2PreCalc CHAPTER P: MIDTERM REVIEW Chapters P, 1, 2, and 3 1. Find the slope of the line that passes through the points: a. (-­‐2, 4) and (-­‐4, -­‐4) b. (1, 3) and (1, -­‐4) c. (5, -­‐1) and (3, -­‐1) 2. Write the equation of the line, in slope-­‐intercept form, that passes through: !
a. (6, -­‐2) and has a slope of − ! b. (1, -­‐4) and has a slope of −3 3. Write the equation of the line, in slope-­‐intercept form, that passes through: a. the points (2, -­‐1) and (-­‐1, 5) b. the points (8, 0) and (4, -­‐8) 4. Write the equation of the line, in slope-­‐intercept form, that passes through (-­‐1, 0) and is parallel to the line 2𝑥 + 𝑦 = 3. 5. Write the equation of the line, in slope-­‐intercept form, that passes through (3, -­‐4) and is perpendicular to the line 2𝑥 + 𝑦 = 3. !
6. Which is the greater slope − ! or 1? And why? 7. Simplify: a. −5 b. −36 c. −16 ∙ −20 d. 3𝑖 + 10𝑖 ! 2PreCalc e. 2 − 3𝑖 − 3 − 4𝑖 MIDTERM REVIEW f. (2 + 𝑖)(1 − 2𝑖) Chapters P, 1, 2, and 3 g. (2 − 3𝑖)! CHAPTER 1: 8. Determine if the following sets of ordered pairs represents a function: a. {(2, 7), (3, 8), (4, 9), (4, 10)} b. {(-­‐3, 1), (-­‐3, 2), (-­‐3, 8)} c. {(1, 8), (-­‐1, 5), (2, 1)} 9. Given f ( x) = 2 x3 − 3 and g ( x) = x + 1 determine: a. f(-­‐3) b. g(2) c. f(g(8)) f
d. (f+g)(3) e. (f • g)(0) f. ( ) (3) g
10. Determine if the following relations are functions, then find the domain and range of each: a. 𝑦 = −3𝑥 + 2 b. 𝑦 = −2𝑥 ! + 3 c. 𝑦 = 3𝑥 ! − 2𝑥 !!!
d. 𝑦 = 𝑥 + 1 − 2 e. 𝑦 = 𝑥 − 3 f. 𝑦 = !!! 11. Determine any intervals that increase, decrease, or are constant. Then list any (relative extrema) min or max points. Last, determine if the function is odd, even, or neither. a. 𝑦 = −𝑥 ! + 4 b. 𝑦 = 2𝑥 ! − 3𝑥 2PreCalc Chapters P, 1, 2, and 3 MIDTERM REVIEW 12. Name the parent function and its equation: a) b) c) d) e) 13. Identify the common function f and describe the sequence of transformations from f (x) to h(x). 1
a. y = ( x + 2)2 + 7 b. y = − x + 3 c. y = x + 2 2
d. y = x + 1 − 7 e. y = 4 x 2 − 7 f. 𝑦 = 5 + (𝑥 − 5)! Describe transformations: 14. Parent function: 𝑓(𝑥) = 𝑥 15. Parent function: 𝑓(𝑥) = 𝑥 ! a. 𝑔 𝑥 = − 𝑥 − 4 + 2 a. 𝑔 𝑥 = 2𝑥 ! − 1 !
b. 𝑔 𝑥 = − ! 𝑥 b. 𝑔 𝑥 = −(𝑥 + 2)! − 3 X 2 4 6 1 Find the inverse of each of the following: Y -­‐1 -­‐3 -­‐5 0 16. {(-­‐1, 1), (3, 6), (-­‐2, 4)} 17. 2PreCalc MIDTERM REVIEW Chapters P, 1, 2, and 3 Use the Horizontal Line Test to determine if the following have inverses. If they pass the HLT, find the inverse function, 𝑓 !! (𝑥): 18. 𝑓 𝑥 = 3𝑥 − 1 19. 𝑓 𝑥 = 2 𝑥 + 1 20. 𝑓 𝑥 = (𝑥 + 1)! − 3 21. 𝑓 𝑥 = (𝑥 − 2)! + 2 Are the following inverse functions? !
