. Recognize polynomial functions from graphs. Understand the relationship between the degree, leading coefficient, leading term, and graphs of polynomials. Determine the end behavior of a polynomial function using the degree and leading coefficient. Find the zeros of a polynomial function from a graph. Recognize multiple roots from equations and from graphs. Factor polynomial functions using zeros and polynomial long division. Understand the relationship between zeros, factors, x-intercepts, and solutions to polynomial equations. Determine the equation of a polynomial function from a graph using zeros and a given point. 5. Rational Functions Details: Standard Form : f(x) = p(x) where q(x) 0. q(x) Divided Form : f(x) = h(x) + r(x) where q(x) 0. q(x) Polynomials : p(x), q(x), h(x), and r(x) Remainder after division : r(x) Horizontal/slant asymptote : h(x) Simplify rational expressions by factoring. Add, subtract, multiply, and divide rational functions. Solve equations involving rational expressions. Recognize rational functions from equations and graphs. Determine the domain of rational functions from equations and graphs. Determine the range of rational functions graph graphs. Find the intercepts of rational functions. Determine equations of vertical asymptotes. Express the behavior near vertical asymptotes using appropriate notation. Recognize gaps and/or holes in rational functions from their symbolic representation. Rewrite rational functions using polynomial and synthetic division. Determine the equations of horizontal & slant asymptotes. Express the end behavior of rational functions using the appropriate notation. 4
6. Exponential Functions General Exponential Function : f(x) = Ca x Continuous Growth/Decay Exponential Function : f(x) = Ce kx Base/Growth/Decay Factor : a where a > 0 and a 1 y-intercept : (0, C) Continuous Rate of Growth/Decay (%) : k Understand and use properties of exponents to simplify expressions. Understand and use rational exponents. Solve exponential equations. Recognize exponential functions from graphs and equations. Evaluate exponential functions. Determine the domain, range, intercepts, and end behavior of exponential functions. Determine the equation of the horizontal asymptote. Graph the inverse of an exponential function. Recognize exponential functions from a table of values. Rewrite exponential functions using a base of e (see continuous exponential model for details). 7. Logarithmic Functions Graph logarithmic functions. Logarithm Base a : f(x) = log a (x) base : a where a > 0 and a 1 Use properties of logarithms appropriately to combine, expand, and/or solve equations. Use logarithms to solve exponential equations. Solve logarithmic equations. Recognize logarithmic functions from graphs and equations. Determine the domain, range, intercepts, and asymptotes of logarithmic functions. Understand the relationship between logarithmic functions and exponential functions. Evaluate logarithmic functions using their inverse relationship to exponential functions. 5
8. Sequences & Series This part of the review packet contains information about sequences and series. After our study of sequences and series, you should be able to apply all the following skills and ideas. Express the terms of a sequence and/or series using an closed formula or recursive formula Recognize number patterns in sequences and/or series AND express those patterns using an closed formula and/or recursive formula. Distinguish between an arithmetic sequence and series AND a geometric sequence and series. Express the nth term of an arithmetic and/or geometric sequence or series using an explicit formula. Recognize and distinguish infinite and finite sequences and/or series. Find the number of terms in a finite sequence and/or series. Find the sum of the first n terms in a geometric and/or arithmetic series. Determine the sum of an infinite geometric series with a common ratio, r whose absolute value is less than 1 AND to distinguish this sum from a finite geometric series. Use sigma notation to write out completely a series AND to express a series that is completely written out using sigma notation. Apply summation properties to determine the actual value of a sum involving n k 2. k=1 n k=1 k and 6