Warm-up for Foundations of Precalculus

Transcription

1 Summer Assignment Warm-up for Foundations of Precalculus Who should complete this packet? Students who will be taking Foundations of Precalculus in the fall of Due Date: The first day of school How many of the problems should I do? ALL OF THEM How should I organize my work? You should show all work in a separate sheets of graph paper Keep your materials in a 3-prong folder or report cover. How will my teacher know that I ve done the work? Your teacher will collect your work on the first day of school. Your teacher may choose to QUIZ or TEST you on this material if he or she feels it is necessary BE PREPARED! How well should I know this material when I return? You should recognize that you ve seen this material before, and you should also be able to answer questions like the ones in this assignment. If the material is revisited in your next class, it will only be for a brief amount of time your teacher will assume that all you need is a quick refresher. Note from your teachers: We feel that this summer work will truly help you succeed this year. We understand that summer is a time for relaxation and fun, but it is imperative that you spend some time before you return reviewing your materials. This packet is mandatory, and you must treat it as you would any other extremely important homework assignment. You will be held accountable for this material. We also highly suggest that you do a bit of it at a time in the weeks leading up to school don t leave it for the last day!!!

2 Warm-up for Foundations of Precalculus Instructions: Complete the problems on graph paper in pencil. At the top of each page of your work, write your name. Complete all the problems carefully. Show enough work to indicate your method of solution. Make sure your work justifies your answer. If a problem requires a graph, you may graph on your calculator, but then sketch the graph on your paper as part of your solution. Please use a ruler. Several problems on this assignment require you to make a diagram. Use a ruler and make an accurate sketch. Keep the packet and your work in a folder. Place your completed work in a report cover or 3-prong folder. Remember, your teacher will collect your work on the first day of school! Late work will be penalized and may NOT be accepted. You will continue to use your 3-Ring binder for your math class throughout the school year. What if I get stuck? - You should check out other additional study materials. Consult a standard algebra textbook. Find a study buddy or a classmate to help you remember the material. Consult the following websites for hints and examples:

3 Warm-up for Foundations of Precalculus page 1 Formatted: Indent: Left: 0", First line: 0" 1. Mr. Bosch is cutting timbers to outline a rectangular flower bed. He is using two 8-foot pieces of timber to outline the bed. He will cut each piece as shown below. a. Write a quadratic equation to represent the situation. Solve the equation graphically and then by factoring. Show your work. Where should he make his cut if the area of each flower bed is to be 15 square feet? b. Do both answers make sense? Why or why not? c. Where should Mr. Bosch cut the timbers to outline a 1-square foot flower bed? Identify a, b, and c and use the quadratic formula to solve. d. Do both answers make sense? Why or why not?. a. Write a quadratic function whose graph has its vertex in the second quadrant. b. Name the vertex, give the equation of the axis of symmetry, and tell whether the graph opens up or down. Justify your reasoning. 3. Find the vertex, axis of symmetry, and all intercepts for each of the following. Also, identify what form the function is written in, and rewrite each in the other two forms a. c. x 8x+ 5= 0 f( x) = x 8x + 5 b. 3x x 17 0 f( x) = 3 x d. + = ( ) x 3.x = 0 f( x) = ( x 4) ( x + 1) 0.5x 3.5x=.7 4. The area of a rectangle is cm. If the length is 3.4 cm longer than the width, find the length of the rectangle to the nearest hundredth of a centimeter.

4 Warm-up for Foundations of Precalculus page 5. Given the polynomial function, Px ( ) ( x 1) ( x ) a. Find the x and y intercepts of the graph. b. Determine the degree of the function. = +, do the following: c. Describe the end behavior of the graph, that is, describe what happens to the y-values as the x-values increase d. Find all local maximums and minimums, and determine if any are absolute maximums or minimums. e. Using the information in parts a d, make an accurate sketch of the graph, being sure to label your axes accordingly. Formatted: Font: Italic 3 6. Given the polynomial function, ( ) ( ) Px ( ) = x x x+ 5, do the following: a. Find the x and y intercepts of the graph. b. Determine the degree of the function. c. Describe the end behavior of the graph, that is, describe what happens to the y-values as the x-values increase Formatted: Indent: Left: 0", Hanging: 0.5" d. Find all local maximums and minimums, and determine if any are absolute maximums or minimums. e. Using the information in parts a d, make an accurate sketch of the graph, being sure to label your axes accordingly. Formatted: Indent: Left: 0", Hanging: 0.5", Tab stops: Not at 0.5" + 3.5" + 3.5" Formatted: Indent: Left: 0", Hanging: 0.5"

