Chapter 5. Quadratic Functions. Solutions to Selected Odd Problems

Chapter 5: Quadratic Functions Solutions to Selected Odd Problems Section 5.6 Section 5.6: Quadratic Inequalities #1 – 12: Use the graph of f(x) to solve a) f(x) = 0 b) f(x) ≥ 0 c) f(x) ≤ 0 d) f(x) > 0 e) f(x) < 0 1) a) f(x) = 0 answer x = -4,3 (x coordinates of x-intercepts) Intervals for b / c (−∞, −4] 𝑎𝑛𝑑 [−4,3]𝑎𝑛𝑑 [3, ∞) b) f(x) > 0 pick intervals where graph is above x-axis (−∞, −4) ∪ (3, ∞) e) f(x) < 0 pick intervals where graph is below x-axis (-4,3) c) f(x) ≥ 0 pick intervals where graph is above or touching the x-axis (−∞, −4] ∪ [3, ∞) e) f(x) ≤ 0 pick intervals where graph is below or touching the x-axis [−4,3] Intervals for d / e (−∞, −4)𝑎𝑛𝑑 (−4,3)𝑎𝑛𝑑 (3, ∞) 90 Chapter 5: Quadratic Functions Solutions to Selected Odd Problems 3) a) f(x) = 0 answer x = -4,3 (x coordinates of x-intercepts) Intervals for b / c (−∞, −4] 𝑎𝑛𝑑 [−4,3]𝑎𝑛𝑑 [3, ∞) b) f(x) > 0 pick intervals where graph is above x-axis (-4,3) c) f(x) < 0 pick intervals where graph is below x-axis (−∞, −4) ∪ (3, ∞) d) f(x) ≥ 0 pick intervals where graph is above or touching the x-axis [−4,3] e) f(x) ≤ 0 pick intervals where graph is below or touching the x-axis (−∞, −4] ∪ [3, ∞) Intervals for d / e (−∞, −4)𝑎𝑛𝑑 (−4,3)𝑎𝑛𝑑 (3, ∞) 91 Chapter 5: Quadratic Functions Solutions to Selected Odd Problems 5) a) f(x) = 0 answer x = 2 (x coordinates of x-intercepts) b) f(x) > 0 pick intervals where graph is above x-axis (−∞, 2) ∪ (2, ∞) c) f(x) < 0 pick intervals where graph is below x-axis 𝑖𝑡 𝑖𝑠 𝑛𝑒𝑣𝑒𝑟 𝑏𝑒𝑙𝑜𝑤 𝑡ℎ𝑒 𝑥 − 𝑎𝑥𝑖𝑠 Solution 5c) No solution d) f(x) ≥ 0 pick intervals where graph is above or touching the x-axis (−∞, ∞) e) f(x) ≤ 0 pick intervals where graph is below x-axis 𝑔𝑟𝑎𝑝ℎ 𝑖𝑠 𝑛𝑒𝑣𝑒𝑟 𝑏𝑒𝑙𝑜𝑤 𝑡ℎ𝑒 𝑥 − 𝑎𝑥𝑖𝑠, 𝑏𝑢𝑡 𝑡ℎ𝑒 𝑔𝑟𝑎𝑝ℎ 𝑡𝑜𝑢𝑐ℎ𝑒𝑠 𝑡ℎ𝑒 𝑥 − 𝑎𝑥𝑖𝑠 𝑎𝑡 𝑥 = 2 Solution 5e) x = 2 92 Chapter 5: Quadratic Functions Solutions to Selected Odd Problems Section 5.6: Quadratic Inequalities 7) a) f(x) = 0 answer x = 3 (x coordinates of x-intercepts) Intervals for b / c (−∞, −3] 𝑎𝑛𝑑[−3, ∞) these two intervals may be combined to the single interval (−∞, ∞) b) f(x) > 0 pick intervals where graph is above xaxis graph is never above the x-axis Solution 7b) No solution c) f(x) < 0 pick intervals where graph is below xaxis (−∞, 3) ∪ (3, ∞) d) f(x) ≥ 0 pick intervals where graph is above or touching the x-axis 𝑇ℎ𝑒 𝑔𝑟𝑎𝑝ℎ 𝑖𝑠 𝑛𝑒𝑣𝑒𝑟 𝑎𝑏𝑜𝑣𝑒 𝑡ℎ𝑒 𝑥 − 𝑎𝑥𝑖𝑠, 𝑏𝑢𝑡 𝑡ℎ𝑒 𝑔𝑟𝑎𝑝ℎ 𝑑𝑜𝑒𝑠 𝑡𝑜𝑢𝑐ℎ 𝑡ℎ𝑒 𝑥 − 𝑎𝑥𝑖𝑠 𝑤ℎ𝑒𝑛 𝑥 = 3 Solution 7d: x = 3 e) f(x) ≤ 0 pick intervals where graph is below x-axis graph is always below or touching the xaxis. Solution 7e) (−∞, ∞) 93 Chapter 5: Quadratic Functions Solutions to Selected Odd Problems 