{"id":15506,"date":"2019-09-04T19:35:52","date_gmt":"2019-09-04T19:35:52","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=15506"},"modified":"2025-02-05T05:18:09","modified_gmt":"2025-02-05T05:18:09","slug":"problem-set-5-transformation-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/problem-set-5-transformation-of-functions\/","title":{"raw":"Problem Set 5: Transformation of Functions","rendered":"Problem Set 5: Transformation of Functions"},"content":{"raw":"1. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal shift from a vertical shift?\r\n\r\n2. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal stretch from a vertical stretch?\r\n\r\n3. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal compression from a vertical compression?\r\n\r\n4. When examining the formula of a function that is the result of multiple transformations, how can you tell a reflection with respect to the x-axis from a reflection with respect to the y-axis?\r\n\r\n5. How can you determine whether a function is odd or even from the formula of the function?\r\n\r\n6.\u00a0Write a formula for the function obtained when the graph of [latex]f\\left(x\\right)=\\sqrt{x}[\/latex] is shifted up 1 unit and to the left 2 units.\r\n\r\n7. Write a formula for the function obtained when the graph of [latex]f\\left(x\\right)=|x|[\/latex]\r\nis shifted down 3 units and to the right 1 unit.\r\n\r\n8.\u00a0Write a formula for the function obtained when the graph of [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] is shifted down 4 units and to the right 3 units.\r\n\r\n9. Write a formula for the function obtained when the graph of [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex] is shifted up 2 units and to the left 4 units.\r\n\r\nFor the following exercises, describe how the graph of the function is a transformation of the graph of the original function [latex]f[\/latex].\r\n\r\n10. [latex]y=f\\left(x - 49\\right)[\/latex]\r\n\r\n11. [latex]y=f\\left(x+43\\right)[\/latex]\r\n\r\n12.\u00a0[latex]y=f\\left(x+3\\right)[\/latex]\r\n\r\n13. [latex]y=f\\left(x - 4\\right)[\/latex]\r\n\r\n14.\u00a0[latex]y=f\\left(x\\right)+5[\/latex]\r\n\r\n15. [latex]y=f\\left(x\\right)+8[\/latex]\r\n\r\n16.\u00a0[latex]y=f\\left(x\\right)-2[\/latex]\r\n\r\n17. [latex]y=f\\left(x\\right)-7[\/latex]\r\n\r\n18.\u00a0[latex]y=f\\left(x - 2\\right)+3[\/latex]\r\n\r\n19. [latex]y=f\\left(x+4\\right)-1[\/latex]\r\n\r\nFor the following exercises, determine the interval(s) on which the function is increasing and decreasing.\r\n\r\n20. [latex]f\\left(x\\right)=4{\\left(x+1\\right)}^{2}-5[\/latex]\r\n\r\n21. [latex]g\\left(x\\right)=5{\\left(x+3\\right)}^{2}-2[\/latex]\r\n\r\n22.\u00a0[latex]a\\left(x\\right)=\\sqrt{-x+4}[\/latex]\r\n\r\n23. [latex]k\\left(x\\right)=-3\\sqrt{x}-1[\/latex]\r\n\r\nFor the following exercises, use the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex]\u00a0to sketch a graph of each transformation of [latex]f\\left(x\\right)[\/latex].\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005101\/CNX_Precalc_Figure_01_05_201.jpg\" alt=\"Graph of f(x) increasing on (0, oo), approaching y = 0 on (-oo,0), passing through the point (1,1).\" width=\"487\" height=\"366\" \/>\r\n\r\n24. [latex]g\\left(x\\right)={2}^{x}+1[\/latex]\r\n\r\n25. [latex]h\\left(x\\right)={2}^{x}-3[\/latex]\r\n\r\n26. [latex]w\\left(x\\right)={2}^{x - 1}[\/latex]\r\n\r\nFor the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.\r\n\r\n27. [latex]f\\left(t\\right)={\\left(t+1\\right)}^{2}-3[\/latex]\r\n\r\n28.\u00a0[latex]h\\left(x\\right)=|x - 1|+4[\/latex]\r\n\r\n29. [latex]k\\left(x\\right)={\\left(x - 2\\right)}^{3}-1[\/latex]\r\n\r\n30.\u00a0[latex]m\\left(t\\right)=3+\\sqrt{t+2}[\/latex]\r\n\r\n31.\u00a0Tabular representations for the functions [latex]f,g[\/latex], and [latex]h[\/latex] are given below. Write [latex]g\\left(x\\right)[\/latex] and [latex]h\\left(x\\right)[\/latex] as transformations of [latex]f\\left(x\\right)[\/latex].\r\n<table id=\"fs-id1165137432561\" class=\"unnumbered\" style=\"line-height: 1.5;\" summary=\"Two rows and six columns. The first row is labeled, \">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>\u22122<\/td>\r\n<td>\u22121<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>\u22122<\/td>\r\n<td>\u22121<\/td>\r\n<td>\u22123<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div id=\"fs-id1165137734659\" class=\"exercise\">\r\n<div id=\"fs-id1165137644805\" class=\"solution\">\r\n<div id=\"fs-id1165135650778\" class=\"exercise\">\r\n<div id=\"fs-id1165135628497\" class=\"solution\">\r\n<div id=\"fs-id1165135421533\" class=\"exercise\">\r\n<div id=\"fs-id1165134234193\" class=\"solution\">\r\n<div id=\"fs-id1165137681998\" class=\"exercise\">\r\n<div id=\"fs-id1165137682000\" class=\"problem\">\r\n<table id=\"fs-id1165135634096\" class=\"unnumbered\" summary=\"Two rows and six columns. The first row is labeled, \">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>\u22121<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>\u22122<\/td>\r\n<td>\u22121<\/td>\r\n<td>\u22123<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"fs-id1165135330589\" class=\"unnumbered\" summary=\"Two rows and six columns. The first row is labeled, \">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>\u22122<\/td>\r\n<td>\u22121<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]h\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>\u22121<\/td>\r\n<td>0<\/td>\r\n<td>\u22122<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165137443424\" class=\"solution\">\r\n<p id=\"fs-id1165134211288\">32.