{"id":1289,"date":"2015-11-12T18:35:30","date_gmt":"2015-11-12T18:35:30","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1289"},"modified":"2017-03-31T22:37:55","modified_gmt":"2017-03-31T22:37:55","slug":"solutions-43","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/solutions-43\/","title":{"raw":"Solutions","rendered":"Solutions"},"content":{"raw":"

Solutions to Try Its<\/h2>\r\n1.\u00a0[latex]f\\left(x\\right)[\/latex]\u00a0is a power function because it can be written as [latex]f\\left(x\\right)=8{x}^{5}[\/latex].\u00a0The other functions are not power functions.\r\n\r\n2.\u00a0As x<\/em>\u00a0approaches positive or negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound: as [latex]x\\to \\pm \\infty , f\\left(x\\right)\\to -\\infty\\\\ [\/latex] because of the negative coefficient.\r\n\r\n3.\u00a0The degree is 6. The leading term is [latex]-{x}^{6}[\/latex]. The leading coefficient is \u20131.\r\n\r\n4.\u00a0As [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty ; as x\\to -\\infty , f\\left(x\\right)\\to -\\infty [\/latex]. It has the shape of an even degree power function with a negative coefficient.\r\n\r\n5.\u00a0The leading term is [latex]0.2{x}^{3}[\/latex], so it is a degree 3 polynomial. As x<\/em>\u00a0approaches positive infinity, [latex]f\\left(x\\right)[\/latex] increases without bound; as x<\/em>\u00a0approaches negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound.\r\n\r\n6.\u00a0y<\/em>-intercept [latex]\\left(0,0\\right)[\/latex]; x<\/em>-intercepts [latex]\\left(0,0\\right),\\left(-2,0\\right)[\/latex], and [latex]\\left(5,0\\right)[\/latex]\r\n\r\n7.\u00a0There are at most 12 x<\/em>-intercepts and at most 11 turning points.\r\n\r\n8.\u00a0The end behavior indicates an odd-degree polynomial function; there are 3 x<\/em>-intercepts and 2 turning points, so the degree is odd and at least 3. Because of the end behavior, we know that the lead coefficient must be negative.\r\n\r\n9.\u00a0The x<\/em>-intercepts are [latex]\\left(2,0\\right),\\left(-1,0\\right)[\/latex], and [latex]\\left(5,0\\right)[\/latex], the y-<\/em>intercept is [latex]\\left(0,\\text{2}\\right)[\/latex], and the graph has at most 2 turning points.\r\n

