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# Midterm Study Guide

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### Midterm Study Guide

1. 1. PrecalculusMidterm Study GuideYou may use your Unit Circle and Chart.1. Find the sine, cosine tangent, cosecant, secant, and cotangent of θ. 15 9 θ3. The angle of elevation from the end of the shadowto the top of the building is 70° and the distance is 180 feet.a) Find the height of the building to the nearest foot.b) Find the length of the shadow to the nearest foot. 154. Find csc (sin-1 /17). 215. Find tan (cos -1 /29).6. Solve the triangle ABC if A = 120°, b = 10, and c = 5. (Find B, C, and a).7. Solve the triangle ABC if A = 135°, b = 8, and c = 10. (Find B, C, and a).8. State the Midline, Amplitude, Period, and Phase Shift of y = sin ( x /3 - π/2). Then graph the function. x π9. Find the asymptotes of y = csc ( /3 - /2). Then graph the function.10. State the Midline, Amplitude, Period, and Phase Shift of x y = tan ( /2 - π) + 2. Find the asymptotes, and then graph the function.11. State the Midline, Amplitude, Period, and Phase Shift of x y = 2 cos ( /2 - π) – 1. Then graph the function. x12. Find the asymptotes of y = 2 sec ( /2 - π) – 1. Then graph the function.13. Find cot [sin-1 (-√3/2) + cos-1 (1/2)].14. Simplify 1 – cos2 x 1 – sin2 x15. If csc x cos x = 6, find the value of tan x.16. If sec x sin x = 2, find the value of cot x.17. Find tan (α - β) if sin α = 12/13 and cos β = 3/5.
2. 2. Note: tan (α ± β) = tan α ± tan β . 1  tan α tan β18. If csc θ = 5/3, find sin 2θ. Note: sin 2θ = 2 sin θ cos θ.19. Find sin π/8. Note: sin α/2 = ±√(1 – cos α) √220. Verify sin x cos x tan x + cos2 x = 1.21. Verify (sin x – 1)(tan x + sec x) = – cos x.22. Give the graphs of y = arcsin x, y = arccos x, and y = arctan x.23. a) Graph the equation y = (x + 2)2 – 1. b) What is the domain and range of the relation? c) Is the relation a function? d) Identify maxima, minima, or inflection points.24. a) Graph the equation y = (x + 2)2. b) What is the domain and range of the relation? c) Is the relation a function? d) Identify maxima, minima, or inflection points.25. a) Find the inverse of f(x) = (x + 2)2 – 1. b) Graph the inverse. c) Is the inverse a function?26. Write an equation of the line perpendicular to the graph of 5x – 3y + 6 = 0 thatpasses through the point (3, 2).27. Write an equation of the line parallel to the graph of 4y + 8 = x that passesthrough the point (2, 3).
3. 3. 28. Solve the system of equations algebraically. 3x – 4y = 10 y=¾x–5 2x + 3y = 19 -3x + 4y = 8 y = 4x – 18 7x – y = 929. a) If A = 4 7, find the determinant of A. -1 3 b) Find the inverse of A, if it exists.30. a) If A = 4 7 and B = 3 5, find A – B if possible. -1 3 -2 1 b) Find BA if possible (Go Rebels!).31. a) If A = 4 7, find AA. -1 3 b) If B = 3 -1 2, can (AA)B be multiplied? If so, multiply. -1 2 132. Give the equation for y = | x | translated 3 units down and reflected over the x-axis.33. Give the equation for the graph of y = x2 reflected over the x-axis and translated 2 units up.34. Graph f(x) ≥ – x2 + 2.35. Graph f(x) ≥ (x + 2)3 – 3.36. The function f(x) = (x3 + 1) has a vertical asymptote at ___________________ (x2 – 1) and a slant asymptote at _____________________________.37. Write an equation of a function with a vertical asymptote at x = 2 and a hole at x = – 1.38. f(x) = 1 has a vertical asymptote at ________________ and a horizontal (x + 3)
4. 4. asymptote at __________________.39. a) Write an equation of a function with roots 5, 2, 2i, and – 2i. b) How many times does the graph cross the x-axis?40. a) Write a polynomial equation with roots 4, 1 + 2i, 1 – 2i. b) How many times does the graph cross the x-axis?41. Factor the quadratic 4x2 + 6x = 3.42. Find all the factors of x3 + 2x2 – x – 2 = 0 if one of the factors is x + 1.43. a) Find the remainder of (2x3 – x2 + x – 2) ÷ (x + 3). b) Is (x + 3) a factor of (2x – x2 + x – 2)?44. Find the value of b so that the remainder of (2x3 – x2 + x + b) ÷ (x – 1) is 0.45. Find the value of k so that the remainder of (x3 + kx2 – x – 7) ÷ (x + 1) is 0.46. Solve the radical equation. 9+√x–1=147. Solve the rational equation. 1 = x+3. x 2x248. Factor the function into partial fractions 7x + 1 . 2 x + 2x – 3