MATH 1314 College Algebra Scott Travis Fall 2014 Review for Exam #2 There are eight sections from Chapters 4 and 5 included in the exam: 4.1, 4.3, 5.1 to 5.6. This review should help you prepare. For each section, I have listed the concepts and skills you need to know. If you have questions about anything, please email me (scott.travis@lonestar.edu) or click on the Ask My Instructor button in an exercise within MyMathLab and I will do my best to help you. 4.1 Linear Functions Know how to graph a linear function, labeling at least two points 4.2 Quadratic Functions Know how to determine whether the graph opens up or down Know how to find the vertex Know how to find the x-intercepts and y-intercepts Know how to find the axis of symmetry Know how to graph a quadratic function, labeling at least five points 5.1 Polynomial Functions Know how to determine whether a function is a polynomial or not Know how to find the degree of a polynomial function Know how to determine the end behavior for the graph of a polynomial function Given a polynomial function that is already factored, know how to find the x-intercepts and the multiplicity of each x-intercept Know how to graph a polynomial function using the end behavior, x-intercepts and multiplicities 5.2 Real Zeros of Polynomial Functions Know how to use synthetic division Using synthetic division, know how to determine whether a number is a real zero of a polynomial function or not Know how to use the degree to determine how many real zeros a polynomial function has Know how to create a list of possible rational zeros for a polynomial function (NOTE: I will give you the Rational Zeros Theorem on the cover sheet of the exam. You do not need to memorize it; you just need to know how to create the list.) Know how to determine how many positive real zeros and how many negative real zeros a polynomial function has (NOTE: I will give you Descartes Rule of Signs on the cover sheet of the exam. You do not need to memorize it; you just need to know how to create the counts. FYI: Remember this is not in the textbook but in the handout I provided in class.) Given a polynomial function, know how to use all of the above tools to find all real zeros of a polynomial function 5.3 Complex Zeros of Polynomial Functions Know what the Conjugate Pairs Theorem states and how that affects the complex zeros of a polynomial function Given a polynomial function and using all of the tools from section 5.2 along with the Conjugate Pairs Theorem, know how to find ALL zeros (real and complex both) of a polynomial function 1 of 11
5.4 and 5.5 Rational Functions Know how to determine the domain, x-intercepts, y-intercepts, vertical asymptotes, and horizontal asymptotes for a rational function Given the domain, x-intercepts, y-intercepts, vertical asymptotes, horizontal asymptotes, and sign chart for a rational function, know how to graph it 5.6 Polynomial and Rational Inequalities Know how to solve a polynomial and rational inequalities Example Problems 1. Graph the function 1 f x x 4. 5 2. Graph the function 4 f x x 1. 3 2 of 11
h x 2x 12x 10. 3. Let 2 a. What direction does the graph open? b. Determine the y-intercept(s), if any. c. Determine the x-intercept(s), if any. d. Determine the vertex. e. Determine the axis of symmetry. f. Sketch the graph of the function. (Label at least 5 points.) j x 3x 6x 9. 4. Let 2 a. What direction does the graph open? b. Determine the y-intercept(s), if any. c. Determine the x-intercept(s), if any. d. Determine the vertex. e. Determine the axis of symmetry. f. Sketch the graph of the function. (Label at least 5 points.) 3 of 11
5. For the function shown below, determine if it is a polynomial function. If it is a polynomial function, then also state the degree. If it is not a polynomial function, then explain why it is not. 5 3 2 3 f x 7x 1.9 x 6.9x 12 6. For the function shown below, determine if it is a polynomial function. If it is a polynomial function, then also state the degree. If it is not a polynomial function, then explain why it is not. 2 8 5 2 f x 3 x 9x 1.4x g x 3x 9x 30x 7. Sketch the graph of 3 2 Clearly label all x-intercepts. x 3x x 5 2 4 of 11
h x x 3x 3x 13x 3x 21x x 15x 4. 8. Sketch the graph of 9 8 7 6 5 Clearly label all x-intercepts. x 2 x 1 x 1 2 3 4 9. Find the quotient and remainder when 3 2 f x x 4x 5 is divided by x 2. Remember: Show your work! 10. Find the quotient and remainder when your work! h x 2x 3x 9x x 3 is divided by x 1. Remember: Show 5 of 11
11. Determine all possible rational zeros for the polynomial function below. f x x 6x 13x 8x 18 7 6 4 2 12. Determine all possible rational zeros for the polynomial function below. g x 3x x 12x x 24 13. Determine the number of positive real zeros and negative real zeros the function might have. f x 2x 5x 5x 20x 12 14. Determine the number of positive real zeros and negative real zeros the function might have. f x x 3x 19x 27x 252 6 of 11
15. Using the information given below for Possible rational zeros: 1, 2, 4, 8 Number of positive real zeros: 3 or 1 Number of negative real zeros: 1 f x x x 2x 4x 8, find all real zeros of f. 16. A polynomial function has degree 5. Three of its zeros are 6, 8i, and 2-i. What are the other zeros? 7 of 11
17. Using the information given below for Possible rational zeros: 1, 2, 5, 10 Number of positive real zeros: 2 or 0 Number of negative real zeros: 2 or 0 g x x 4x x 14x 10, find all zeros of g. 8 of 11
18. f x x 2 x 6 x 2 x 3 2 3x 9x 6 3 x 1 x 2 a. State the domain of f. b. State the x-intercept(s), if any. c. State the y-intercept(s), if any. d. State the vertical asymptote(s), if any. e. State the horizontal asymptotes(s), if any. 19. g x 4 x 7 4x 28 16 4 4 3 x x x x x a. State the domain of g. b. State the x-intercept(s), if any. c. State the y-intercept(s), if any. d. State the vertical asymptote(s), if any. e. State the horizontal asymptotes(s), if any. 9 of 11
20. Sketch the graph of the rational function which has the following characteristics. Clearly label all intercepts and asymptotes in your sketch. x-intercept: 1 y-intercept: 3 8 Sign Chart: vertical asymptotes: x 2, x 4 horizontal asymptote: y 3 + + + + + + + + -2 1 4 21. Sketch the graph of the rational function f which has the following characteristics. Clearly label all intercepts and asymptotes in your sketch. x-intercepts: 0,2 y-intercept:0 vertical asymptote: x 3 horizontal asymptote: none Sign Chart: + + + + + + + + + + + + 0 2 3 10 of 11
22. Solve the following inequality. 3 2 x 4x 5x 23. Solve the following inequality. 3 3x 6x 0 x 3 11 of 11