\u00a0Tabular representations for the functions [latex]f,g[\/latex], and [latex]h[\/latex] are given below. Write [latex]g\\left(x\\right)[\/latex] and [latex]h\\left(x\\right)[\/latex] as transformations of [latex]f\\left(x\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137734475\" class=\"exercise\">\r\n<div id=\"fs-id1165137734477\" class=\"problem\">\r\n<table id=\"fs-id1165134558032\" class=\"unnumbered\" summary=\"Two rows and six columns. The first row is labeled, \">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>\u22122<\/td>\r\n<td>\u22121<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>\u22121<\/td>\r\n<td>\u22123<\/td>\r\n<td>4<\/td>\r\n<td>2<\/td>\r\n<td>1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"fs-id1165134380916\" class=\"unnumbered\" summary=\"Two rows and six columns. The first row is labeled, \">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>\u22123<\/td>\r\n<td>\u22122<\/td>\r\n<td>\u22121<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>\u22121<\/td>\r\n<td>\u22123<\/td>\r\n<td>4<\/td>\r\n<td>2<\/td>\r\n<td>1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"fs-id1165137894261\" class=\"unnumbered\" summary=\"Two rows and six columns. The first row is labeled, \">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>\u22122<\/td>\r\n<td>\u22121<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]h\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>\u22122<\/td>\r\n<td>\u22124<\/td>\r\n<td>3<\/td>\r\n<td>1<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137570566\">For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.<\/p>\r\n\r\n<div id=\"fs-id1165137431229\" class=\"exercise\">\r\n<div id=\"fs-id1165137431231\" class=\"problem\">\r\n\r\n<span id=\"fs-id1165135543438\">33.\r\n<img class=\"alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005102\/CNX_Precalc_Figure_01_05_210.jpg\" alt=\"Graph of an absolute function with vertex at (3,-2), decreasing on (-oo,3) and increasing on (3,oo).\" width=\"487\" height=\"317\" \/><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135516945\" class=\"solution\">\r\n<p id=\"fs-id1165135516948\">34.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135481230\" class=\"exercise\">\r\n<div id=\"fs-id1165135481232\" class=\"problem\">\r\n\r\n<span id=\"fs-id1165137851362\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005102\/CNX_Precalc_Figure_01_05_211.jpg\" alt=\"Graph of a parabola with vertex at (1,-3), decreasing on (-oo,1) and increasing on (1,oo).\" width=\"487\" height=\"314\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137817635\" class=\"exercise\">\r\n<div id=\"fs-id1165137817637\" class=\"problem\">\r\n\r\n<span id=\"fs-id1165133341017\">35.\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005102\/CNX_Precalc_Figure_01_05_212.jpg\" alt=\"Graph of a square root function originating at (-3,-1), increasing on [-3,oo).\" width=\"487\" height=\"317\" \/><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135190190\" class=\"solution\">\r\n<p id=\"fs-id1165135190193\">36.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165132929618\" class=\"exercise\">\r\n<div id=\"fs-id1165132929620\" class=\"problem\">\r\n\r\n<span id=\"fs-id1165135203675\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005103\/CNX_Precalc_Figure_01_05_213.jpg\" alt=\"Graph of an absolute function with vertex at (-2, 2), decreasing on (-oo,-2) and increasing on (-2,oo).\" width=\"487\" height=\"409\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135487204\" class=\"exercise\">\r\n<div id=\"fs-id1165135487206\" class=\"problem\">\r\n\r\n<span id=\"fs-id1165135535017\">37.\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005103\/CNX_Precalc_Figure_01_05_214.jpg\" alt=\"Graph of a parabola with vertex at (2,0), decreasingon (-inf., 2), increasing on (2, inf.)\" width=\"487\" height=\"409\" \/><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135560630\" class=\"solution\">\r\n<p id=\"fs-id1165135560633\">38.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135190411\" class=\"exercise\">\r\n<div id=\"fs-id1165135190413\" class=\"problem\">\r\n\r\n<span id=\"fs-id1165133277626\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005103\/CNX_Precalc_Figure_01_05_215.jpg\" alt=\"Graph of a square root function originating at (-3,0) increasing on [-3, inf) and passing through (1,2).\" width=\"487\" height=\"226\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133103936\" class=\"exercise\">\r\n<div id=\"fs-id1165133103938\" class=\"problem\">\r\n\r\n<span id=\"fs-id1165134362846\">39.\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005103\/CNX_Precalc_Figure_01_05_216f.jpg\" alt=\"Graph of an absolute function with vertex at (-3,-2), decreasing on (-inf., -3) and increasing on (-3,inf.), passing through (0,1).\" width=\"487\" height=\"379\" \/><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165133030812\" class=\"solution\">\r\n<p id=\"fs-id1165133030814\">40.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134043550\" class=\"exercise\">\r\n<div id=\"fs-id1165134043552\" class=\"problem\">\r\n\r\n<span id=\"fs-id1165134036728\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005104\/CNX_Precalc_Figure_01_05_217F.jpg\" alt=\"Graph of a square root function originating at (-2, -2) increasing on [-2, inf) and passing through (2,0).\" width=\"487\" height=\"378\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165134187273\">For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.<\/p>\r\n\r\n<div id=\"fs-id1165134187277\" class=\"exercise\">\r\n<div id=\"fs-id1165134187279\" class=\"problem\">\r\n\r\n<span id=\"fs-id1165137834957\">41.\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005104\/CNX_Precalc_Figure_01_05_218.jpg\" alt=\"Graph of a square root function originating at (0,0) decreasing on (0,inf) passing through (4,-2).