9) a) f(x) = 0 graph has no x-intercepts 9a) No Solution. There is only one interval for parts b – e as there are no x-intercepts. The interval is (−∞, ∞) Since the graph is always below the x-axis the interval (−∞, ∞) will be the solution to parts c and e. The graph never touches, nor is ever above the x-axis so parts b and d have no solution. b) c) d) e) f(x) > 0 f(x) < 0 f(x) ≥ 0 f(x) ≤ 0 No Solution. solution (−∞, ∞) No Solution. solution (−∞, ∞) 11) a) f(x) = 0 graph has no x-intercepts 11a) No Solution. There is only one interval for parts b – e as there are no x-intercepts. The interval is (−∞, ∞) Since the graph is always above the x-axis the interval (−∞, ∞) will be the solution to parts band d. The graph never touches, nor is ever below the x-axis so parts c and e have no solution. b) f(x) > 0 solution (−∞, ∞) c) f(x) < 0 No Solution. d) f(x) ≥ 0 solution (−∞, ∞) e) f(x) ≤ 0 No Solution. 94 Chapter 5: Quadratic Functions Solutions to Selected Odd Problems 13) f(x) = x2 + 6x – 16 13a) f(x) = 0 x2 + 6x – 16 = 0 (x + 8)(x – 2) = 0 x = -8 x = 2 x -5 -4 -3 -2 -1 Solution 13a) x = -8,2 y -21 -24 -25 -24 -21 Solution 13b) Solution 13c) Solution 13d) Solution 13e) (−∞, −8) ∪ (2, ∞) (-8,2) (−∞, −8] ∪ [2, ∞) [-8,2] 95 Chapter 5: Quadratic Functions Solutions to Selected Odd Problems Here is another way to solve 13b – 13e 13b and 13c intervals for solutions (−∞, −8) 𝑎𝑛𝑑 (−8,2) 𝑎𝑛𝑑 (2, ∞) Intervals Number to Plug into function check (pick any number In the interval) x = -9 f(-9) = (-9)2 + 6(-9) - 16 = 11 (−∞, −8) (-8,2) x=0 f(0) = (0)2 + 6(0) -16 = -16 x=3 f(3) = (3)2 + 6(3) -16 = 11 (2, ∞) Solves f(x) > 0 Or f(x) < 0 f(x) > 0 f(x) < 0 f(x) > 0 Solution 13b) (−∞, −8) ∪ (2, ∞) Solution 13c) (-8,2) 13d and 13e intervals for solutions (−∞, −8] 𝑎𝑛𝑑 [−8,2] 𝑎𝑛𝑑 [2, ∞) Intervals Number to Plug into function check (pick any number In the interval) x = -9 f(-9) = (-9)2 + 6(-9) - 16 = 11 (−∞, −8] [-8,2] x=0 f(0) = (0)2 + 6(0) -16 = -16 x=3 f(3) = (3)2 + 6(3) -16 = 11 [2, ∞) Solves f(x) ≥ 0 Or f(x) ≤ 0 f(x) ≥ 0 f(x) ≤ 0 f(x) ≥ 0 Solution 13d) (−∞, −8] ∪ [2, ∞) Solution 13e) [-8,2] 96 Chapter 5: Quadratic Functions Solutions to Selected Odd Problems 15) f(x) = x2 - 6x + 5 15a) f(x) = 0 x2 – 6x + 5 = 0 (x – 1)(x – 5) = 0 x=1 x=d x 1 2 3 4 5 Solution 15a) x = 1,5 y 0 -3 -4 -3 0 Solution 15b) Solution 15c) Solution 15d) Solution 15e) (−∞, 1) ∪ (5, ∞) (1,5) (−∞, 1] ∪ [5, ∞) [1,5] 97 Chapter 5: Quadratic Functions Solutions to Selected Odd Problems Here is an alternate way to solve 15b-15e 15b and 15c intervals for solutions (−∞, 1) 𝑎𝑛𝑑 (1,5) 𝑎𝑛𝑑 (5, ∞) Intervals Number to Plug into function check (pick any