Hello, future Honors Pre-Calculus victims! June 2015

Transcription

1 Hello, future Honors Pre-Calculus victims! June 05 The packet attached to this letter contains a series of problems that will overview the Algebra II skills you should have mastered in order to have a good start in Pre-Calculus. Completing this packet over the summer will help you practice and maintain these skills. I expect that you will work through these as best you can, and that your work will be neat and accurate. Moreover, you should work on this periodically throughout the summer, as it will take several hours to complete. This assignment will be checked on the first day of classes; after spending our first two days reviewing it, you will be tested on the material. That grade will count as your first test grade of quarter one. Be sure to present your work in an organized and systematic way. In this course, the way you arrive at an answer is often as important as the answer itself. Please be mindful of this when you are working through this packet; provide sufficient work to justify your answers. For your convenience, problems in this packet are preceded by the section number in parentheses in your Algebra II notes, indicating when the concept was covere If you retained your notes from Algebra II, you can refer to those sections for a refresher as neede Also, you will need a Texas Instruments TI-8, TI-8 Plus, TI-84, TI-84 Plus or TI-84 Silver Edition graphing calculator for this packet as well as for the course. Lastly, I recommend that you purchase the appropriate materials for this course: A -ring binder with dividers: Homework, Class Notes, Assessments. A pack of filler paper for the Homework and Class Notes sections of your binder A graph paper notebook I look forward to meeting you in the fall. Expect an exciting and challenging learning experience! Sincerely, Mr. Brown brownr@mersorg

2 Directions: Prerequisite Algebra II skills for Honors Pre-Calculus. All work must be done NEATLY, on separate paper. You are expected to number each problem, show work clearly, and circle your final answers.. All graphs must be done on graph paper. (print some from if necessary). Questions with the icon indicate that a calculator may not be use 4. A number in parentheses that immediately follows the problem number indicates the section of your Algebra II notes to which you should refer, if neede I. Point-Slope Equation of a Line. (.) Provide an equation, in point-slope form, of the line that contains the points 5, and, 8 7. (.4) Use the linear regression capability of the graphing calculator to find the slope-intercept equation of the line in (a). (.4) Verify your answer to (b) by solving that equation for y and comparing it to the linear regression equation you got in part (a).. (.) Write a point-slope equation of the line whose x-intercept is and y-intercept is -4.. (.) Write a point-slope equation of the line perpendicular to the line x 4y 5 that shares its y-intercept. Write a point-slope equation of the line that is parallel to the line in (a) and contains the point (, -7) Sketch three lines: the given line in (a), your answer to (a), and your answer to (b). 4. (.) Write an equation of a line that has an x-intercept but no y-intercept and only runs through quadrants II and III. Then, sketch the line. What is the slope of your line? Write an equation of a line that is perpendicular to your line in (a). Then, state its slope and sketch it. 5. (.4) A school district purchases a high-volume copy machine for $5,000. After 0 years, the equipment will have to be replaced (due to wear and tear). Its value at that time is expected to be $000. Write a linear equation giving the value v of the equipment as a function of the number of years t since the machine was purchase Write a sentence in which you describe the slope of your linear model in the context of this problem.

3 II. Operations with Functions, Composition of Functions. (., 5.5, 5., 7.) Given the function f ( x) x x, evaluate: i. f () ii. f iii. ( f (0)) 4 evaluate, and fully simplify: f ( 4 h) f 4 i. f ( ) ii. f ( 4 ) iii. h For what values of x is i. f ( x) 0? ii. f ( x) 48? f iv. f f 0, where h 0 (why?) 7. (5.5, 7.) If f ( x) x x and g ( x) 7x 5, find the values of x for which f(x) = 0 g(x) = 0 f(x) = g(x) f gx 0 e. g gx 0 8. (.,., 7.) Please refer to the graph of y = g(x) shown. Is g(x) a function? Briefly explain. State the value of each: i. g() ii. g(-) iii. g(g(0)) For what values of x is g(x) = 0? Write a possible equation of g(x) i. in factored form ii. in standard form III. Graphs of Functions, Relations, and Inverses 9. For each relation: i. Sketch its graph. ii. (.) State its domain and range. iii. (.) Tell whether or not it is a function. (.) y (.5) 4 x y e. (.) x x x (5.) x g. (.) xx j. (7.8) y (5.) y x y f. (.) y x y h. (7.8) y x 5 i. (7.8) y x 5 y x 5 k. (8.) x y l. (8.) y log x 0. (7.7) i. Find the equation of the inverse f ii. Sketch f(x), f (x) of each function. (x), and the line of identity y = x for each. iii. State the domain and range of f(x) and f 4 x (x). x f ( x) x

4 IV. Quadratic Functions. (5.7) i. Rewrite each equation in vertex form y k ax h by the method of completing the square. ii. State the coordinates of the vertex and the equation of the axis of symmetry of each parabol x 0x 4 f ( x) x x 9 x 8x 7. (5.) Use the maximum/minimum feature of the grapher to find the vertex of each parabola in #. Use these answers to confirm that your answers in # are correct (5.) The path of a diver jumping off of a diving board is given by the equation f ( x) x x, where y 9 9 is the diver s height (in feet) above the surface of the water and x is the horizontal distance (in feet) of the diver from the end of the diving boar How high above the water is the. diving board? What is the maximum height of the diver? Find the diver s horizontal distance from the end of the diving board when the diver hits the water. V. Polynomial Functions 4. (5.8,.) Find all real zeroes of each function using quadratic formula or square roots. Answers must be expressed in simplest radical form. x x 4 f ( x) x 0 x 4x 9 5. (.) Write a polynomial function, in standard form, that has the given zeroes: x,, 4 x 5, 5, 0 x 4, multiplicity. (.) Use polynomial long division to find the quotient: 4 x 7x x 54x 5 Use your answer from part (a) to factor 4x 7x x 5 completely. Use your answer from part (b) to state all real zeroes of the function f ( x) 4x 7x x (5.4,.4) Algebraically determine all real zeroes of each function by using factoring techniques such as splitting the middle, trial & error, Greatest Common Factor, and/or grouping: x 0x 8 f ( x) x 8 x x 0 f ( x) 4x 0x 5x e. f ( x) x x 50x 75 f. f ( x) x x 4 8. (.) Use synthetic division to find the quotient: x 4x x 8 x Use your answer to part (a) to find all real zeroes of the function f ( x) x 4x x 8.

5 VI. Operations with Radicals, Rationalizing the Denominator 9. (5,5, 7., 7.) Simplify each of the following radical expressions. Express all answers in simplest radical form, with rationalized denominators (i.e. radicals may not be left in the denominator of any fraction.) 7 e. f. 8 8 g. 0. (7.) Write each expression in simplest radical form by rationalizing the denominator with the conjugate (5.5, 5.8) Use square roots or quadratic formula to find all complex zeroes. Express answers in simplest radical form, with rationalized denominators. x 0 x 4x 5 0 x x 0 VII. Operations with Imaginary Numbers, Rationalizing the Denominator (For these problems, i represents the imaginary number i ). Given the function f ( x) x x, evaluate and fully simplify each. Express answers in a + bi form. i. f (i) ii. f ( i) iii. f ( i) iv. f ( i). (5.) Show that each number satisfies the equation x 4 0x 9 0 by substituting them and simplifying: i. x i ii. x i iii. x i iv. x i 4. (5.) Find all real zeroes of the equation x 4 0x 9 0 by factoring. Do your answers look familiar? 5. (5.) Simplify each fraction by rationalizing the denominator. Express answers in a + bi form. i i i i e. i f. i g. i h. i