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Transcript
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, βˆ’2) and that is perpendicular to the
line 2π‘₯ + 4𝑦 = 1.
2. Find the inverse function of 𝑓(π‘₯) =
2π‘₯βˆ’5
π‘₯βˆ’3
, if it exists.
3. Find all three zeros (or roots) of the polynomial 𝑃(π‘₯) = π‘₯ 3 + π‘₯.
4. Solve for x: log(28 + π‘₯) = log(2 βˆ’ π‘₯) + log(2 βˆ’ π‘₯).
5. Solve for x: 73π‘₯+1 = 100.
5π‘₯+21
6. Find all horizontal and vertical asymptotes, if any, of π‘Ÿ(π‘₯) = π‘₯ 2 +10π‘₯+25.
7. If 𝑓(π‘₯) = 3π‘₯ 2 + 1, simplify the following expression:
𝑓(π‘₯+β„Ž)βˆ’π‘“(π‘₯)
β„Ž
.
8. Identify the conic section given by the following equation and find its center:
π‘₯ 2 + 𝑦 2 βˆ’ 4π‘₯ + 10𝑦 + 25 = 0
9. Find the rectangular coordinates for the point whose polar coordinates are (√2, βˆ’πœ‹/4).
10. Find functions 𝑓(π‘₯) and 𝑔(π‘₯) so that 𝑓(𝑔(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 𝑔(π‘₯).
𝑓(π‘₯) = π‘₯ 2 ; shift 2 units to the left and reflect in the x-axis.
12. Find the equation of the quadratic function 𝑓(π‘₯) whose graph has vertex (3,5) and y-intercept 23.
13. Find the value of b if π‘™π‘œπ‘”3 𝑏 = βˆ’2.
14. Find the sum of the first five terms of the arithmetic sequence whose first term is 25 and common
difference is βˆ’2.
15. Find all possible rational zeros (or roots) of the polynomial 𝑓(π‘₯) = 2π‘₯ 3 βˆ’ 7π‘₯ 2 + 10π‘₯ βˆ’ 6.
16. Find the formula for a polynomial 𝑓 of degree 3 with integer coefficients such that 𝑓(1) = βˆ’2 and
both 1 βˆ’ 𝑖 and 3 are zeros (or roots) of the polynomial.
17. Find the value of tanβˆ’1 (βˆ’1).
18. Find polar coordinates for the point whose rectangular coordinates are (0, βˆ’3).
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 (βˆ’3, βˆ’2) and (5, 4).
21. Write the complex number βˆ’1 + π‘–βˆš3 in the polar form π‘Ÿ(cos πœƒ + 𝑖 βˆ™ sin πœƒ).
22. Use DeMoivre’s Theorem to compute (βˆ’1 + π‘–βˆš3)12 .
23. Find the two foci of the ellipse with equation
π‘₯2 𝑦2
+
=1
36 25
24. Evaluate the expression
3
βˆ‘(π‘˜ 3 + 2π‘˜)
π‘˜=0
25. A sequence is defined recursively by 𝐹𝑛 = πΉπ‘›βˆ’1 + πΉπ‘›βˆ’2 . If 𝐹0 = 0 and 𝐹1 = 1, find 𝐹6 .
In problems 26-31, sketch the graph of the given function or equation.
26.
π‘₯2
4
𝑦2
+ 16 = 1
27. 𝑓(π‘₯) =
π‘₯
2π‘₯βˆ’3
π‘₯βˆ’2
28. 𝑦 = 𝑒 + 2
3, if π‘₯ ≀ 0
29. 𝑦 = { 2
βˆ’π‘₯ βˆ’ 2, if π‘₯ > 0
30. 𝑦 = π‘₯ 3 + 2π‘₯ 2 βˆ’ 5π‘₯ βˆ’ 3
31. 𝑦 = 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‘₯βˆ’5
1.2π‘₯ βˆ’ 𝑦 = 4
2. 𝑓 βˆ’1 (π‘₯) =
4. π‘₯ = βˆ’3
5. π‘₯ = 3 log 7 βˆ’ 3
6. Horizontal asymptote is
𝑦 = 0. Vertical asymptote
is π‘₯ = βˆ’5.
7. 6π‘₯ + 3β„Ž
8. A circle with center
(2, βˆ’5).
9. (1, βˆ’1)
10. 𝑓(π‘₯) = π‘₯ 3 ,
𝑔(π‘₯) = 2π‘₯ + 1
11. 𝑔(π‘₯) = βˆ’(π‘₯ + 2)2
12. 𝑓(π‘₯) = 2π‘₯ 2 βˆ’ 12π‘₯ + 23
13. 1/9
14. 105
15. ±1, ±2, ±3, ±6, ± 2 , ± 2
16.
𝑓(π‘₯) = π‘₯ 3 βˆ’ 5π‘₯ 2 + 8π‘₯ βˆ’ 6
17. βˆ’ 4
18. (3,
19. 45
20.
(π‘₯ βˆ’ 1)2 + (𝑦 βˆ’ 1)2 = 25
21. 2(π‘π‘œπ‘ 
22. 212
23. (√11, 0), (βˆ’βˆš11, 0)
24. 48
26.
27.
29.
30.
32. 4
33. 3
2
π‘₯βˆ’2
1
πœ‹
3. π‘₯ = 0, π‘₯ = ±π‘–
1
25. 8
28.
31.
3πœ‹
2
3
)
2πœ‹
3
+ 𝑖 βˆ™ 𝑠𝑖𝑛
2πœ‹
3
)