State whether each sentence is true or false. If false, replace the underlined word, phrase, expression, or number to make a true sentence. 1. A triangle with sides having measures of 3, 4, and 6 is a right triangle. If a, b, and c are the sides of a right triangle, then c 2 = a 2 + b 2, where c is the greatest number. Since 6 2 = 36 and 3 2 + 4 2 = 25, 6 2 3 2 + 4 2. Thus, a triangle with sides having measures of 3, 4, and 6 is not a right triangle. So, the statement is false. Since 3 2 + 4 2 = 25 and 5 2 = 25, a triangle with sides having measures of 3, 4, and 5 would be a right triangle. 2. The expressions and are equivalent. false; 3. The expressions 2 + and are conjugates. Binomials of the form true. and and are called conjugates. So, the binomials are conjugates. The statement is 4. In the expression 5, the radicand is 2. The expression under the radical sign is called the radicand. In the expression, the radicand is 2. The statement is true. 5. The shortest side of a right triangle is the hypotenuse. In a right triangle, the side opposite the right angle is the hypotenuse. This side is always the longest. So, the statement is false. The longest side of a right triangle is the hypotenuse. 6. The cosine of an angle is found by dividing the measure of the side opposite to the angle by the hypotenuse. The sine of an angle is found by dividing the measure of the side opposite of the angle by the hypotenuse. So, the statement is false. The cosine of an angle is found by dividing the measure of the side adjacent to the angle by the hypotenuse. 7. The domain of the function is. The domain of the function the statement is false. is {x x 0}. So, 8. After the first step in solving = x + 5, you would have 2x + 4 = x 2 + 10x + 25. To solve = x + 5, you first need to square each side of the equation. After the first step in solving = x + 5, you would have 2x + 4 = x 2 + 10x + 25. The statement is true. 9. The converse of the Pythagorean Theorem is true. If c 2 a 2 + b 2, then the triangle is not a right triangle. So, the converse of the Pythagorean Theorem is true. The statement is true. 10. The range of the function is. The range of the function the statement is false. is {y y 0}. So, esolutions Manual - Powered by Cognero Page 1
Graph each function. Compare to the parent graph. State the domain and range. 11. y = + 3 Make a table. x 0 0.5 1 2 3 4 y 3 3.7 4 4.4 4.7 5 12. y = + 2 Make a table. x 0 0.5 1 2 3 4 y 2 2.7 3 3.4 3.7 4 Plot the points on a coordinate system and draw a smooth curve that connects then. Plot the points on a coordinate systems and draw a smooth curve that connects them. The value 3 is being added to the parent function, so the graph is translated up 3 units from the parent graph. Another way to identify the translation is to note that the y-values in the table are 3 more than the corresponding y-values for the parent function. The domain is {x x 0} and the range is {y y 3}. The value 2 is being added to the parent function, so the graph is translated up 2 units from the parent graph. Another way to identify the translation is to note that the y-values in the table are 2 greater than the corresponding y-values for the parent function. The domain is {x x 0} and the range is {y y 2}. esolutions Manual - Powered by Cognero Page 2
13. y = 5 x 0 0.5 1 2 y 0 3.5 5 7.1 15. y = x 1 1.5 2 3 4 y 0 0.7 1 1.4 1.7 The parent function is multiplied by a value less than 1, so the graph is a vertical stretch of and a reflection across the x-axis. Another way to identify the stretch is to notice that the y- values in the table are 5 times the corresponding y- values for the parent function. The domain is {x x 0} and the range is {y y 0}. 14. y = 6 x 0 0.5 1 2 3 4 y 6 5 4 5.3 4.6 4.3 The value 1 is being subtracted from the square root of the parent function, so the graph is translated 1 unit right from the parent graph. Another way to identify the translation is to note that the x-values in the table are 1 more than the corresponding x-values for the parent function. The domain is {x x 1}, and the range is {y y 0}. 16. y = + 5 x 0 0.5 1 2 3 4 y 5 5.7 6 6.4 6.7 7 The value 6 is being subtracted from the parent function, so the graph is translated down 6 units from the parent graph. Another way to identify the translation is to note that the y-values in the table are 6 less than the corresponding y-values for the parent function. The domain is {x x 0} and the range is {y y 6}. The value 5 is being added to the parent function, so the graph is translated up 5 units from the parent graph. Another way to identify the translation is to note that the y-values in the table are 5 greater than the corresponding y-values for the parent function. The domain is {x x 0} and the range is {y y 5}. esolutions Manual - Powered by Cognero Page 3
