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Unformatted text preview: College Algebra / Test 2 Review Erika Noffsinger / Fall 2009 1. Let f(x) = a) b) c) d) e) f) g) h) 1 3 . ( x 2) 2 What is the parent function of f(x)? y = _____________ Graph the parent function (you don’t need to plot points – make a general sketch). What are the horizontal and vertical asymptotes of the parent function? What is the domain and range of the parent function? What transformations are you performing on the parent function to get the graph of 1 3 ? f(x) = ( x 2) 2 Graph f(x). What are the horizontal and vertical asymptotes of f(x)? What is the domain and range of f(x)? 2. Let P(x) = x4 + x3 + 7x2 + 9x – 18 a) Use the rational roots theorem to list all possible rational roots. b) Use Descarte’s rule of signs to count the number of possible positive and negative roots. c) Use synthetic division to find the zeroes of P(x). d) Write P(x) in linear factored form (some of your linear factors may have complex roots.) 3. Let p(x) = – x3 +x2 – 8x – 10. a) Show that x = 1 + 3i is a zero of p(x). b) Find the remaining zeros of p(x). 4. f(x) = x3 ‐ 8x2 + 22x ‐20 a) Find the possible rational roots of f(x). b) Find the roots of f(x). c) Write f(x) as a product and linear and quadratic factors (where quadratic factors do not have real roots). 5. Given 1‐i, 1+i, are roots of f(x) and f(2) = 1 find the equation of f(x). 6. Let s(x) = – (x – 2)2(x + 3)(x + 1)3 a) What are the end behaviors of the graph of s(x)? b) What are the x – intercepts? State their multiplicity. c) What is the y – intercept? d) Sketch a general shape of s(x), plot any additional points, if needed, to see what is happening to the graph. e) Using the graph of s(x): State when s(x) > 0 and when s(x) < 0. 7. Let h(x) = x3 , answer the following: x2 1 a) What is the vertical asymptote(s)? b) Is there a horizontal asymptote? If so, where is it? c) Is there an oblique asymptote? If so, where is it? d) Does the graph cross any asymptotes if they exist? Show your work. d) Graph the general sketch of h(x), plotting additional points if needed. 8. An automobile manufacturer can produce up to 300 cars per day. The profit made from the sale of these vehicles can be modeled by the function P(x) = ‐10x2 + 3500x – 66,000, where P(x) is the profit in dollars and x is the number of automobiles made and sold. Based on this model: a) Find the y – intercept and explain what it means in this context. b) Find the x – intercepts and explain what they mean in this context. c) How many cars should be made and sold to maximize profit? d) What is the maximum profit? 9. For q(x) = x3 + 8x2 – 20x a) Identify the end behaviors of the graph of q(x). b) Find the x and y intercepts of q(x). c) Graph a general sketch of q(x). 10. a) b) c) d) e) f) g) h) i) j) For f(x) = ‐ (x+2)2 + 4 answer the following questions. What is the vertex of f(x)? Is it a maximum or minimum point? Why? Find the x – intercept(s) if they exist, if they don’t exist explain why. What is the axis of symmetry? What is the y – intercept? What is the point symmetrical to the y – intercept? Graph f(x). What is the domain of f(x)? What is the range of f(x)? Find the inverse of f(x), when f(x) > ‐2 stating any necessary restrictions. Show that f(x) and f – 1 (x) found in part f) are inverses of one another. 11. Graph the following: a) f ( x) x2 x2 1 b) h(x) = 2 x+1‐ 3 c) g(x) = log3x d) List the asymptotes of part a), b) and c) if they exist. 12. Suppose the lost city of Atlantis had a population of 25,000 in 1998 and 27,000 in 1999. a) If the country’s population is growing exponentially, find an exponential growth function for the population of Atlantis for year t . Use t 0 to represent the year 1998. (Use 6 decimal places for k .) b) Use your function to predict in what year the population will reach 100,000. Do your work by hand (algebraically). 13. You found a suitcase full of money. There was $1,972 in the suitcase. After all the excitement of finding the money you decide to invest it for 6 years? You can either invest it at 3.75% compounded continuously or at 4.1% compounded 3 times a year. Which investment gives you a greater return? 14. Solve the following equations: a) 2 x
2 5 x 1 16 b) e2x = 5 c) log 27 x 1 3 d) log x 64 3 e) logx + log(x – 3) = 1 f) log(4x ‐ 1) = log(x+1) + log2 g) logx + log(x‐3) = 1 15. Let f(x) = – 2 x+3 + 2 a) What transformations are being preformed on the parent function, y = 2x, to graph f(x)? b) Graph of f(x). c) What is the domain and range of f(x)? 16. Let f(x) = x 4 and g(x) = 2 x 5 . Find f g ( x ) and state the domain. 16. 2x 1 0 2x 4 1 x x 17. g(x) = a) Find g ‐1(x). b) Check your work on part a) by showing g ‐ 1(g(x)) = x 18. Find the domain of g(x) = 1 x 4 x 12
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