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2015 Math III End of Year Review Linear Write the slope-intercept form of the equation. 1. Verbal Description 3. When x is increased by twice y, the result is 4. 4. The opposite of half the value of x, when increased by 1 results in the value of y. Absolute Values 5. π¦ = |π₯ + 1| 6. π¦ = β|βπ₯ β 2| + 1 Symbolic 2. through (β2, β3); slope = 4 Graph Piecewise π(π₯) = { π₯ + 3, 2π₯, 7. π(2) ππ π₯ β€ 0 ππ π₯ > 0 8. π(β4) 9. π(0) 1 10. π(2) 11. Graph Quadratic 12. A) What is the factored form of π₯ 2 + 9? 12.B) What about factored form of 18x2 +27x +10 Composition of Functions You have a coupon for 20% off a meal at your favorite restaurant. You also have a $15 gift card to the restaurant. 13. Write a function representing the cost, x, of the meal just with the coupon. 14. Write a function representing the cost, x, of the meal just with the gift card. 15. Does it save you more money if the coupon is applied first or if the gift card is applied first? How much is the savings? Polynomials 16. Look at the expression 3ππ β ππ + π + π(π β π) to identify its elements. Coefficients: Constant: Factor: Terms: 17. An equation is shown π(π‘ 2 + 1) β π₯ = π₯(π‘ 3 β π‘ 2 + 2π‘ β 1) Does the function π depend on the variable π₯? a. Yes, the function π is dependent on π₯. b. No, the function π is not dependent on π₯. c. It is impossible to determine whether or not π is dependent on π₯. 18. The polynomial π(π₯) contains the points (3,0) and (8,1). Which statement is true according to the Remainder Theorem? a. (π₯ + 3) is a factor of π(π₯). b. (π₯ β 3) is a factor of π(π₯). c. (π₯ + 8) is a factor of π(π₯). d. (π₯ β 8) is a factor of π(π₯). Simplify the following polynomials. 19. (8π₯ 2 + 3π₯ + 4) + (5π₯ 2 β 6π₯ β 8) 20. (5π₯ 2 β 6π₯ + 1) β (π₯ 2 β 5π₯ + 3) 21. 2π₯(π₯ 2 β π₯ + 5) β 5π₯(3π₯ β 2) 22. (2π + 1)(π 2 + 7π β 9) 23. Fluffy Pet Products sells a pet toy that brings in revenue represented by the function π(π₯) = β0.05π₯ 2 + 100π₯ + 100, where x is the number of pet toys. The production cost for the pet toy is represented by the function π(π₯) = 0.42π₯ + 50. Which function represents the profit Fluffy Pet Products earns on the pet toy? a. b. c. d. 0.05π₯ 2 β 99.58π₯ β 50 β0.47π₯ 2 + 50π₯ + 100 β0.05π₯ 2 + 99.58π₯ + 50 β0.05π₯ 2 + 99.58π₯ + 150 Find the quotient and the remainder and state if the given divisor is a factor of the polynomial. 24. (π4 β 1) ÷ (π β 1) 25. Use the graph below to answer the following questions: a. At what x-value does the function have a relative maximum? b. At what x-value does the function have a relative minimum? c. Along what interval is the function decreasing? d. Along what intervals is the function increasing? e. Is the degree of this function even or odd? How can you tell? f. Is the leading coefficient of this function positive or negative? How can you tell? 26. Graph the following: π(π₯) = (π₯ β 4)2 (π₯ + 2) 27. How many solutions to the equation π₯ 3 β 1 = 0 are imaginary? 28. Which function is an even function? a. π(π₯) = 4π₯ b. π(π₯) = π₯ 2 + 2π₯ + 1 c. π(π₯) = π₯ 3 + π₯ d. π(π₯) = 5π₯ 4 + 3π₯ 2 + 1 Average Rate of Change 29. The table represents the number of people who held bookstore memberships in 2005 and 2008. Years Number of Members 2005 330,000 2008 480,000 What is the average rate of change per year? a. 37,500 per year b. 50,000 per year c. 150,000 per year d. 270,000 per year Inverse Functions 30. What is the inverse of the function π(π₯) = 4π₯ 2 for π₯ β₯ 0? 1 a. π β1 (π₯) = 2 βπ₯ π₯ b. π β1 (π₯) = 4 π₯ c. π β1 (π₯) = 2 1 d. π β1 (π₯) = ± 2 βπ₯ Radicals Sketch the graph and state the domain and range. 31. π(π) = ββπ β π + π 32. Solve the equation βπ + π β ππ = βπ 33. Solve the equation βππ β π = βπ Rational ππ 34. Simplify π+π ÷ π (πβπ)(π+π) (π+π)π π 35. Simplify π+π β ππ+π 36. Solve ππ βππβππ ππ βπβππ πβπ = ππ βπβππ + π βπ 37. Identify the vertical asymptotes of π(π) = ππ βππ+π 38. Graph π(π) = π πβπ +π Trigonometry 39. Solve the triangle and find the area. 40. Solve the triangle and find the area. 41. Identify the function graphed below. a. sin(2π₯) 1 b. sin (2 π₯) c. 2 cos(π₯) d. cos(2π₯) 5π 42. What is the value of sin ( 2 )? 1 π 43. Identify the amplitude, period, midline, and phase shift π(π₯) = 2 sin[4 (π₯ β 4 )] β 1 Logarithmic 44. Which represents the solution to 5 β 32π₯ = 20? a. log 3 4 b. 2 log 3 2 1 c. 2 log 4 d. 1 2 log 3 4 45. The population growth of a city can be modeled by the equation π = 130,000 π 0.09π‘ , where π is the population in thousands and π‘ is years since 1995. In what year does the model predict that the city reaches a population of approximately 600,000 people? a. 2012 b. 2014 c. 2016 d. 2017 46. Graph and identify the domain and range π(π₯) = log(π₯) + 1 Expand each logarithm. π₯2 47. log 8 (π¦ 6 ) 48. ln(π₯βπ¦) Condense each expression to a single logarithm. log 6 50. ln 8 β 2 ln 2 49. 2 log 3 7 + 33 51. Sally currently pays a $3,034 premium for health insurance. If the premium increases at an annual rate of 9.2% per year, how many years will it take for the premium to be $7,315.47? Critical Thinking 52. The sum of Vinceβs age and Emyβs age is 34. Emy is 2 years less than twice Vinceβs age. Which equation represents this situation if π£ represents Vinceβs age? a. (2 β 2π£) + π£ = 34 b. (2 β 2π£) β π£ = 34 c. (2π£ β 2) + π£ = 34 d. (2π£ β 2) β π£ = 34 53. An equation is shown. β(56π¦ 3 π₯ 4 )0 + 3π₯ + 4 = π₯ + π¦ Solve for π¦. 54. A function is shown. Which word describes the graph of this function? 52 β π₯ 3 β (3π₯ 2 )3 π¦= 125 β π₯ 6 β (2π₯)2 a. Exponential b. Linear c. Logarithmic d. Quadratic 55. An employer uses a function to calculate each employeeβs salary. The input is the number of hours and the output is the salary. Since the employeesβ time clock is in 15-minute intervals, what is a good domain for this function? 1 a. Non-negative multiples of 4 b. Non-negative rational numbers c. Non-negative multiples of 15 d. Non-negative integers