17. GEOMETRY The function s = can be used to find the length of a side of a square given its area. Use this function to determine the length of a side of a square with an area of 90 square inches. Round to the nearest tenth if necessary. 22. The side length of the square is about 9.5 inches. 23. 18. Simplify. 24. 19. 20. 21. esolutions Manual - Powered by Cognero Page 4
25. 27. 28. WEATHER To estimate how long a thunderstorm will last, use, where t is the time in hours 26. and d is the diameter of the storm in miles. A storm is 10 miles in diameter. How long will it last? Substitution 10 for d. 29. The storm will last about 2.15 hours. To convert 2.15 hours to hours and minutes, multiply the number of minutes in an hour by the decimal part. Because 60 0.15 = 9, 2.15 hours is equal to 2 hours and 9 minutes. Simplify each expression. esolutions Manual - Powered by Cognero Page 5
30. 31. 36. MOTION The velocity of a dropped object when it hits the ground can be found using, where v is the velocity in feet per second, g is the acceleration due to gravity, and d is the distance in feet the object drops. Find the speed of a penny when it hits the ground, after being dropped from 984 feet. Use 32 feet per second squared for g. Substitute 32 for g and 984 for d. 32. The speed of the penny when it hits the ground is about 250.95 feet per second. 33. 37. Solve each equation. Check your solution. 34. Because the square root of a number cannot be negative, there is no solution. 35. esolutions Manual - Powered by Cognero Page 6
38. 40. Check: Check: 41. 39. Check: Check: Because 5 does not satisfy the original equation, 12 is the only solution. esolutions Manual - Powered by Cognero Page 7
42. Determine whether each set of measures can be lengths of the sides of a right triangle. 44. 6, 8, 10 Since the measure of the longest side is 10, let c = 10, a = 6, and b = 8. Then determine whether c 2 = Check: Yes, because c 2 = a 2 + b 2, a triangle with side lengths 6, 8, and 10 is a right triangle. 45. 3, 4, 5 Because 10 does not satisfy the original equation, 5 is the only solution. 43. FREE FALL Assuming no air resistance, the time t in seconds that it takes an object to fall h feet can be Since the measure of the longest side is 5, let c = 5, a = 3, and b = 4. Then determine whether c 2 = a 2 + b 2. determined by. If a skydiver jumps from an airplane and free falls for 10 seconds before opening the parachute, how many feet does she fall? Substitute 10 for t. Yes, because c 2 = a 2 + b 2, a triangle with side lengths 3, 4, and 5 is a right triangle. 46. 12, 16, 21 Since the measure of the longest side is 21, let c = 21, a = 12, and b = 16. Then determine whether c 2 = The skydiver will fall 1600 feet before opening the parachute. No, because c 2 a 2 + b 2, a triangle with side lengths 12, 16, and 21 is not a right triangle. esolutions Manual - Powered by Cognero Page 8
47. 10, 12, 15 Since the measure of the longest side is 15, let c = 15, a = 10, and b = 12. Then determine whether c 2 = 50. 5, 12, 13 Since the measure of the longest side is 13, let c = 13, a = 5, and b = 12. Then determine whether c 2 = No, because c 2 a 2 + b 2, a triangle with side lengths 10, 12, and 15 is not a right triangle. 48. 2, 3, 4 Since the measure of the longest side is 4, let c = 4, a = 2, and b = 3. Then determine whether c 2 = a 2 + b 2. Yes, because c 2 = a 2 + b 2, a triangle with side lengths 5, 12, and 13 is a right triangle. 51. 15, 19, 23 Since the measure of the longest side is 23, let c = 23, a = 15, and b = 19. Then determine whether c 2 = No, because c 2 a 2 + b 2, a triangle with side lengths 2, 3, and 4 is not a right triangle. 49. 7, 24, 25 Since the measure of the longest side is 25, let c = 25, a = 7, and b = 24. Then determine whether c 2 = No, because c 2 a 2 + b 2, a triangle with side lengths 15, 19, and 23 is not a right triangle. 52. LADDER A ladder is leaning on a building. The base of the ladder is 10 feet from the building, and the ladder reaches up 15 feet on the building. How long is the ladder? Use the Pythagorean Theorem, substituting 10 for a and 15 for b. Yes, because c 2 = a 2 + b 2, a triangle with side lengths 7, 24, and 25 is a right triangle. The ladder is approximately 18.0 feet long. esolutions Manual - Powered by Cognero Page 9
Find the values of the three trigonometric ratios for angle A. 55. RAMPS How long is the ramp? 53. You know the measure of the side opposite the angle and the measure of the angle. Use the sine ratio. The ramp is 6 feet long. 54. esolutions Manual - Powered by Cognero Page 10