Math Algebra Chapter 2 Solution

Advanced Math Midterm Review Name: Key! Chapter 2 Find the distance between the set of points. 1. (6, 4) and (-8, 11) d = √((-8-6)² + (11-4)²) = √196 + 49 = √245 = √49 * √5 = 7√5 2. (40, 32) and (36, 20) d = √((40-36)² + (32-20)²) = √16 + 144 = √160 = √16 * √10 = 4√10 Find the midpoint between the set of points. 3. (6, 4) and (-8, 11) M = ((6+(-8))/2, (4+11)/2) = (-1, 7.5) 4. (40, 32) and (36, 20) M = ((40+36)/2, (32+20)/2) = (38, 26) 5. (5, 2) and (-6, 9) M = ((5+(-6))/2, (2+9)/2) = (-1/2, 11/2) 6. (3, 5) and (-6, 2) M = ((3+(-6))/2, (5+2)/2) = (-3/2, 7/2) State whether the equation defines y as a function of x. 7. 2x + 4y = 8 Yes, function 8. y = x² Yes, function 9. y = ±x Not a function 10. y = |x+3| Yes, function State whether the set of ordered pairs defines y as a function of x. 11. (5, 2), (3, 7), (4, 3), (5, 6) Not a function 12. (6, 3), (-7, 3), (8, 3), (9, 3) Yes, function (ok for y's to repeat!) Evaluate the function. 13. Given f(x) = √(x² + 5) a. f(-2) = √((-2)² + 5) = √9 = 3 b. f(0) = √(0² + 5) = √5 c. f(3) = √(3² + 5) = √14 14. Given f(x) = 3x² + 1 a. f(-2) = 3(-2)² + 1 = 13 b. f(0) = 3(0)² + 1 = 1 c. f(3) = 3(3)² + 1 = 28 CH.2 Write each interval as an inequality and graph on a number line. 15. (-2, 5) -2 < x < 5 [Image of a number line with open circles at -2 and 5, and a line between them] 16. (3, ∞) x > 3 [Image of a number line with an open circle at 3 and a line extending to the right] 17. (-∞, 6] x ≤ 6 [Image of a number line with a closed circle at 6 and a line extending to the left] 18. [-1, 1] -1 ≤ x ≤ 1 [Image of a number line with closed circles at -1 and 1, and a line between them] Write each inequality in interval notation. 19. -4 < x ≤ 1 (-4, 1] 20. x ≤ 3 or x ≥ 6 (-∞, 3] U [6, ∞) 21. -8 < x < -2 (-8, -2) 22. -3 ≤ x ≤ 2 [-3, 2] 23. x < 6 (-∞, 6) 24. -5 < x < ∞ (-5, ∞) Determine the domain of the function represented by the given equation. Write your answer in INTERVAL NOTATION. 25. f(x) = 2x + 5 (-∞, ∞) 26. f(x) = x² - 5x (-∞, ∞) 27. f(x) = √(6 + x) 6 + x ≥ 0 x ≥ -6 [-6, ∞) 28. f(x) = √(x - 3) x - 3 ≥ 0 x ≥ 3 [3, ∞) 29. f(x) = 4 / (x - 6) x - 6 ≠ 0 x ≠ 6 (-∞, 6) U (6, ∞) 30. f(x) = 7 / (x + 8) x + 8 ≠ 0 x ≠ -8 (-∞, -8) U (-8, ∞) Find the real value or values of a in the domain of f for which f(a) equals the given number. 31. f(x) = 3x - 2; f(a) = 10 10 = 3a - 2 12 = 3a a = 4 32. f(x) = 2 - 5x; f(a) = 7 7 = 2 - 5a 5 = -5a a = -1 33. f(x) = x² + 2x - 2; f(a) = 1 1 = a² + 2a - 2 0 = a² + 2a - 3 0 = (a + 3)(a - 1) a = -3, 1 34. f(x) = |x + 2|; f(a) = 6 6 = |a + 2| 6 = a + 2 or -6 = a + 2 4 = a or -8 = a a = -8, 4 CH.2 Find the zeros, or x-intercepts, of f(x). 35. f(x) = 3x + 2 0 = 3x + 2 -2/3 = x 36. f(x) = 6 + 2x 0 = 6 + 2x -6 = 2x -3 = x 37. f(x) = x² - 5x - 24 0 = (x - 8)(x + 3) x - 8 = 0 or x + 3 = 0 x = 8, -3 38. f(x) = x² + 7x - 21 0 = (x + 7)(x - 3) x + 7 = 0 or x - 3 = 0 x = -7, 3 Find the slope of the line through the given points. 39. (8, -2) and (-4, 10) m = (10 - (-2))/(-4 - 8) = 12/-12 = -1 40. (-2, -7) and (4, -9) m = (-9 - (-7))/(4 - (-2)) = -2/6 = -1/3 Graph y as a function of x by finding the slope and y-intercept. 41. y = -3x + 5 Slope: -3 y-intercept: 5 [Image of a line with slope -3 and y-intercept 5] 42. y = 4x - 2 Slope: 4 y-intercept: -2 [Image of a line with slope 4 and y-intercept -2] 43. 2x - 3y = 12 -3y = -2x + 12 y = (2/3)x - 4 Slope: 2/3 y-intercept: -4 [Image of a line with slope 2/3 and y-intercept -4] 44. -3x + 5y = 15 5y = 3x + 15 y = (3/5)x - 3 Slope: 3/5 y-intercept: -3 [Image of a line with slope 3/5 and y-intercept -3]