## Math 116: Study Guide - Final Exam

1. Eight graphs, corresponding to the eight basic graphs on the front cover of the text, are shown. For each graph, identify the basic graph and the equation of the graph shown.
2. Find the inverse of a function. Pay special attention to those with restrictions necessary to make it one-to-one.
3. Solve a equation for x.
4. Simplify an expression involving i.
5. Find a function with the indicated zeros. Pay special attention to those involving radical or complex roots.
6. Use synthetic division to divide a polynomial by a binomial.
7. Given a polynomial function, give the total number of real or complex zeros, the maximum number of positive and negative roots, the right and left hand behavior of the graph, where the graph crosses and touches the x-axis, the y-intercept, the domain, the form of any possible rational zeros, and all real and complex roots.
8. Given a ration function, give the domain, the behavior at a point where there is a common factor between the numerator and denominator, the behavior at each vertical asymptote and x-intercept, and the horizontal asymptote.
9. Combine several logarithms into a single logarithm.
10. Expand a single logarithm into the sum, difference, and constant multiples of several logarithms.
11. Solve the logarithmic equation.
12. Identify the conic section or degenerate case. Nine parts.
13. Eliminate the parameter from a set of parametric equations.
14. Complete the square and place the conic section into standard form. Then sketch the graph.
15. Sketch the graph of the conic section that is already in standard form.
16. Find the partial fraction decomposition for the expression.
17. Maximize an objective function subject to the given constraints.
18. Solve a system of equations by graphing.
19. Solve a system of equations by substitution.
20. Solve a system of equations by addition or elimination.
21. Solve a system of equations by Gauss Jordan Elimination using matrices.
22. Solve a system of equations by Cramer's Rule.
23. Solve a system of equations by Matrix Algebra. You can use the calculator, but write the matrices entered into the calculator and the expression evaluated with the calculator.
24. Setup and solve a system of linear equations which will find the equation of a parabola passing through the given points. I suggest using matrix inverses to find the solution.
25. Multiply two matrices.
26. Find the inverse of a matrix.
27. Find the determinant of a 3x3 matrix. The vertical lines do not mean take the absolute value.
28. Solve a matrix equation. Watch out for commutativity and division.
29. Write the solution from a reduced matrix.
30. Write the first five terms of the sequence with the given general term.
31. Find the sum of an arithmetic series.
32. Simplify the ratio of the factorials.
33. Expand a binomial using the binomial expansion theorem.
34. Write whether or not the simplification is valid. Seventeen parts. Capital letters represent matrices and lower case letters represent real numbers. Several of these deal with inverses (when do two things inverse out).
35. Identify the term, rule, or theorem defined, or which is applicable. You should know the following: Fundamental Theorem of Arithmetic, Fundamental Theorem of Algebra, Fundamental Theorem of Linear Programming, Fundamental Counting Principle, Descartes' Rule of Signs, Rational Root Theorem.; Definitions of circle, ellipse, parabola, hyperbola, combination, permutation. Eight parts.
36. Identify each statement as true or false. Concentrate on: Elementary row operations, Matrix multiplication, Melodic properties of logs, Determinants of special matrices, Relationship of a matrix inverse to its determinant, Relationship of a matrix inverse to its size, Limit definition of e, One-to-one functions in regards to line tests, Matrix division. Nine parts.

### Notes:

• Make sure you use the proper technique to solve the systems of equations. All the systems are 2x2.
• This test is open notebook. Your old tests may be in your notebook. This study guide should certainly be in the notebook.
• For the terms, rules, and/or theorems, write them all out on one piece of paper and stick that in your notebook so you don't have to go looking for things.
• Make sure your notes on the areas covered on the test are full. If your notes aren't complete, supplement.
• You may want to organize your notes - perhaps index the sections with tabs that will be used on the test. Another idea would be to put all the notes for the test on at the beginning. Indexing would be better, as you may not get everything that's on the test, and then your notes would be out of order. Some people like to take the study guide and indicate what section of the book that applies to so they know where to go in their notes.
• The test has been arranged so that it is is primarily in the order the material was presented in class.
• Answer as many questions as you can without using your notes. You will not have adequate time to research every question in your notes.
 # 1 2 3 4 5 6 7 8 9 10 11 12 Pts 16 4 4 4 4 4 11 7 3 3 5 9 # 13 14 15 16 17 18 19 20 21 22 23 24 Pts 4 5 4 4 4 5 5 5 5 5 5 5 # 25 26 27 28 29 30 31 32 33 34 35 36 Total Pts 4 4 4 4 4 4 4 4 4 17 8 9 200