\" width=\"487\" height=\"225\" \/><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134190724\" class=\"solution\">\r\n<p id=\"fs-id1165134190726\">42.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137930320\" class=\"exercise\">\r\n<div id=\"fs-id1165137930323\" class=\"problem\"><span id=\"fs-id1165135487154\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005104\/CNX_Precalc_Figure_01_05_219.jpg\" alt=\"Graph of a square root function originating at (0,0) decreasing on (-inf., 0) passing through (-1,1).\" width=\"487\" height=\"222\" \/><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1165134177109\">For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.<\/p>\r\n\r\n<div id=\"fs-id1165134177113\" class=\"exercise\">\r\n<div id=\"fs-id1165137806559\" class=\"problem\">\r\n\r\n<span id=\"fs-id1165137806566\">43.\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005104\/CNX_Precalc_Figure_01_05_220.jpg\" alt=\"Graph of a parabola concave down, vertex at (-1,2).\" width=\"487\" height=\"283\" \/><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137889877\" class=\"solution\">\r\n<p id=\"fs-id1165134261684\">44.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135384400\" class=\"exercise\">\r\n<div id=\"fs-id1165137693606\" class=\"problem\">\r\n\r\n<span id=\"fs-id1165137693612\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005105\/CNX_Precalc_Figure_01_05_221.jpg\" alt=\"Graph of a cubic function passing through (2,1) .\" width=\"487\" height=\"251\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137892243\" class=\"exercise\">\r\n<div id=\"fs-id1165134352554\" class=\"problem\">\r\n\r\n<span id=\"fs-id1165134352561\">45.\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005105\/CNX_Precalc_Figure_01_05_222.jpg\" alt=\"Graph of a square root function concave down, originating at (0,1) decreasing on (-inf., 0).\" width=\"487\" height=\"191\" \/><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134159665\" class=\"solution\">\r\n<p id=\"fs-id1165134159668\">46.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135411377\" class=\"exercise\">\r\n<div id=\"fs-id1165135411379\" class=\"problem\">\r\n\r\n<span id=\"fs-id1165133155251\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005105\/CNX_Precalc_Figure_01_05_223.jpg\" alt=\"Graph of an absolute function downward facing, vertex at (2,3).\" width=\"487\" height=\"285\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165132924966\">For the following exercises, determine whether the function is odd, even, or neither.<\/p>\r\n\r\n<div id=\"fs-id1165132924969\" class=\"exercise\">\r\n<div id=\"fs-id1165132924971\" class=\"problem\">\r\n<p id=\"fs-id1165137812602\">47. [latex]f\\left(x\\right)=3{x}^{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135671506\" class=\"solution\">\r\n<p id=\"fs-id1165135671508\">48.\u00a0[latex]g\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137828008\" class=\"exercise\">\r\n<div id=\"fs-id1165137828010\" class=\"problem\">\r\n<p id=\"fs-id1165133408839\">49. [latex]h\\left(x\\right)=\\frac{1}{x}+3x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134187165\" class=\"exercise\">\r\n<div id=\"fs-id1165134187167\" class=\"problem\">\r\n<p id=\"fs-id1165134271332\">50. [latex]f\\left(x\\right)={\\left(x - 2\\right)}^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134039317\" class=\"exercise\">\r\n<div id=\"fs-id1165134039319\" class=\"problem\">\r\n<p id=\"fs-id1165135238484\">51. [latex]g\\left(x\\right)=2{x}^{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137679220\" class=\"solution\">\r\n<p id=\"fs-id1165137679222\">52.\u00a0[latex]h\\left(x\\right)=2x-{x}^{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137571611\">For the following exercises, describe how the graph of each function is a transformation of the graph of the original function [latex]f[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165137599981\" class=\"exercise\">\r\n<div id=\"fs-id1165137599983\" class=\"problem\">\r\n<p id=\"fs-id1165137599985\">53. [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135208810\" class=\"solution\">\r\n<p id=\"fs-id1165135400954\">54.\u00a0[latex]g\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135412892\" class=\"exercise\">\r\n<div id=\"fs-id1165135412894\" class=\"problem\">\r\n<p id=\"fs-id1165135412896\">55. [latex]g\\left(x\\right)=4f\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135193808\" class=\"solution\">\r\n<p id=\"fs-id1165135193810\">56.\u00a0[latex]g\\left(x\\right)=6f\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135440224\" class=\"exercise\">\r\n<div id=\"fs-id1165135440226\" class=\"problem\">\r\n<p id=\"fs-id1165135440229\">57. [latex]g\\left(x\\right)=f\\left(5x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135481187\" class=\"solution\">\r\n<p id=\"fs-id1165137558467\">58.\u00a0[latex]g\\left(x\\right)=f\\left(2x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133307633\" class=\"exercise\">\r\n<div id=\"fs-id1165133307635\" class=\"problem\">\r\n<p id=\"fs-id1165133307637\">59. [latex]g\\left(x\\right)=f\\left(\\frac{1}{3}x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137749379\" class=\"solution\">\r\n<p id=\"fs-id1165137749381\">60.\u00a0[latex]g\\left(x\\right)=f\\left(\\frac{1}{5}x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137664915\" class=\"exercise\">\r\n<div id=\"fs-id1165137664917\" class=\"problem\">\r\n<p id=\"fs-id1165137664919\">61. [latex]g\\left(x\\right)=3f\\left(-x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135301694\" class=\"solution\">\r\n<p id=\"fs-id1165135301696\">62.