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1 Algebra / Trig Final Exa Study Guide (Fall Seester) Moncada/Dunphy Inforation About the Final Exa The final exa is cuulative, covering Appendix A (A.1-A.5) and Chapter 1. All probles will be ultiple choice with one correct answer out of 4. You ay use your calculator for the whole final exa. It will be reset before the test! The study guide below has all of the topics we have covered and you will be working selected review probles. The probles ay not cover every concept, so a good strategy is to study those topics you do not iediately know. Each day we will have a war-up review and a set of probles to work. You will turn in a packet with all war-ups and probles on the day of the final exa. The final exa accounts for 0% of your total seester average. While your grade is not just a nueric coputation, your average does play a huge part in deterining your grade. You MUST be present on the specified day and tie. There will NOT be any ake-ups! BRING YOUR GRAPHING CALCULATOR!!! APPENDIX A.1: Properties of Real Nubers Hierarchy of real nubers and the subsets for integers, whole nubers, and natural nubers. Recognize rational and irrational nubers when represented as decials. Express intervals on the nuber line graphically using interval notation with [ ] instead of and ( ) instead of. Write intervals using [ ] and ( ) and inequality notation and how to convert words into intervals. Deterine the distance between two points using absolute value. Definition of expression and what it eans to siplify an expression. Use the order of operations to siplify an expression State the "rules of algebra" given "For all real nubers a, b, c..." and one of the properties. For exaple, "State the transitive property for all real nubers a, b, c." Identify the properties that were used to siplify an algebraic stateent. Add, subtract, and ultiply signed nubers Divide integers and fractions using the definition of division: ultiplication by the reciprocal APPENDIX A.: Exponents and Radicals Know the exponent rules for any real nubers a and b and integers and n: n n a a a a b a b n n a a 0 a n a a a a 1 n a b b Siplify expressions containing exponents according to a variety of instructions. For exaple, know how to express an answer with only positive exponents or without any fractions. Definition of scientific notation: x 10 n, where is in the interval [1, 10) and n is an integer. Convert fro decial to scientific notation and vice versa. Know how to ove the decial point and to set the exponent properly depending on the size of the nuber. Significant digits: All non-zero nubers are significant. Zeros are significant only when they do soething other than place the decial. For exaple, 5.0 has SD whereas 50 has 1 SD; 4008 has 4 SD whereas 4000 has 1 SD. Know the definition of the n th root of a nuber: the solution to the equation Understand the structure of a radical: index radicand Siplify radicals by looking for factors of the radicand that are perfect n th powers. Apply the absolute value restriction to a siplified radical: when the index is even and the resulting root has a variable with an odd exponent. NEVER restrict an odd-index radical (e.g. cube root). n x b.

2 Rationalize the denoinator of a radical by finding a ultiplier that "copletes the perfect n th power". Add radicals by finding like radicals (sae index and radicand). Multiply binoials containing radicals using the distributive property (soeties called FOIL). Rationalize a binoial denoinator by ultiplying by the conjugate. n n Understand the relationship between rational exponents and radicals: n b b b. Convert a radical into exponential for and vice versa. Siplify expressions with rational exponents using the relationship above. Siplify expressions containing radicals by first converting into exponential for, using exponent rules to siplify the new expression, then converting back to radical for at the end. Siplify expressions containing real nuber exponents using the standard rules of exponents. APPENDIX A.3: Polynoials and Factoring Siplify a polynoial and write in standard for (decreasing degree of lowest alphabet variable) State the degree of a onoial and the degree of a polynoial Multiply polynoials using the distributive property Recognize special products to ake the ultiplication process sipler Factor a difference of squares: a b ( a b)( a b) Factor a su or difference of cubes: a 3 b 3 ( a b) a ab b and a 3 b 3 ( a b) a ab b Factor a perfect square trinoial: a ab b a b Factor a trinoial: ax bx c using the a c ethod Factor a trinoial in quadratic for: a* b* c using the a c ethod Factor a polynoial by grouping ters to for coon factors so that becoes coon factors.. Look for difference of squares, perfect square trinoials, and other groupings that for APPENDIX A.4: Rational Expressions Definitions: doain of an expression; equivalent expressions State the doain of a rational expression or an expression involving roots. Siplify a rational expression by prie factoring the nuerator and denoinator, dividing coon factors, and stating restrictions. Understand the difference between the doain of a rational expression and restrictions stated on an equivalent expression. "Siplify then ultiply" rational expressions and state any restrictions on the siplified for Turn a division proble into ultiplication by the reciprocal Find the least coon denoinator (LCD) of two or ore rational expressions and convert all fractions to have the LCD. Siplify a coplex fraction using "ethod A" (siplify nu & den, then divide by ultiplying by the reciprocal or "ethod B" (ultiply by LCD/LCD, where the LCD is for all denoinators in the coplex fraction). Siplify a coplex fraction involving rational exponents by factoring out the sallest exponent and siplifying the resulting expression. Siplify a coplex fraction involving rational exponents by ultiplying by a factor that akes all exponents whole nubers.