Solutions to Odd-Numbered Exercises<\/h2>\r\n1.\u00a0The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.\r\n\r\n3.\u00a0As x<\/em>\u00a0decreases without bound, so does [latex]f\\left(x\\right)[\/latex].\u00a0As x<\/em>\u00a0increases without bound, so does [latex]f\\left(x\\right)[\/latex].\r\n\r\n5.\u00a0The polynomial function is of even degree and leading coefficient is negative.\r\n\r\n7.\u00a0Power function\r\n\r\n9.\u00a0Neither\r\n\r\n11.\u00a0Neither\r\n\r\n13.\u00a0Degree = 2, Coefficient = \u20132\r\n\r\n15.\u00a0Degree =4, Coefficient = \u20132\r\n\r\n17.\u00a0[latex]\\text{As }x\\to \\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to -\\infty ,f\\left(x\\right)\\to \\infty [\/latex]\r\n\r\n19.\u00a0[latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to -\\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to -\\infty [\/latex]\r\n\r\n21.\u00a0[latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to -\\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to -\\infty [\/latex]\r\n\r\n23.\u00a0[latex]\\text{As }x\\to \\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to -\\infty ,f\\left(x\\right)\\to -\\infty [\/latex]\r\n\r\n25. y<\/em>-intercept is [latex]\\left(0,12\\right)[\/latex], t<\/em>-intercepts are [latex]\\left(1,0\\right);\\left(-2,0\\right);\\text{and }\\left(3,0\\right)[\/latex].\r\n\r\n27.\u00a0y<\/em>-intercept is [latex]\\left(0,-16\\right)[\/latex]. x<\/em>-intercepts are [latex]\\left(2,0\\right)[\/latex] and [latex]\\left(-2,0\\right)[\/latex].\r\n\r\n29.\u00a0y<\/em>-intercept is [latex]\\left(0,0\\right)[\/latex].i x-intercepts are [latex]\\left(0,0\\right),\\left(4,0\\right)[\/latex], and [latex]\\left(-2, 0\\right)[\/latex].\r\n\r\n31. 3\r\n\r\n33. 5\r\n\r\n35. 3\r\n\r\n37. 5\r\n\r\n39.\u00a0Yes. Number of turning points is 2. Least possible degree is 3.\r\n\r\n41.\u00a0Yes. Number of turning points is 1. Least possible degree is 2.\r\n\r\n43.\u00a0Yes. Number of turning points is 0. Least possible degree is 1.\r\n\r\n45.\u00a0Yes. Number of turning points is 0. Least possible degree is 1.\r\n\r\n47.\u00a0[latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty [\/latex]\r\n
x<\/em><\/th>\r\nf<\/em>(x<\/em>)<\/th>\r\n<\/tr><\/thead>
10<\/td>\r\n9,500<\/td>\r\n<\/tr>
100<\/td>\r\n99,950,000<\/td>\r\n<\/tr>
\u201310<\/td>\r\n9,500<\/td>\r\n<\/tr>
\u2013100<\/td>\r\n99,950,000<\/td>\r\n<\/tr><\/tbody><\/table>\r\n49.\u00a0[latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to -\\infty [\/latex]\r\n
x<\/em><\/th>\r\nf<\/em>(x<\/em>)<\/th>\r\n<\/tr><\/thead>
10<\/td>\r\n\u2013504<\/td>\r\n<\/tr>
100<\/td>\r\n\u2013941,094<\/td>\r\n<\/tr>
\u201310<\/td>\r\n1,716<\/td>\r\n<\/tr>
\u2013100<\/td>\r\n1,061,106<\/td>\r\n<\/tr><\/tbody><\/table>\r\n51.\u00a0The y<\/em>-intercept is [latex]\\left(0, 0\\right)[\/latex].\u00a0The x<\/em>-intercepts are [latex]\\left(0, 0\\right),\\text{ }\\left(2, 0\\right)[\/latex]. [latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty [\/latex]\r\n\"Graph\r\n\r\n53.\u00a0The y<\/em>-intercept is [latex]\\left(0,0\\right)[\/latex]. The x<\/em>-intercepts are [latex]\\left(0, 0\\right),\\text{ }\\left(5, 0\\right),\\text{ }\\left(7, 0\\right)[\/latex]. [latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to -\\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty [\/latex]\r\n\"Graph\r\n\r\n55.\u00a0The y<\/em>-intercept is [latex]\\left(0, 0\\right)[\/latex]. The x<\/em>-intercept is [latex]\\left(-4, 0\\right),\\text{ }\\left(0, 0\\right),\\text{ }\\left(4, 0\\right)[\/latex]. [latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to -\\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty [\/latex]\r\n\"\"\r\n\r\n57.\u00a0The y<\/em>-intercept is [latex]\\left(0, -81\\right)[\/latex].\u00a0The x<\/em>-intercept are [latex]\\left(3, 0\\right),\\text{ }\\left(-3, 0\\right)[\/latex]. [latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty [\/latex]\r\n\"Graph\r\n\r\n59.\u00a0The y<\/em>-intercept is [latex]\\left(0, 0\\right)[\/latex]. The x<\/em>-intercepts are [latex]\\left(-3, 0\\right),\\text{ }\\left(0, 0\\right),\\text{ }\\left(5, 0\\right)[\/latex]. [latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to -\\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty [\/latex]\r\n\"Graph\r\n\r\n61.\u00a0[latex]f\\left(x\\right)={x}^{2}-4[\/latex]\r\n\r\n63.\u00a0[latex]f\\left(x\\right)={x}^{3}-4{x}^{2}+4x[\/latex]\r\n\r\n65.\u00a0[latex]f\\left(x\\right)={x}^{4}+1[\/latex]\r\n\r\n67.\u00a0[latex]V\\left(m\\right)=8{m}^{3}+36{m}^{2}+54m+27[\/latex]\r\n\r\n69.\u00a0[latex]V\\left(x\\right)=4{x}^{3}-32{x}^{2}+64x[\/latex]","rendered":"