\u00a0[latex]g\\left(x\\right)=-f\\left(3x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165135637428\">For the following exercises, write a formula for the function [latex]g[\/latex] that results when the graph of a given toolkit function is transformed as described.<\/p>\r\n\r\n<div id=\"fs-id1165135195127\" class=\"exercise\">\r\n<div id=\"fs-id1165135195130\" class=\"problem\">\r\n<p id=\"fs-id1165135195132\">63. The graph of [latex]f\\left(x\\right)=|x|[\/latex] is reflected over the [latex]y[\/latex] <em>-<\/em>axis and horizontally compressed by a factor of [latex]\\frac{1}{4}[\/latex]\u00a0.<\/p>\r\n64.\u00a0The graph of [latex]f\\left(x\\right)=\\sqrt{x}[\/latex] is reflected over the [latex]x[\/latex] -axis and horizontally stretched by a factor of 2.\r\n<div id=\"fs-id1165137634443\" class=\"exercise\">\r\n<div id=\"fs-id1165137634445\" class=\"problem\">\r\n<p id=\"fs-id1165137634448\">65. The graph of [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex] is vertically compressed by a factor of [latex]\\frac{1}{3}[\/latex], then shifted to the left 2 units and down 3 units.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137812539\" class=\"solution\">\r\n<p id=\"fs-id1165137812541\">66.\u00a0The graph of [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] is vertically stretched by a factor of 8, then shifted to the right 4 units and up 2 units.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137642586\" class=\"exercise\">\r\n<div id=\"fs-id1165137642588\" class=\"problem\">\r\n<p id=\"fs-id1165137642590\">67. The graph of [latex]f\\left(x\\right)={x}^{2}[\/latex] is vertically compressed by a factor of [latex]\\frac{1}{2}[\/latex], then shifted to the right 5 units and up 1 unit.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165133047549\" class=\"solution\">\r\n<p id=\"fs-id1165133047551\">68. The graph of [latex]f\\left(x\\right)={x}^{2}[\/latex] is horizontally stretched by a factor of 3, then shifted to the left 4 units and down 3 units.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137668699\">For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.<\/p>\r\n\r\n<div id=\"fs-id1165137668704\" class=\"exercise\">\r\n<div id=\"fs-id1165137668706\" class=\"problem\">\r\n<p id=\"fs-id1165137668708\">69. [latex]g\\left(x\\right)=4{\\left(x+1\\right)}^{2}-5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137445949\" class=\"solution\"><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135199465\" class=\"exercise\">\r\n<div id=\"fs-id1165135199468\" class=\"problem\">\r\n<p id=\"fs-id1165137526795\">70. [latex]g\\left(x\\right)=5{\\left(x+3\\right)}^{2}-2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134069304\" class=\"exercise\">\r\n<div id=\"fs-id1165134069306\" class=\"problem\">\r\n<p id=\"fs-id1165134069308\">71. [latex]h\\left(x\\right)=-2|x - 4|+3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137762365\" class=\"solution\"><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137758532\" class=\"exercise\">\r\n<div id=\"fs-id1165137758534\" class=\"problem\">\r\n<p id=\"fs-id1165135693772\">72. [latex]k\\left(x\\right)=-3\\sqrt{x}-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135351654\" class=\"exercise\">\r\n<div id=\"fs-id1165135351656\" class=\"problem\">\r\n<p id=\"fs-id1165135351658\">73. [latex]m\\left(x\\right)=\\frac{1}{2}{x}^{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137817390\" class=\"solution\"><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137832423\" class=\"exercise\">\r\n<div id=\"fs-id1165137832425\" class=\"problem\">\r\n<p id=\"fs-id1165137832427\">74. [latex]n\\left(x\\right)=\\frac{1}{3}|x - 2|[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134155168\" class=\"exercise\">\r\n<div id=\"fs-id1165134155170\" class=\"problem\">\r\n<p id=\"fs-id1165134155172\">75. [latex]p\\left(x\\right)={\\left(\\frac{1}{3}x\\right)}^{3}-3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135253220\" class=\"solution\">\r\n<p id=\"fs-id1165135253222\">76.\u00a0[latex]q\\left(x\\right)={\\left(\\frac{1}{4}x\\right)}^{3}+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137861993\" class=\"exercise\">\r\n<div id=\"fs-id1165137861995\" class=\"problem\">\r\n<p id=\"fs-id1165137861997\">77. [latex]a\\left(x\\right)=\\sqrt{-x+4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135433479\" class=\"solution\"><\/div>\r\n<\/div>\r\n<p id=\"fs-id1165134338807\">For the following exercises, use the graph below\u00a0to sketch the given transformations.<span id=\"fs-id1165134338818\">\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005106\/CNX_Precalc_Figure_01_05_233.jpg\" alt=\"Graph of a polynomial.\" width=\"731\" height=\"566\" \/><\/span><\/p>\r\n\r\n<div id=\"fs-id1165135706785\" class=\"exercise\">\r\n<div id=\"fs-id1165135208393\" class=\"problem\">\r\n<p id=\"fs-id1165135208395\">78. [latex]g\\left(x\\right)=f\\left(x\\right)-2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135432954\" class=\"exercise\">\r\n<div id=\"fs-id1165135432956\" class=\"problem\">\r\n<p id=\"fs-id1165135432958\">79. [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137936918\" class=\"solution\"><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137722436\" class=\"exercise\">\r\n<div id=\"fs-id1165137722438\" class=\"problem\">\r\n<p id=\"fs-id1165137722441\">80. [latex]g\\left(x\\right)=f\\left(x+1\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134269560\" class=\"exercise\">\r\n<div id=\"fs-id1165134269563\" class=\"problem\">\r\n<p id=\"fs-id1165134269565\">81. [latex]g\\left(x\\right)=f\\left(x - 2\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134211351\" class=\"solution\"><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<p>1. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal shift from a vertical shift?<\/p>\n<p>2. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal stretch from a vertical stretch?<\/p>\n<p>3. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal compression from a vertical compression?<\/p>\n<p>4. When examining the formula of a function that is the result of multiple transformations, how can you tell a reflection with respect to the x-axis from a reflection with respect to the y-axis?<\/p>\n<p>5. How can you determine whether a function is odd or even from the formula of the function?<\/p>\n<p>6.\u00a0Write a formula for the function obtained when the graph of [latex]f\\left(x\\right)=\\sqrt{x}[\/latex] is shifted up 1 unit and to the left 2 units.<\/p>\n<p>7. Write a formula for the function obtained when the graph of [latex]f\\left(x\\right)=|x|[\/latex]<br \/>\nis shifted down 3 units and to the right 1 unit.<\/p>\n<p>8.\u00a0Write a formula for the function obtained when the graph of [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] is shifted down 4 units and to the right 3 units.<\/p>\n<p>9. Write a formula for the function obtained when the graph of [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex] is shifted up 2 units and to the left 4 units.<\/p>\n<p>For the following exercises, describe how the graph of the function is a transformation of the graph of the original function [latex]f[\/latex].<\/p>\n<p>10. [latex]y=f\\left(x - 49\\right)[\/latex]<\/p>\n<p>11. [latex]y=f\\left(x+43\\right)[\/latex]<\/p>\n<p>12.\u00a0[latex]y=f\\left(x+3\\right)[\/latex]<\/p>\n<p>13. [latex]y=f\\left(x - 4\\right)[\/latex]<\/p>\n<p>14.\u00a0[latex]y=f\\left(x\\right)+5[\/latex]<\/p>\n<p>15. [latex]y=f\\left(x\\right)+8[\/latex]<\/p>\n<p>16.\u00a0[latex]y=f\\left(x\\right)-2[\/latex]<\/p>\n<p>17. [latex]y=f\\left(x\\right)-7[\/latex]<\/p>\n<p>18.\u00a0[latex]y=f\\left(x - 2\\right)+3[\/latex]<\/p>\n<p>19. [latex]y=f\\left(x+4\\right)-1[\/latex]<\/p>\n<p>For the following exercises, determine the interval(s) on which the function is increasing and decreasing.<\/p>\n<p>20. [latex]f\\left(x\\right)=4{\\left(x+1\\right)}^{2}-5[\/latex]<\/p>\n<p>21. [latex]g\\left(x\\right)=5{\\left(x+3\\right)}^{2}-2[\/latex]<\/p>\n<p>22.\u00a0[latex]a\\left(x\\right)=\\sqrt{-x+4}[\/latex]<\/p>\n<p>23. [latex]k\\left(x\\right)=-3\\sqrt{x}-1[\/latex]<\/p>\n<p>For the following exercises, use the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex]\u00a0to sketch a graph of each transformation of [latex]f\\left(x\\right)[\/latex].<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005101\/CNX_Precalc_Figure_01_05_201.jpg\" alt=\"Graph of f(x) increasing on (0, oo), approaching y = 0 on (-oo,0), passing through the point (1,1).\" width=\"487\" height=\"366\" \/><\/p>\n<p>24. [latex]g\\left(x\\right)={2}^{x}+1[\/latex]<\/p>\n<p>25. [latex]h\\left(x\\right)={2}^{x}-3[\/latex]<\/p>\n<p>26. [latex]w\\left(x\\right)={2}^{x - 1}[\/latex]<\/p>\n<p>For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.<\/p>\n<p>27. [latex]f\\left(t\\right)={\\left(t+1\\right)}^{2}-3[\/latex]<\/p>\n<p>28.\u00a0[latex]h\\left(x\\right)=|x - 1|+4[\/latex]<\/p>\n<p>29. [latex]k\\left(x\\right)={\\left(x - 2\\right)}^{3}-1[\/latex]<\/p>\n<p>30.\u00a0[latex]m\\left(t\\right)=3+\\sqrt{t+2}[\/latex]<\/p>\n<p>31.\u00a0Tabular representations for the functions [latex]f,g[\/latex], and [latex]h[\/latex] are given below. Write [latex]g\\left(x\\right)[\/latex] and [latex]h\\left(x\\right)[\/latex] as transformations of [latex]f\\left(x\\right)[\/latex].<\/p>\n<table id=\"fs-id1165137432561\" class=\"unnumbered\" style=\"line-height: 1.5;\" summary=\"Two rows and six columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>\u22122<\/td>\n<td>\u22121<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>\u22122<\/td>\n<td>\u22121<\/td>\n<td>\u22123<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1165137734659\" class=\"exercise\">\n<div id=\"fs-id1165137644805\" class=\"solution\">\n<div id=\"fs-id1165135650778\" class=\"exercise\">\n<div id=\"fs-id1165135628497\" class=\"solution\">\n<div id=\"fs-id1165135421533\" class=\"exercise\">\n<div id=\"fs-id1165134234193\" class=\"solution\">\n<div id=\"fs-id1165137681998\" class=\"exercise\">\n<div id=\"fs-id1165137682000\" class=\"problem\">\n<table id=\"fs-id1165135634096\" class=\"unnumbered\" summary=\"Two rows and six columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>\u22121<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>\u22122<\/td>\n<td>\u22121<\/td>\n<td>\u22123<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"fs-id1165135330589\" class=\"unnumbered\" summary=\"Two rows and six columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>\u22122<\/td>\n<td>\u22121<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]h\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>\u22121<\/td>\n<td>0<\/td>\n<td>\u22122<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137443424\" class=\"solution\">\n<p id=\"fs-id1165134211288\">32.