3 APPENDIX A.5: Solving Equations Know the eaning of equation and solve an equation. Solve linear equations using the properties of equality to create an equivalent equation of the for x c. Be able to recognize when there is no solution and when the solution is all the reals (fro an identity). Know how to eliinate all fractions by ultiplying the entire equation by the LCD. CAUTION: Multiplying by an LCD with variables is not guaranteed to create an equivalent equation, so you MUST check all potential solutions in the ORIGINAL equation. Know the definition of the roots of an equation. Solve polynoial equations by converting the to standard for (expression = 0), factoring the expression, setting each factor equal to zero, and solving for each root. Know how to derive the quadratic forula by copleting the square on ax bx c 0. Solve equations of the for ax bx c 0 by copleting the square or the quadratic forula. NOTE: You ust be able to deonstrate how to solve by copleting the square. Be able to solve equations that are in quadratic for: a * b(*) c 0, where * represents an expression. Solve radical equations by isolating the radical, squaring both sides, and isolating the variable. CAUTION: Squaring both sides of a radical equation is not guaranteed to ake an equivalent equation. You ust check all potential solutions in the ORIGINAL equation. Make sure to state if a potential solution is really an extraneous root. CHAPTER 1: Functions and Their Graphs Section 1.1: The coordinate plane Midpoint forula: x x, y M y Use the distance forula to find the length of a segent or to verify that three points for a right triangle. Use the idpoint forula to find the idpoint of a line segent or to perfor another task that requires the idpoint. Know the coordinate for of transforations and how to transfor a point in the coordinate plane. Distance forula: d x x y y o Translation: x, y x a, y b o Reflection in x-axis: x, y x, y o Reflection in y-axis: x, y x, y o Reflection in origin: x, y x, y Section 1.: Graphs of equations Know the definition of an equation in two variables and the solution to an equation in two variables. Sketch the graph of an equation in two variables using the point plotting ethod. Understand the eaning of x-intercept and y-intercept and be able to find the algebraically for an equation in two variables. Know the eaning of syetry and be able to deterine fro a graph or algebraically if an equation has x-axis syetry, y-axis syetry, origin syetry, or no syetry. Section 1.3: Graphs of lines and linear equations Know the definition of the slope of a line and how to find the slope through two points. Pay particular attention to the special cases of a horizontal line (slope = 0) and a vertical line (slope undefined). Know slope-intercept: y x b. Be able to graph the equation of a line in slope-intercept for or a vertical line in "x-for". Be able to write the equation of a line in slope-intercept for when given two points on the line.

4 Deterine if two lines are parallel or perpendicular and know how to create the equation of a line parallel or perpendicular to a given line passing through a given point. Solve word probles related to linear equations and interpret the results in the context of the proble. Section 1.5: Analyzing functions by their graphs In general, know how to answer the "7 questions", though I do not expect you to recite the actual questions. Be able to identify the doain and range of a relationship (function or not) directly fro its graph. Deterine if a graph represents a function using the vertical line test. Find the zeros of a function algebraically and graphically using the calculator (CALC, :zero). Estiate the intervals where a function is increasing, decreasing, or constant graphically. Deterine all axia and inia of a function graphically using the calculator (CALC, 3:iniu or CALC, 4:axiu) Be able to create a function fro a word proble and find its axiu or iniu using the calculator. Section 1.6: Graphs of parent functions Know how to graph all of the parent functions using the 3 point ethod. Recognize that the squaring, cubic, square root, and absolute value functions all have (0, 0) and (1, 1) in coon. The third point depends on the function itself. Be able to graph the greatest integer function with correct use of and on the endpoints of the steps. Section 1.7: Transforations of parent functions Know how to convert a given function into standard graphing for: ( ) ( ) g x a f b x c d where b is factored out. Know the effects of a, b, c, and d. o a causes a vertical stretch if a 1, a vertical shrink if 0 a 1, and a reflection in the x-axis if a < 0. o b causes a horizontal shrink if b 1, a horizontal stretch if 0 b 1, and a reflection in the y-axis if b < 0. o c causes a horizontal shift to the right if c > 0 and to the left if c < 0. Reeber, x c x ( c). o d causes a vertical shift up if d > 0 and down if d < 0. Be able to graph a transfored function using the three step procedure: 1. Write the function in standard graphing for.. Locate the "new origin" by applying any translations to (0, 0). x, y on f( x ) becoes 3. Plot additional points fro the new origin using the basic relationship 1 x, ay b on gx. ( ) Know how to convert a word description of transforations into function notation. Be able to describe the transforations of a parent function f( x ) when given a function gx. ( )

5 Practice Probles *You DO NOT need to turn in this packet when you turn in your work* A.1 and A. P. A4-A6 33, 35, 37, 41, 43, 45, 47, 53, 55, 63, 67, 69, 77, 83, 85, 89, 97, 105, 107, 109, 117 A.3 and A.4 P. A36 A37 75, 77, 91, 103, 113, 139, 155, 161, 171, 01, 07 P. A46 A47 35, 37, 53, 55, 59, 65, 73, 79 A.5 and A.6 P. A60 A61 1, 5, 7, 39, 41, 53, 65, 73, 85, 95, 115, 119, 13, 19, 131, 139, 149, 153 P. A69 A70 49, Odd P , 5, 49, 59, 61, 63, 65, 71, 73, 87, 89, 97, 99, odd, odd, 13, 17