Solutions to Try Its<\/h2>\n

1.\u00a0[latex]f\\left(x\\right)[\/latex]\u00a0is a power function because it can be written as [latex]f\\left(x\\right)=8{x}^{5}[\/latex].\u00a0The other functions are not power functions.<\/p>\n

2.\u00a0As x<\/em>\u00a0approaches positive or negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound: as [latex]x\\to \\pm \\infty , f\\left(x\\right)\\to -\\infty\\\\[\/latex] because of the negative coefficient.<\/p>\n

3.\u00a0The degree is 6. The leading term is [latex]-{x}^{6}[\/latex]. The leading coefficient is \u20131.<\/p>\n

4.\u00a0As [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty ; as x\\to -\\infty , f\\left(x\\right)\\to -\\infty[\/latex]. It has the shape of an even degree power function with a negative coefficient.<\/p>\n

5.\u00a0The leading term is [latex]0.2{x}^{3}[\/latex], so it is a degree 3 polynomial. As x<\/em>\u00a0approaches positive infinity, [latex]f\\left(x\\right)[\/latex] increases without bound; as x<\/em>\u00a0approaches negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound.<\/p>\n

6.\u00a0y<\/em>-intercept [latex]\\left(0,0\\right)[\/latex]; x<\/em>-intercepts [latex]\\left(0,0\\right),\\left(-2,0\\right)[\/latex], and [latex]\\left(5,0\\right)[\/latex]<\/p>\n

7.\u00a0There are at most 12 x<\/em>-intercepts and at most 11 turning points.<\/p>\n

8.\u00a0The end behavior indicates an odd-degree polynomial function; there are 3 x<\/em>-intercepts and 2 turning points, so the degree is odd and at least 3. Because of the end behavior, we know that the lead coefficient must be negative.<\/p>\n

9.\u00a0The x<\/em>-intercepts are [latex]\\left(2,0\\right),\\left(-1,0\\right)[\/latex], and [latex]\\left(5,0\\right)[\/latex], the y-<\/em>intercept is [latex]\\left(0,\\text{2}\\right)[\/latex], and the graph has at most 2 turning points.<\/p>\n

Solutions to Odd-Numbered Exercises<\/h2>\n

1.\u00a0The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.<\/p>\n

3.\u00a0As x<\/em>\u00a0decreases without bound, so does [latex]f\\left(x\\right)[\/latex].\u00a0As x<\/em>\u00a0increases without bound, so does [latex]f\\left(x\\right)[\/latex].<\/p>\n

5.\u00a0The polynomial function is of even degree and leading coefficient is negative.<\/p>\n

7.\u00a0Power function<\/p>\n

9.\u00a0Neither<\/p>\n

11.\u00a0Neither<\/p>\n

13.\u00a0Degree = 2, Coefficient = \u20132<\/p>\n

15.\u00a0Degree =4, Coefficient = \u20132<\/p>\n

17.\u00a0[latex]\\text{As }x\\to \\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to -\\infty ,f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n

19.\u00a0[latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to -\\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to -\\infty[\/latex]<\/p>\n

21.\u00a0[latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to -\\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to -\\infty[\/latex]<\/p>\n

23.\u00a0[latex]\\text{As }x\\to \\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to -\\infty ,f\\left(x\\right)\\to -\\infty[\/latex]<\/p>\n

25. y<\/em>-intercept is [latex]\\left(0,12\\right)[\/latex], t<\/em>-intercepts are [latex]\\left(1,0\\right);\\left(-2,0\\right);\\text{and }\\left(3,0\\right)[\/latex].<\/p>\n

27.\u00a0y<\/em>-intercept is [latex]\\left(0,-16\\right)[\/latex]. x<\/em>-intercepts are [latex]\\left(2,0\\right)[\/latex] and [latex]\\left(-2,0\\right)[\/latex].<\/p>\n