22. 𝑓 𝑥 = ! 𝑥 − 4 𝑎𝑛𝑑 𝑔 𝑥 = 4𝑥 + 16 Chapter 2: 23. Find the vertex, the direction of opening, and the axis of symmetry for each quadratic function. a. 𝑓 𝑥 = −2(𝑥 + 1)! − 1 b. 𝑓 𝑥 = (𝑥 + 4)! c. 𝑓 𝑥 = −𝑥 ! + 5 24. Write the equation for the quadratic function whose graph contains the given vertex and point. a. Point (-­‐1, 3) Vertex (0, 2) b. Point (-­‐5, 2) Vertex (-­‐1, 4) 25. Describe the end behavior of the polynomial using: lim 𝑓 𝑥 𝑎𝑛𝑑 lim 𝑓 𝑥 a) 𝑓 𝑥 = −2𝑥 ! − 2𝑥 !→!!
!→!
b) 𝑓 𝑥 = 4𝑥 ! + 2𝑥 ! − 6 c) 𝑓 𝑥 = 𝑥 + 5 2PreCalc MIDTERM REVIEW Chapters P, 1, 2, and 3 26. Find the zeros of the function algebraically. a. 𝑓 𝑥 = 𝑥 ! − 2𝑥 ! b. 𝑓 𝑥 = −2𝑥 ! + 12𝑥 ! − 16𝑥 ! 27. Divide by synthetic division: a. 𝑓 𝑥 = 2𝑥 ! − 7𝑥 ! + 4𝑥 + 5; 𝑥 − 3 b. 𝑓 𝑥 = 𝑥 ! + 3𝑥 ! + 𝑥 ! − 3𝑥 + 3; 𝑥 + 2 28. Is the 1st polynomial a factor of the 2nd? a. 𝑥 − 2; 𝑥 ! − 4𝑥 ! + 8𝑥 − 8 b. 𝑥 + 3: 𝑥 ! + 2𝑥 ! − 4𝑥 − 2 29. Is the integer a zero of the polynomial? a. Is -­‐2 a zero of 2𝑥 ! − 6𝑥 ! − 12𝑥 + 16? B. Is 3 a zero of 2𝑥 ! + 6𝑥 ! − 26𝑥 − 30? 30. Write an equation of the polynomial function, given the roots: a. 3, -­‐1 b. 5, − 5, 0 31. List all possible rational roots. a. 𝑓 𝑥 = 3𝑥 ! − 5𝑥 ! + 3𝑥 ! − 7𝑥 + 2 b. 𝑓 𝑥 = 4𝑥 ! − 2𝑥 ! + 𝑥 − 24 2PreCalc Chapters P, 1, 2, and 3 MIDTERM REVIEW For 32 – 34, find all vertical asymptotes, horizontal asymptotes, and any holes. 32. 𝑦 =
!!
!!!
33. 𝑦 =
!!!!
!!!!
34. 𝑦 =
!!!
! ! !!
Chapter 3: Determine if the following is an exponential growth or exponential decay. Then find the constant percentage rate (r): 35. 𝑓 𝑥 = 20(1.025)! 36. 𝑓 𝑥 = 2(.75)! 37. 𝑓 𝑥 = (3)! Expand. 38. 𝑙𝑜𝑔! 𝑥 ! 𝑦 ! 39. ln (!!!)!
!
Condense. 40. 𝑙𝑜𝑔! 𝑦 − 2𝑙𝑜𝑔! 𝑥 41. 𝑙𝑜𝑔! 13 + 𝑙𝑜𝑔! 𝑥 Simplify, using properties of logs. 42. 𝑙𝑜𝑔! 32 43. 𝑙𝑜𝑔 10 44. 𝑙𝑜𝑔! 27 2PreCalc !
MIDTERM REVIEW Chapters P, 1, 2, and 3 45. 𝑙𝑜𝑔! !"# 46. 11!"!!! (.!) 47. 5!"!! ! Using the calculator change of base formula, simplify. 48. log 4 9 49. log12 200 Solve for x. Use any method. 50. 2! = 256 51. 𝑙𝑜𝑔! 81 = 4 52. 𝑒 !!!! = 51 53. 4 12! = 190 54. 2𝑙𝑛4𝑥 = 15 55. 𝑙𝑜𝑔! 𝑥 − 𝑙𝑜𝑔! 8 − 5𝑥 = 2 Find the exponential function that satisfies the given conditions: 56. Initial value = 225, increasing at a rate of 5.3% per year 57. Initial population = 45 grams, half-­‐life of 15 years