\u00a0Tabular representations for the functions [latex]f,g[\/latex], and [latex]h[\/latex] are given below. Write [latex]g\\left(x\\right)[\/latex] and [latex]h\\left(x\\right)[\/latex] as transformations of [latex]f\\left(x\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137734475\" class=\"exercise\">\n<div id=\"fs-id1165137734477\" class=\"problem\">\n<table id=\"fs-id1165134558032\" class=\"unnumbered\" summary=\"Two rows and six columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>\u22122<\/td>\n<td>\u22121<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>\u22121<\/td>\n<td>\u22123<\/td>\n<td>4<\/td>\n<td>2<\/td>\n<td>1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"fs-id1165134380916\" class=\"unnumbered\" summary=\"Two rows and six columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>\u22123<\/td>\n<td>\u22122<\/td>\n<td>\u22121<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>\u22121<\/td>\n<td>\u22123<\/td>\n<td>4<\/td>\n<td>2<\/td>\n<td>1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"fs-id1165137894261\" class=\"unnumbered\" summary=\"Two rows and six columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>\u22122<\/td>\n<td>\u22121<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]h\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>\u22122<\/td>\n<td>\u22124<\/td>\n<td>3<\/td>\n<td>1<\/td>\n<td>0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137570566\">For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.<\/p>\n<div id=\"fs-id1165137431229\" class=\"exercise\">\n<div id=\"fs-id1165137431231\" class=\"problem\">\n<p><span id=\"fs-id1165135543438\">33.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005102\/CNX_Precalc_Figure_01_05_210.jpg\" alt=\"Graph of an absolute function with vertex at (3,-2), decreasing on (-oo,3) and increasing on (3,oo).\" width=\"487\" height=\"317\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1165135516945\" class=\"solution\">\n<p id=\"fs-id1165135516948\">34.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135481230\" class=\"exercise\">\n<div id=\"fs-id1165135481232\" class=\"problem\">\n<p><span id=\"fs-id1165137851362\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005102\/CNX_Precalc_Figure_01_05_211.jpg\" alt=\"Graph of a parabola with vertex at (1,-3), decreasing on (-oo,1) and increasing on (1,oo).\" width=\"487\" height=\"314\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137817635\" class=\"exercise\">\n<div id=\"fs-id1165137817637\" class=\"problem\">\n<p><span id=\"fs-id1165133341017\">35.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005102\/CNX_Precalc_Figure_01_05_212.jpg\" alt=\"Graph of a square root function originating at (-3,-1), increasing on [-3,oo).\" width=\"487\" height=\"317\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1165135190190\" class=\"solution\">\n<p id=\"fs-id1165135190193\">36.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132929618\" class=\"exercise\">\n<div id=\"fs-id1165132929620\" class=\"problem\">\n<p><span id=\"fs-id1165135203675\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005103\/CNX_Precalc_Figure_01_05_213.jpg\" alt=\"Graph of an absolute function with vertex at (-2, 2), decreasing on (-oo,-2) and increasing on (-2,oo).\" width=\"487\" height=\"409\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135487204\" class=\"exercise\">\n<div id=\"fs-id1165135487206\" class=\"problem\">\n<p><span id=\"fs-id1165135535017\">37.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005103\/CNX_Precalc_Figure_01_05_214.jpg\" alt=\"Graph of a parabola with vertex at (2,0), decreasingon (-inf., 2), increasing on (2, inf.)\" width=\"487\" height=\"409\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1165135560630\" class=\"solution\">\n<p id=\"fs-id1165135560633\">38.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135190411\" class=\"exercise\">\n<div id=\"fs-id1165135190413\" class=\"problem\">\n<p><span id=\"fs-id1165133277626\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005103\/CNX_Precalc_Figure_01_05_215.jpg\" alt=\"Graph of a square root function originating at (-3,0) increasing on [-3, inf) and passing through (1,2).\" width=\"487\" height=\"226\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133103936\" class=\"exercise\">\n<div id=\"fs-id1165133103938\" class=\"problem\">\n<p><span id=\"fs-id1165134362846\">39.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005103\/CNX_Precalc_Figure_01_05_216f.jpg\" alt=\"Graph of an absolute function with vertex at (-3,-2), decreasing on (-inf., -3) and increasing on (-3,inf.), passing through (0,1).\" width=\"487\" height=\"379\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1165133030812\" class=\"solution\">\n<p id=\"fs-id1165133030814\">40.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134043550\" class=\"exercise\">\n<div id=\"fs-id1165134043552\" class=\"problem\">\n<p><span id=\"fs-id1165134036728\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005104\/CNX_Precalc_Figure_01_05_217F.jpg\" alt=\"Graph of a square root function originating at (-2, -2) increasing on [-2, inf) and passing through (2,0).\" width=\"487\" height=\"378\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165134187273\">For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.<\/p>\n<div id=\"fs-id1165134187277\" class=\"exercise\">\n<div id=\"fs-id1165134187279\" class=\"problem\">\n<p><span id=\"fs-id1165137834957\">41.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005104\/CNX_Precalc_Figure_01_05_218.jpg\" alt=\"Graph of a square root function originating at (0,0) decreasing on (0,inf) passing through (4,-2).\" width=\"487\" height=\"225\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1165134190724\" class=\"solution\">\n<p id=\"fs-id1165134190726\">42.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137930320\" class=\"exercise\">\n<div id=\"fs-id1165137930323\" class=\"problem\"><span id=\"fs-id1165135487154\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005104\/CNX_Precalc_Figure_01_05_219.jpg\" alt=\"Graph of a square root function originating at (0,0) decreasing on (-inf., 0) passing through (-1,1).\" width=\"487\" height=\"222\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1165134177109\">For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.