5 Warm-up for Foundations of Precalculus page Graph f( x) = x + 6x 3x 1 and gx ( ) = 3x+ x 3x+ on the same axes. Make an accurate sketch of both graphs. Find the coordinates of the point P where the two functions intersect. Round to the nearest hundredth. Circle and label this point on your graph. Formatted: Indent: Left: 0", First line: 0", Tab stops: Not at 0.5" + 3.5" + 3.5" 8. Determine whether each polynomial is factored correctly. If it is correct, sketch the graph. If it is incorrect, write the correct factorization and sketch the graph. a. x + 9x 5= ( x+ 1)( x 5) b. x 3 1= ( x 1)( x + x+ 1) 9. Write a polynomial function f( x ) of second degree. a. Describe how you find the inverse of your function, then find the inverse. b. Tell in your own words how the graph of a function and its inverse are related. c. Graph the inverse you found in part a. Is it a function? Why or why not? 10. Find the inverse of each function. Sketch the graph of the function and its inverse on the same axes. a. f( x) 5 = x b. f( x) x 3 = + c. y = 3 x 1

6 Warm-up for Foundations of Precalculus page a. Graph the function y = x. Sketch the graph, label the y-intercept and other points on the graph. b. Graph the function y = x. Sketch the graph, label the y-intercept and other points on the graph. c. How are the two graphs related? What is an alternate way to write the function in part b? 1. Exactly eight years ago, an investor deposited $1700 into an account that pays 5.75% annual interest, compounded continuously. If there were no additional deposits or withdrawals, what rt amount is in the account today? Use the formula A = Pe. Formatted: Indent: Left: 0", Hanging: 0.5" 13. According to a commonly used rule, the Rule of 7, an estimation of the number of years it will take to double an investment invested an n% is given by 7 n. a. Estimate the number of years it would require for $4000 invested at 8% to be worth $8000. b. Find the number of years it would take for a savings account of $4000 to double if invested at 8% compounded annually. c. How long will it take a $4000 savings account to double when invested at 8% compounded continuously? d. Is the Rule of 7 a better estimate for the time required to double an investment deposited at a fixed rate compounded annually or continuously? e. Find a more accurate rule for estimating the time required to double money invested at a constant rate compounded continuously. 14. A ship carrying 1000 passengers has the misfortune to be shipwrecked on a small island; the passengers are never rescued. The natural resources of the island limit the population to The population gets closer and closer to this limiting value, but never reaches it. The population of the island after time t in years is given by the function: 5780 Pt () = t e a. Find the population after 0, 1, 5, 10, and 0 years. b. How long does it before the population reaches 5000? c. Graph the function, and show your answer to parts a & b on the graph.

7 Warm-up for Foundations of Precalculus page Find: a. the 4 th term of the sequence 4., 6.3, 8.4,. b. the first four terms of an arithmetic sequence if a 1 = and d = The 4 th term of an arithmetic sequence is 1, and the 7 th term is Find the 1 st term and the 1 st term. 17. Find the first five terms and the twelfth term of the geometric sequence in which a 1 = 1.1 and r =. 18. Find the first five terms of the sequence in which a 1 = 4. and a = n 1 0.9a + + n a. Explain in your own words what is meant by the term arithmetic sequence. b. Write an arithmetic sequence. c. Write the formula for the nth term of your arithmetic sequence. Then find the 30 th term. 0. a. Explain in your own words what is meant by the term geometric sequence. b. Write the formula for the nth term of your geometric sequence. Then find the 10 th term. 1. Change each radian measure to degree measure. a. 11π 7 b. 8.5

8 Warm-up for Foundations of Precalculus page 6. Each of the following graphs passes through the point P = (1, ). Match the correct function to each graph. Identify a characteristic that enables you to determine which graph. a. y = x b. y = sin(30 x) + 1 c. y = log x d. e. 1 3 y = x+ = y x x

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