29.\u00a0y<\/em>-intercept is [latex]\\left(0,0\\right)[\/latex].i x-intercepts are [latex]\\left(0,0\\right),\\left(4,0\\right)[\/latex], and [latex]\\left(-2, 0\\right)[\/latex].<\/p>\n

31. 3<\/p>\n

33. 5<\/p>\n

35. 3<\/p>\n

37. 5<\/p>\n

39.\u00a0Yes. Number of turning points is 2. Least possible degree is 3.<\/p>\n

41.\u00a0Yes. Number of turning points is 1. Least possible degree is 2.<\/p>\n

43.\u00a0Yes. Number of turning points is 0. Least possible degree is 1.<\/p>\n

45.\u00a0Yes. Number of turning points is 0. Least possible degree is 1.<\/p>\n

47.\u00a0[latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n\n\n\n\n\n\n\n\n
x<\/em><\/th>\nf<\/em>(x<\/em>)<\/th>\n<\/tr>\n<\/thead>\n
10<\/td>\n9,500<\/td>\n<\/tr>\n
100<\/td>\n99,950,000<\/td>\n<\/tr>\n
\u201310<\/td>\n9,500<\/td>\n<\/tr>\n
\u2013100<\/td>\n99,950,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

49.\u00a0[latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to -\\infty[\/latex]<\/p>\n\n\n\n\n\n\n\n\n
x<\/em><\/th>\nf<\/em>(x<\/em>)<\/th>\n<\/tr>\n<\/thead>\n
10<\/td>\n\u2013504<\/td>\n<\/tr>\n
100<\/td>\n\u2013941,094<\/td>\n<\/tr>\n
\u201310<\/td>\n1,716<\/td>\n<\/tr>\n
\u2013100<\/td>\n1,061,106<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

51.\u00a0The y<\/em>-intercept is [latex]\\left(0, 0\\right)[\/latex].\u00a0The x<\/em>-intercepts are [latex]\\left(0, 0\\right),\\text{ }\\left(2, 0\\right)[\/latex]. [latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex]
\n\"Graph<\/p>\n

53.\u00a0The y<\/em>-intercept is [latex]\\left(0,0\\right)[\/latex]. The x<\/em>-intercepts are [latex]\\left(0, 0\\right),\\text{ }\\left(5, 0\\right),\\text{ }\\left(7, 0\\right)[\/latex]. [latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to -\\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex]
\n\"Graph<\/p>\n

55.\u00a0The y<\/em>-intercept is [latex]\\left(0, 0\\right)[\/latex]. The x<\/em>-intercept is [latex]\\left(-4, 0\\right),\\text{ }\\left(0, 0\\right),\\text{ }\\left(4, 0\\right)[\/latex]. [latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to -\\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex]
\n\"\"<\/p>\n

57.\u00a0The y<\/em>-intercept is [latex]\\left(0, -81\\right)[\/latex].\u00a0The x<\/em>-intercept are [latex]\\left(3, 0\\right),\\text{ }\\left(-3, 0\\right)[\/latex]. [latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex]
\n\"Graph<\/p>\n

59.\u00a0The y<\/em>-intercept is [latex]\\left(0, 0\\right)[\/latex]. The x<\/em>-intercepts are [latex]\\left(-3, 0\\right),\\text{ }\\left(0, 0\\right),\\text{ }\\left(5, 0\\right)[\/latex]. [latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to -\\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex]
\n\"Graph<\/p>\n

61.\u00a0[latex]f\\left(x\\right)={x}^{2}-4[\/latex]<\/p>\n

63.\u00a0[latex]f\\left(x\\right)={x}^{3}-4{x}^{2}+4x[\/latex]<\/p>\n

65.\u00a0[latex]f\\left(x\\right)={x}^{4}+1[\/latex]<\/p>\n

67.\u00a0[latex]V\\left(m\\right)=8{m}^{3}+36{m}^{2}+54m+27[\/latex]<\/p>\n

69.\u00a0[latex]V\\left(x\\right)=4{x}^{3}-32{x}^{2}+64x[\/latex]<\/p>\n\n\t\t\t

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