<\/p>\n<div id=\"fs-id1165134177113\" class=\"exercise\">\n<div id=\"fs-id1165137806559\" class=\"problem\">\n<p><span id=\"fs-id1165137806566\">43.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005104\/CNX_Precalc_Figure_01_05_220.jpg\" alt=\"Graph of a parabola concave down, vertex at (-1,2).\" width=\"487\" height=\"283\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1165137889877\" class=\"solution\">\n<p id=\"fs-id1165134261684\">44.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135384400\" class=\"exercise\">\n<div id=\"fs-id1165137693606\" class=\"problem\">\n<p><span id=\"fs-id1165137693612\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005105\/CNX_Precalc_Figure_01_05_221.jpg\" alt=\"Graph of a cubic function passing through (2,1) .\" width=\"487\" height=\"251\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137892243\" class=\"exercise\">\n<div id=\"fs-id1165134352554\" class=\"problem\">\n<p><span id=\"fs-id1165134352561\">45.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005105\/CNX_Precalc_Figure_01_05_222.jpg\" alt=\"Graph of a square root function concave down, originating at (0,1) decreasing on (-inf., 0).\" width=\"487\" height=\"191\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1165134159665\" class=\"solution\">\n<p id=\"fs-id1165134159668\">46.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135411377\" class=\"exercise\">\n<div id=\"fs-id1165135411379\" class=\"problem\">\n<p><span id=\"fs-id1165133155251\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005105\/CNX_Precalc_Figure_01_05_223.jpg\" alt=\"Graph of an absolute function downward facing, vertex at (2,3).\" width=\"487\" height=\"285\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165132924966\">For the following exercises, determine whether the function is odd, even, or neither.<\/p>\n<div id=\"fs-id1165132924969\" class=\"exercise\">\n<div id=\"fs-id1165132924971\" class=\"problem\">\n<p id=\"fs-id1165137812602\">47. [latex]f\\left(x\\right)=3{x}^{4}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135671506\" class=\"solution\">\n<p id=\"fs-id1165135671508\">48.\u00a0[latex]g\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137828008\" class=\"exercise\">\n<div id=\"fs-id1165137828010\" class=\"problem\">\n<p id=\"fs-id1165133408839\">49. [latex]h\\left(x\\right)=\\frac{1}{x}+3x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134187165\" class=\"exercise\">\n<div id=\"fs-id1165134187167\" class=\"problem\">\n<p id=\"fs-id1165134271332\">50. [latex]f\\left(x\\right)={\\left(x - 2\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134039317\" class=\"exercise\">\n<div id=\"fs-id1165134039319\" class=\"problem\">\n<p id=\"fs-id1165135238484\">51. [latex]g\\left(x\\right)=2{x}^{4}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137679220\" class=\"solution\">\n<p id=\"fs-id1165137679222\">52.\u00a0[latex]h\\left(x\\right)=2x-{x}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137571611\">For the following exercises, describe how the graph of each function is a transformation of the graph of the original function [latex]f[\/latex].<\/p>\n<div id=\"fs-id1165137599981\" class=\"exercise\">\n<div id=\"fs-id1165137599983\" class=\"problem\">\n<p id=\"fs-id1165137599985\">53. [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135208810\" class=\"solution\">\n<p id=\"fs-id1165135400954\">54.\u00a0[latex]g\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135412892\" class=\"exercise\">\n<div id=\"fs-id1165135412894\" class=\"problem\">\n<p id=\"fs-id1165135412896\">55. [latex]g\\left(x\\right)=4f\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135193808\" class=\"solution\">\n<p id=\"fs-id1165135193810\">56.\u00a0[latex]g\\left(x\\right)=6f\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135440224\" class=\"exercise\">\n<div id=\"fs-id1165135440226\" class=\"problem\">\n<p id=\"fs-id1165135440229\">57. [latex]g\\left(x\\right)=f\\left(5x\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135481187\" class=\"solution\">\n<p id=\"fs-id1165137558467\">58.\u00a0[latex]g\\left(x\\right)=f\\left(2x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133307633\" class=\"exercise\">\n<div id=\"fs-id1165133307635\" class=\"problem\">\n<p id=\"fs-id1165133307637\">59. [latex]g\\left(x\\right)=f\\left(\\frac{1}{3}x\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137749379\" class=\"solution\">\n<p id=\"fs-id1165137749381\">60.\u00a0[latex]g\\left(x\\right)=f\\left(\\frac{1}{5}x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137664915\" class=\"exercise\">\n<div id=\"fs-id1165137664917\" class=\"problem\">\n<p id=\"fs-id1165137664919\">61. [latex]g\\left(x\\right)=3f\\left(-x\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135301694\" class=\"solution\">\n<p id=\"fs-id1165135301696\">62.\u00a0[latex]g\\left(x\\right)=-f\\left(3x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135637428\">For the following exercises, write a formula for the function [latex]g[\/latex] that results when the graph of a given toolkit function is transformed as described.<\/p>\n<div id=\"fs-id1165135195127\" class=\"exercise\">\n<div id=\"fs-id1165135195130\" class=\"problem\">\n<p id=\"fs-id1165135195132\">63. The graph of [latex]f\\left(x\\right)=|x|[\/latex] is reflected over the [latex]y[\/latex] <em>&#8211;<\/em>axis and horizontally compressed by a factor of [latex]\\frac{1}{4}[\/latex]\u00a0.<\/p>\n<p>64.\u00a0The graph of [latex]f\\left(x\\right)=\\sqrt{x}[\/latex] is reflected over the [latex]x[\/latex] -axis and horizontally stretched by a factor of 2.