number In the interval) x=0 f(0) = (0)2 - 6(0) + 5 = 5 (−∞, 1) (1,4) x=2 f(2) = (2)2 - 6(2) + 5 = -3 x=6 f(6) = (5)2 - 5(6) + 5 = 5 (4, ∞) Solves f(x) > 0 Or f(x) < 0 f(x) > 0 f(x) < 0 f(x) > 0 Solution 15b) (−∞, 1) ∪ (5, ∞) Solution 15c) (1,5) 15d and 15e intervals for solutions (−∞, 1] 𝑎𝑛𝑑 [1,5] 𝑎𝑛𝑑 [5, ∞) Intervals Number to Plug into function check (pick any number In the interval) x=0 f(0) = (0)2 - 6(0) + 5 = 5 (−∞, 1] [1,4] x=2 f(2) = (2)2 - 6(2) + 5 = -3 x=5 f(6) = (5)2 - 5(6) + 5 = 5 [4, ∞) Solves f(x) ≥ 0 Or f(x) ≤ 0 f(x) ≥ 0 f(x) ≤ 0 f(x) ≥ 0 Solution 15d) (−∞, 1] ∪ [5, ∞) Solution 15e) [1,5] 98 Chapter 5: Quadratic Functions Solutions to Selected Odd Problems 17) f(x) = (x-3)2 - 4 17a) (x – 3)2 – 4 = 0 +4 +4 (x – 3)2 = 4 √(𝑥 − 3)2 = ±√4 𝑥 − 3 = ±2 x = 3 + 2 and x = 3 – 2 x = 5,2 x 1 2 3 4 5 Solution 17a) x = 5,2 y 0 -3 -4 -3 0 Solution 17b) Solution 17c) Solution 17d) Solution 17e) (−∞, 2) ∪ (5, ∞) (2,5) (−∞, 2] ∪ [5, ∞) [2,5] 99 Chapter 5: Quadratic Functions Solutions to Selected Odd Problems Here is an alternate way to solve 17b – 17e 17b and 17c intervals for solutions (−∞, 2) 𝑎𝑛𝑑 (2,5) 𝑎𝑛𝑑 (5, ∞) Intervals Number to Plug into function check (pick any number In the interval) x=0 f(0) = (0-3)2 – 4 = 5 (−∞, 2) (2,5) x=3 f(3) = (3-3)2 – 4 = -4 x=6 f(6) = (6-3)2 – 4 = 5 (5, ∞) Solves f(x) > 0 Or f(x) < 0 f(x) > 0 f(x) < 0 f(x) > 0 Solution 17b) (−∞, 2) ∪ (5, ∞) Solution 17c) (2,5) 17d and 17e intervals for solutions (−∞, 2] 𝑎𝑛𝑑 [2,5] 𝑎𝑛𝑑 [5, ∞) Intervals Number to Plug into function check (pick any number In the interval) x=0 f(0) = (0-3)2 – 4 = 5 (−∞, 2] [2,5] x=3 f(3) = (3-3)2 – 4 = -4 x=6 f(6) = (6-3)2 – 4 = 5 [5, ∞) Solves f(x) ≥ 0 Or f(x) ≤ 0 f(x) ≥ 0 f(x) ≤ 0 f(x) ≥ 0 Solution 17d) (−∞, 2] ∪ [5, ∞) Solution 17e) [2,5] 100 Chapter 5: Quadratic Functions Solutions to Selected Odd Problems 19) f(x) = (x + 4)2 - 49 19a) (x + 4)2 – 49 = 0 +49 +49 (x + 4)2 = 49 √(𝑥 + 4)2 = ±√49 𝑥 + 4 = ±7 x = - 4 +7 x = -4 -7 x=3 x = -11 x -6 -5 -4 -3 -2 Solution 19a) x = 3,-11 y -45 -48 -49 -45 -40 Solution 19b) Solution 19c) Solution 19d) Solution 19e) (−∞, −11) ∪ (3, ∞) (-11,3) (−∞, −11] ∪ [3, ∞) [-11,3] 101 Chapter 5: Quadratic Functions Solutions to Selected Odd Problems Here is an alternate way to solve 19b – 19e 19d and 19e intervals for solutions (−∞, −11] 𝑎𝑛𝑑 [−11,3] 𝑎𝑛𝑑 [3, ∞) Intervals Number to Plug into function check (pick any number In the interval) x = -12 f(-12) = (-12+4)2 – 49 =15 (−∞, −11] [-11,3] x=0 f(-0) = (-0+4)2 – 49 =-33 x=4 f(4) = (4+4)2 – 49 =15 [3, ∞) Solves f(x) ≥ 0 Or f(x) ≤ 0 f(x) ≥ 0 f(x) ≤ 0 f(x) ≥ 0 Solution 19d) (−∞, −11] ∪ [3, ∞) Solution 19e) [-11,3] 19b and 19c