College Algebra

Math 3—College Algebra—Final Examβ€”Summer 2021 Instructor: Su Lan Wong Section # 53218 Name: ____________________________________ Q1. (6 points) For (π‘₯ + 3)2 + (𝑦 βˆ’ 4)2 = 25 a) Find the center (β„Ž, π‘˜) and radius π‘Ÿ of the circle. b) Graph the circle π‘₯βˆ’4 Q2. (6 points) Solve: 2π‘₯+1 ≀ 5, write the solution set in interval notation. 1 Math 3—College Algebra—Final Examβ€”Summer 2021 Section # 53218 Instructor: Su Lan Wong Name: ____________________________________ 2 Q3. (6 points) Let 𝑓(π‘₯) = 4π‘₯ βˆ’ π‘₯ π‘Žπ‘›π‘‘ 𝑔(π‘₯) = 2π‘₯ βˆ’ 5. Find and simplify each of the following expressions. b) a) (π‘“π‘œπ‘”)(π‘₯) Q4. (6 points) Let 𝑓(π‘₯) = 𝑔(π‘₯+β„Ž)βˆ’π‘”(π‘₯) β„Ž 2π‘₯βˆ’1 π‘₯+5 , find 𝑓 βˆ’1 (π‘₯). Give the domain and range of both 𝑓 π‘Žπ‘›π‘‘ 𝑓 βˆ’1. 2 Math 3—College Algebra—Final Examβ€”Summer 2021 Section # 53218 Instructor: Su Lan Wong Name: ____________________________________ Q5. (6 points) An equation of a hyperbola is given. (a) Find the vertices, foci and asymptotes of the hyperbola. (b) sketch a graph of the hyperbola. 𝑦2 π‘₯2 βˆ’ =1 9 16 Q6. (5 points) Evaluate the sum if possible. 1 1 1 1+ + + +β‹― 6 36 216 Q7. (5 points) Expand the binomial by using the binomial theorem. (2π‘₯ + 𝑦 2 )5 3 Math 3—College Algebra—Final Examβ€”Summer 2021 Section # 53218 Instructor: Su Lan Wong Name: ____________________________________ Q8. (5 points) The initial size of a culture of bacteria is 8600. After 1 hour the bacteria count is 10,000. a) Find a function that models the number of bacteria 𝑛(𝑑) after t hours. b) After how many hours will the number of bacteria doubles? Q9. (5 points) Solve the logarithmic equation for π‘₯. π‘™π‘œπ‘”9 (π‘₯ βˆ’ 5) + π‘™π‘œπ‘”9 (π‘₯ + 3) = 1 4 Math 3—College Algebra—Final Examβ€”Summer 2021 Section # 53218 Instructor: Su Lan Wong Name: ____________________________________ Q10. (6 points) Find the partial fraction decomposition 6π‘₯ βˆ’ 7 (π‘₯ βˆ’ 2)(π‘₯ + 3) Q11. (6 points) Solve the system by using addition method. { 3π‘₯ 2 βˆ’ 𝑦 2 = βˆ’4 π‘₯ 2 + 2𝑦 2 = 36 5 Math 3—College Algebra—Final Examβ€”Summer 2021 Instructor: Su Lan Wong Section # 53218 Name: ____________________________________ Q12. (8 points) Given 𝑃(π‘₯) = π‘₯ 4 + π‘₯ 3 + 7π‘₯ 2 + 9π‘₯ βˆ’ 18 a) List all possible rational zeros (without testing to see whether they actually are zeros) b) Determine the possible number of positive and negative real zeros using Descartes’ Rule of Signs. c) Is (π‘₯ + 2) a factor of 𝑃(π‘₯)? d) Find all complex zeros of 𝑃(π‘₯) e) Sketch the graph of P. 6 Math 3—College Algebra—Final Examβ€”Summer 2021 Instructor: Su Lan Wong Q13. (8 points) Given a) b) c) d) e) π‘Ÿ(π‘₯ ) = Section # 53218 Name: ____________________________________ π‘₯+2 π‘₯ 2 +2π‘₯βˆ’3 , Find the domain Find π‘₯ βˆ’ π‘–π‘›π‘‘π‘’π‘Ÿπ‘π‘’π‘π‘‘π‘  π‘Žπ‘›π‘‘ 𝑦 βˆ’ π‘–π‘›π‘‘π‘’π‘Ÿπ‘π‘’π‘π‘‘ Find vertical asymptotes and horizontal asymptotes Sketch the graph of π‘Ÿ(π‘₯) Solve the inequality π‘Ÿ(π‘₯) β‰₯ 0 7 Math 3—College Algebra—Final Examβ€”Summer 2021 Instructor: Su Lan Wong Name: ____________________________________ Q14. (8 points) Given 𝑓(π‘₯) = 𝑒 π‘₯ + 3 a) b) c) d) e) Section # 53218 Sketch the graph of the function State its domain and range in interval notation Find 𝑓 βˆ’1 (π‘₯), and state its domain and range. Graph 𝑓 βˆ’1 (π‘₯) on the same graph as part (a). State the asymptote for 𝑓(π‘₯) π‘Žπ‘›π‘‘ 𝑓 βˆ’1 (π‘₯) 8 Math 3—College Algebra—Final Examβ€”Summer 2021 Instructor: Su Lan Wong Section # 53218 Name: ____________________________________ Q15. ( 6 points) Graph the solution set. If there is no solution, indicate that the solution set is the empty set. { 𝑦 β‰₯ π‘₯2 π‘₯+𝑦 β‰₯6 9 Math 3—College Algebra—Final Examβ€”Summer 2021 Section # 53218 Instructor: Su Lan Wong Name: ____________________________________ Q16. (8 points) Determine whether the equation represents an ellipse, a parabola, or a hyperbola. If the graph is an ellipse, find the center, foci, and vertices. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. 9π‘₯ 2 βˆ’ 54π‘₯ + 𝑦 2 + 2𝑦 = βˆ’46 10
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