<\/p>\n<div id=\"fs-id1165137634443\" class=\"exercise\">\n<div id=\"fs-id1165137634445\" class=\"problem\">\n<p id=\"fs-id1165137634448\">65. The graph of [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex] is vertically compressed by a factor of [latex]\\frac{1}{3}[\/latex], then shifted to the left 2 units and down 3 units.<\/p>\n<\/div>\n<div id=\"fs-id1165137812539\" class=\"solution\">\n<p id=\"fs-id1165137812541\">66.\u00a0The graph of [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] is vertically stretched by a factor of 8, then shifted to the right 4 units and up 2 units.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137642586\" class=\"exercise\">\n<div id=\"fs-id1165137642588\" class=\"problem\">\n<p id=\"fs-id1165137642590\">67. The graph of [latex]f\\left(x\\right)={x}^{2}[\/latex] is vertically compressed by a factor of [latex]\\frac{1}{2}[\/latex], then shifted to the right 5 units and up 1 unit.<\/p>\n<\/div>\n<div id=\"fs-id1165133047549\" class=\"solution\">\n<p id=\"fs-id1165133047551\">68. The graph of [latex]f\\left(x\\right)={x}^{2}[\/latex] is horizontally stretched by a factor of 3, then shifted to the left 4 units and down 3 units.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137668699\">For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.<\/p>\n<div id=\"fs-id1165137668704\" class=\"exercise\">\n<div id=\"fs-id1165137668706\" class=\"problem\">\n<p id=\"fs-id1165137668708\">69. [latex]g\\left(x\\right)=4{\\left(x+1\\right)}^{2}-5[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137445949\" class=\"solution\"><\/div>\n<\/div>\n<div id=\"fs-id1165135199465\" class=\"exercise\">\n<div id=\"fs-id1165135199468\" class=\"problem\">\n<p id=\"fs-id1165137526795\">70. [latex]g\\left(x\\right)=5{\\left(x+3\\right)}^{2}-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134069304\" class=\"exercise\">\n<div id=\"fs-id1165134069306\" class=\"problem\">\n<p id=\"fs-id1165134069308\">71. [latex]h\\left(x\\right)=-2|x - 4|+3[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137762365\" class=\"solution\"><\/div>\n<\/div>\n<div id=\"fs-id1165137758532\" class=\"exercise\">\n<div id=\"fs-id1165137758534\" class=\"problem\">\n<p id=\"fs-id1165135693772\">72. [latex]k\\left(x\\right)=-3\\sqrt{x}-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135351654\" class=\"exercise\">\n<div id=\"fs-id1165135351656\" class=\"problem\">\n<p id=\"fs-id1165135351658\">73. [latex]m\\left(x\\right)=\\frac{1}{2}{x}^{3}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137817390\" class=\"solution\"><\/div>\n<\/div>\n<div id=\"fs-id1165137832423\" class=\"exercise\">\n<div id=\"fs-id1165137832425\" class=\"problem\">\n<p id=\"fs-id1165137832427\">74. [latex]n\\left(x\\right)=\\frac{1}{3}|x - 2|[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134155168\" class=\"exercise\">\n<div id=\"fs-id1165134155170\" class=\"problem\">\n<p id=\"fs-id1165134155172\">75. [latex]p\\left(x\\right)={\\left(\\frac{1}{3}x\\right)}^{3}-3[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135253220\" class=\"solution\">\n<p id=\"fs-id1165135253222\">76.\u00a0[latex]q\\left(x\\right)={\\left(\\frac{1}{4}x\\right)}^{3}+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137861993\" class=\"exercise\">\n<div id=\"fs-id1165137861995\" class=\"problem\">\n<p id=\"fs-id1165137861997\">77. [latex]a\\left(x\\right)=\\sqrt{-x+4}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135433479\" class=\"solution\"><\/div>\n<\/div>\n<p id=\"fs-id1165134338807\">For the following exercises, use the graph below\u00a0to sketch the given transformations.<span id=\"fs-id1165134338818\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005106\/CNX_Precalc_Figure_01_05_233.jpg\" alt=\"Graph of a polynomial.\" width=\"731\" height=\"566\" \/><\/span><\/p>\n<div id=\"fs-id1165135706785\" class=\"exercise\">\n<div id=\"fs-id1165135208393\" class=\"problem\">\n<p id=\"fs-id1165135208395\">78. [latex]g\\left(x\\right)=f\\left(x\\right)-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135432954\" class=\"exercise\">\n<div id=\"fs-id1165135432956\" class=\"problem\">\n<p id=\"fs-id1165135432958\">79. [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137936918\" class=\"solution\"><\/div>\n<\/div>\n<div id=\"fs-id1165137722436\" class=\"exercise\">\n<div id=\"fs-id1165137722438\" class=\"problem\">\n<p id=\"fs-id1165137722441\">80. [latex]g\\left(x\\right)=f\\left(x+1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134269560\" class=\"exercise\">\n<div id=\"fs-id1165134269563\" class=\"problem\">\n<p id=\"fs-id1165134269565\">81. [latex]g\\left(x\\right)=f\\left(x - 2\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134211351\" class=\"solution\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-15506\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169554,"menu_order":16,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-15506","chapter","type-chapter","status-publish","hentry"],"part":10705,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/15506","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/169554"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/15506\/revisions"}],"predecessor-version":[{"id":15566,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/15506\/revisions\/15566"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/10705"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/15506\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=15506"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=15506"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=15506"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=15506"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}