intervals for solutions (−∞, −11) 𝑎𝑛𝑑 (−11,3) 𝑎𝑛𝑑 (3, ∞) Intervals Number to Plug into function check (pick any number In the interval) x = -12 f(-12) = (-12+4)2 – 49 =15 (−∞, −11) (-11,3) x=0 f(-0) = (-0+4)2 – 49 =-33 x=4 f(4) = (4+4)2 – 49 =15 (3, ∞) Solves f(x) > 0 Or f(x) < 0 f(x) > 0 f(x) < 0 f(x) > 0 Solution 19b) (−∞, −11) ∪ (3, ∞) Solution 19c) (-11,3) 102 Chapter 5: Quadratic Functions Solutions to Selected Odd Problems 21) f(x) = -2(x – 5)2 + 50 21a) -2(x – 5)2 + 50 = 0 _________-50 -50 -2(x – 5)2 = - 50 divide both sides by -2 (x – 5)2 = 25 √(𝑥 − 5)2 = ±√25 x – 5 = ±5 x=5+5 x = 5–5 x = 10 x=0 Solution 21a) x =0, 10 x 3 4 5 6 7 y 42 48 50 48 42 Solution 21b) Solution 21c) Solution 21d) Solution 21e) (0,10) (−∞, 0) ∪ (10, ∞) [0,10] (−∞, 0] ∪ [10, ∞) 103 Chapter 5: Quadratic Functions Solutions to Selected Odd Problems Here is an alternate way to solve 21b – 21e 21d and 21e intervals for solutions (−∞, 0] 𝑎𝑛𝑑 [0,10] 𝑎𝑛𝑑 [10, ∞) Intervals Number to Plug into function check (pick any number In the interval) x = -1 f(-1) = -2(-1-5)2 + 50 = -22 (−∞, 0] [0,10] x=1 f(1) = -2(1-5)2 + 50 = 18 x = 11 f(11) = -2(11-5)2 + 50 = -22 [10, ∞) Solves f(x) ≥ 0 Or f(x) ≤ 0 f(x) ≤ 0 f(x) ≥ 0 f(x) ≤ 0 Solution 21d) [0,10] Solution 21e) (−∞, 0] ∪ [10, ∞) 21b and 21c intervals for solutions (−∞, 0) 𝑎𝑛𝑑 (0,10) 𝑎𝑛𝑑 (10, ∞) Intervals Number to Plug into function check (pick any number In the interval) x = -1 f(-1) = -2(-1-5)2 + 50 = -22 (−∞, 0) (0,10) x=1 f(1) = -2(1-5)2 + 50 = 18 x = 11 f(11) = -2(11-5)2 + 50 = -22 (10, ∞) Solves f(x) > 0 Or f(x) < 0 f(x) < 0 f(x) > 0 f(x) < 0 Solution 21b) (0,10) Solution 21c) (−∞, 0) ∪ (10, ∞) 104 Chapter 5: Quadratic Functions Solutions to Selected Odd Problems 23) f(x) = -2(x + 6)2 23a) -2(x+6)2 = 0 -2(x+6)(x+6) = 0 -2 = 0 x+6=0 no solution x = -6 x -8 -7 -6 -5 -4 x+6 =0 x = -6 Solution 23a) x = -6 y -8 -2 0 -2 -8 Solution 23b) Solution 23c) Solution 23d) Solution 23e) No solution (−∞, −6) ∪ (−6, ∞) x = -6 (−∞, ∞) 105 Chapter 5: Quadratic Functions Solutions to Selected Odd Problems Here is an alternate way to solve 23b – 23e 23b and 23c intervals for solutions (−∞, −6) 𝑎𝑛𝑑 (−6, ∞) (can’t be combined to the interval (−∞, ∞) because of the round parenthesis excluding the -6) Intervals Number to Plug into function Solves f(x) > 0 check Or (pick any f(x) < 0 number In the interval) x = -7 f(-7) = -2(-7+6)2 = - 2 f(x) < 0 (−∞, −6) x=0 f(0) = -2(0+6)2 = -72 f(x) < 0 (−6, ∞) Solution 23b) No solution since on positive values in plug into function column Solution 23c) (−∞, −6) ∪ (−6, ∞) 23d and 23e intervals for solutions (−∞, −6] 𝑎𝑛𝑑 [−6, ∞) since both intervals have closed brackets