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SAMPLE PLACEMENT TEST TO BYPASS MATH 1404 Department of Mathematics and Statistics, University of Houston-Downtown 1. Find an equation of the line that passes through the point ( 1, β2) and that is perpendicular to the line 2π₯ + 4π¦ = 1. 2. Find the inverse function of π(π₯) = 2π₯β5 π₯β3 , if it exists. 3. Find all three zeros (or roots) of the polynomial π(π₯) = π₯ 3 + π₯. 4. Solve for x: log(28 + π₯) = log(2 β π₯) + log(2 β π₯). 5. Solve for x: 73π₯+1 = 100. 5π₯+21 6. Find all horizontal and vertical asymptotes, if any, of π(π₯) = π₯ 2 +10π₯+25. 7. If π(π₯) = 3π₯ 2 + 1, simplify the following expression: π(π₯+β)βπ(π₯) β . 8. Identify the conic section given by the following equation and find its center: π₯ 2 + π¦ 2 β 4π₯ + 10π¦ + 25 = 0 9. Find the rectangular coordinates for the point whose polar coordinates are (β2, βπ/4). 10. Find functions π(π₯) and π(π₯) so that π(π(π₯)) = (2π₯ + 1)3. 11. A function f is given, and the indicated transformations are applied to its graph, in the given order. Write the equation for the final transformed function π(π₯). π(π₯) = π₯ 2 ; shift 2 units to the left and reflect in the x-axis. 12. Find the equation of the quadratic function π(π₯) whose graph has vertex (3,5) and y-intercept 23. 13. Find the value of b if πππ3 π = β2. 14. Find the sum of the first five terms of the arithmetic sequence whose first term is 25 and common difference is β2. 15. Find all possible rational zeros (or roots) of the polynomial π(π₯) = 2π₯ 3 β 7π₯ 2 + 10π₯ β 6. 16. Find the formula for a polynomial π of degree 3 with integer coefficients such that π(1) = β2 and both 1 β π and 3 are zeros (or roots) of the polynomial. 17. Find the value of tanβ1 (β1). 18. Find polar coordinates for the point whose rectangular coordinates are (0, β3). 19. Find the sum of the first four terms of the geometric sequence whose first term is 3 and common ratio is 2. 20. Find the equation of the circle that has a diameter with endpoints at (β3, β2) and (5, 4). 21. Write the complex number β1 + πβ3 in the polar form π(cos π + π β sin π). 22. Use DeMoivreβs Theorem to compute (β1 + πβ3)12 . 23. Find the two foci of the ellipse with equation π₯2 π¦2 + =1 36 25 24. Evaluate the expression 3 β(π 3 + 2π) π=0 25. A sequence is defined recursively by πΉπ = πΉπβ1 + πΉπβ2 . If πΉ0 = 0 and πΉ1 = 1, find πΉ6 . In problems 26-31, sketch the graph of the given function or equation. 26. π₯2 4 π¦2 + 16 = 1 27. π(π₯) = π₯ 2π₯β3 π₯β2 28. π¦ = π + 2 3, if π₯ β€ 0 29. π¦ = { 2 βπ₯ β 2, if π₯ > 0 30. π¦ = π₯ 3 + 2π₯ 2 β 5π₯ β 3 31. π¦ = log 1/2 (π₯ + 3) 32. A polynomial of degree 4 can have at most how many x-intercepts? 33. A polynomial of degree 4 can have at most how many local extrema? ANSWERS: 3π₯β5 1.2π₯ β π¦ = 4 2. π β1 (π₯) = 4. π₯ = β3 5. π₯ = 3 log 7 β 3 6. Horizontal asymptote is π¦ = 0. Vertical asymptote is π₯ = β5. 7. 6π₯ + 3β 8. A circle with center (2, β5). 9. (1, β1) 10. π(π₯) = π₯ 3 , π(π₯) = 2π₯ + 1 11. π(π₯) = β(π₯ + 2)2 12. π(π₯) = 2π₯ 2 β 12π₯ + 23 13. 1/9 14. 105 15. ±1, ±2, ±3, ±6, ± 2 , ± 2 16. π(π₯) = π₯ 3 β 5π₯ 2 + 8π₯ β 6 17. β 4 18. (3, 19. 45 20. (π₯ β 1)2 + (π¦ β 1)2 = 25 21. 2(πππ 22. 212 23. (β11, 0), (ββ11, 0) 24. 48 26. 27. 29. 30. 32. 4 33. 3 2 π₯β2 1 π 3. π₯ = 0, π₯ = ±π 1 25. 8 28. 31. 3π 2 3 ) 2π 3 + π β π ππ 2π 3 )