and the end / start with the same number the two intervals can be condensed to the single interval (−∞, ∞) Intervals Number to Plug into function Solves f(x) ≥ 0 check Or (pick any f(x) ≤ 0 number In the interval) x=0 f(0) = -2(0+6)2 = - 72 f(x) ≤ 0 (−∞, ∞) Solution 23d) This is an evil trick, there are no values that you can plug into the function to get a positive result, thus there is no solution to f(x) > 0, but you do get 0 when you plug -6 in for x. so -6 is the only solution to f(x) ≥ 0 Solution 23d) x = -6 Solution 23e) (−∞, ∞) 106 Chapter 5: Quadratic Functions Solutions to Selected Odd Problems 25) f(x) = (x – 3)2 + 9 25a) x 1 2 3 4 5 (x – 3)2 + 9 = 0 -9 -9 (x-3)2 = -9 √(𝑥 − 3)2 =±√−9 𝑥 − 3 = ±3𝑖 (no solution because of the i) Solution 25a) No solution y 13 10 9 10 13 Solution 25b) Solution 25c) Solution 25d) Solution 25e) (−∞, ∞) No solution (−∞, ∞) No solution 107 Chapter 5: Quadratic Functions Solutions to Selected Odd Problems Here is an alternate way to solve 25b – 25e 25b and 25c No check numbers so the only interval to test is (−∞, ∞) Intervals Number to Plug into function check (pick any number In the interval) x=0 f(0) = (0-3)2 + 9 = 18 (−∞, ∞) Solves f(x) ≥ 0 Or f(x) ≤ 0 f(x) > 0 Solution 25b) (−∞, ∞) Solution 25c) No solution 25d and 25e No check numbers so the only interval to test is (−∞, ∞) Intervals Number to Plug into function check (pick any number In the interval) x=0 f(0) = (0-3)2 + 9 = 18 (−∞, ∞) Solves f(x) ≥ 0 Or f(x) ≤ 0 f(x) ≥ 0 Solution 25d) (−∞, ∞) Solution 25e) No solution 108 Chapter 5: Quadratic Functions Solutions to Selected Odd Problems 27) f(x) = x2 + 2x + 4 27a) x2 + 2x + 4 = 0 a = a b =2 c=4 𝑥= 𝑥= 𝑥= −2±√22 −4(1)(4) 2(1) −2±√−12 2 −2±2√3𝑖 2 𝑥 = −1 ± √3𝑖 (no solution because of the i) Solution 27a) No solution x -3 -2 -1 0 1 y 7 4 3 4 7 Solution 27b) Solution 27c) Solution 27d) Solution 27e) (−∞, ∞) No solution (−∞, ∞) No solution 109 Chapter 5: Quadratic Functions Solutions to Selected Odd Problems 110 Chapter 5: Quadratic Functions Solutions to Selected Odd Problems Here is an alternate way to solve 27b – 27e 25b and 25c No check numbers so the only interval to test is (−∞, ∞) Intervals (−∞, ∞) Number to check (pick any number In the interval) x=0 Plug into function Solves f(x) ≥ 0 Or f(x) ≤ 0 f(0) = (0)2 + 2(0) + 4 = 4 f(x) > 0 Solution 27b) (−∞, ∞) Solution 27c) No solution 27d and 27e No check numbers so the only interval to test is (−∞, ∞) Intervals Number to Plug into function check (pick any number In the interval) x=0 f(0) = (0)2 + 2(0) + 4 = 4 (−∞, ∞) Solves f(x) ≥ 0 Or f(x) ≤ 0 f(x) ≥ 0 Solution 27d) (−∞, ∞) Solution 27e) No solution 111