{"id":185,"date":"2026-01-12T16:01:05","date_gmt":"2026-01-12T16:01:05","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/?post_type=chapter&p=185"},"modified":"2026-01-12T16:01:05","modified_gmt":"2026-01-12T16:01:05","slug":"parametric-curves-and-their-applications-get-stronger-answer-key","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/parametric-curves-and-their-applications-get-stronger-answer-key\/","title":{"raw":"Parametric Curves and Their Applications: Get Stronger Answer Key","rendered":"Parametric Curves and Their Applications: Get Stronger Answer Key"},"content":{"raw":"

Fundamentals of Parametric Equations<\/span><\/h2>\r\n
    \r\n \t
  1. \"A\r\norientation: bottom to top<\/li>\r\n \t
  2. \"A\r\norientation: left to right<\/li>\r\n \t
  3. \"Half<\/li>\r\n \t
  4. \"A<\/li>\r\n \t
  5. \"A<\/li>\r\n \t
  6. \"An<\/li>\r\n \t
  7. \"An<\/li>\r\n \t
  8. \"A\r\nAsymptotes are [latex]y=x[\/latex] and [latex]y=\\text{-}x[\/latex]<\/li>\r\n \t
  9. \"A<\/li>\r\n \t
  10. \"A<\/li>\r\n \t
  11. [latex]y=\\dfrac{\\sqrt{x+1}}{2}[\/latex]; domain: [latex]x\\in \\left[1,\\infty \\right)[\/latex].<\/li>\r\n \t
  12. [latex]\\dfrac{{x}^{2}}{16}+\\dfrac{{y}^{2}}{9}=1[\/latex]; domain [latex]x\\in \\left[-4,4\\right][\/latex].<\/li>\r\n \t
  13. [latex]y=3x+2[\/latex]; domain: all real numbers.<\/li>\r\n \t
  14. [latex]{\\left(x - 1\\right)}^{2}+{\\left(y - 3\\right)}^{2}=1[\/latex]; domain: [latex]x\\in \\left[0,2\\right][\/latex].<\/li>\r\n \t
  15. [latex]y=\\sqrt{{x}^{2}-1}[\/latex]; domain: [latex]x\\in \\left[-\\infty,1\\right][\/latex].<\/li>\r\n \t
  16. [latex]{y}^{2}=\\dfrac{1-x}{2}[\/latex]; domain: [latex]x\\in \\left[2,\\infty \\right)\\cup \\left(\\text{-}\\infty ,-2\\right][\/latex].<\/li>\r\n \t
  17. [latex]y=\\text{ln}x[\/latex]; domain: [latex]x\\in \\left(1,\\infty \\right)[\/latex].<\/li>\r\n \t
  18. [latex]y=\\text{ln}x[\/latex]; domain: [latex]x\\in \\left(0,\\infty \\right)[\/latex].<\/li>\r\n \t
  19. [latex]{x}^{2}+{y}^{2}=4[\/latex]; domain: [latex]x\\in \\left[-2,2\\right][\/latex].<\/li>\r\n \t
  20. line<\/li>\r\n \t
  21. parabola<\/li>\r\n \t
  22. circle<\/li>\r\n \t
  23. ellipse<\/li>\r\n \t
  24. hyperbola<\/li>\r\n \t
  25. \"A\r\nThe equations represent a cycloid.<\/li>\r\n \t
  26. \"A<\/li>\r\n<\/ol>\r\n

    Calculus with Parametric Curves<\/span><\/h2>\r\n
      \r\n \t
    1. [latex]0[\/latex]<\/li>\r\n \t
    2. [latex]\\dfrac{-3}{5}[\/latex]<\/li>\r\n \t
    3. [latex]\\text{Slope}=0[\/latex]; [latex]y=8[\/latex].<\/li>\r\n \t
    4. Slope is undefined; [latex]x=2[\/latex].<\/li>\r\n \t
    5. [latex]\\tan{t}=\\left(-2\\right)[\/latex] [latex]\\left(\\dfrac{4}{\\sqrt{5}},\\dfrac{-8}{\\sqrt{5}}\\right),\\left(\\dfrac{4}{\\sqrt{5}},\\dfrac{-8}{\\sqrt{5}}\\right)[\/latex].<\/li>\r\n \t
    6. No points possible; undefined expression.<\/li>\r\n \t
    7. [latex]y=\\text{-}\\left(\\dfrac{4}{e}\\right)x+5[\/latex]<\/li>\r\n \t
    8. [latex]y=-2x + 3[\/latex]<\/li>\r\n \t
    9. [latex]\\dfrac{\\pi }{4},\\dfrac{5\\pi }{4},\\dfrac{3\\pi }{4},\\dfrac{7\\pi }{4}[\/latex]<\/li>\r\n \t
    10. [latex]\\dfrac{dy}{dx}=\\text{-}\\tan\\left(t\\right)[\/latex]<\/li>\r\n \t
    11. [latex]\\dfrac{dy}{dx}=\\dfrac{3}{4}[\/latex] and [latex]\\dfrac{{d}^{2}y}{d{x}^{2}}=0[\/latex], so the curve is neither concave up nor concave down at [latex]t=3[\/latex]. Therefore the graph is linear and has a constant slope but no concavity.<\/li>\r\n \t
    12. [latex]\\dfrac{dy}{dx}=4,\\dfrac{{d}^{2}y}{d{x}^{2}}=-6\\sqrt{3}[\/latex]; the curve is concave down at [latex]\\theta =\\dfrac{\\pi }{6}[\/latex].<\/li>\r\n \t
    13. No horizontal tangents. Vertical tangents at [latex]\\left(1,0\\right),\\left(-1,0\\right)[\/latex].<\/li>\r\n \t
    14. [latex]\\text{-}{\\sec}^{3}\\left(\\pi t\\right)[\/latex]<\/li>\r\n \t
    15. Horizontal [latex]\\left(0,-9\\right)[\/latex]; vertical [latex]\\left(\\pm2,-6\\right)[\/latex].<\/li>\r\n \t
    16. [latex]1[\/latex]<\/li>\r\n \t
    17. [latex]0[\/latex]<\/li>\r\n \t
    18. [latex]4[\/latex]<\/li>\r\n \t
    19. Concave up on [latex]t>0[\/latex].<\/li>\r\n \t
    20. [latex]\\dfrac{e\\frac{1}{2}-1}{2}[\/latex]<\/li>\r\n \t
    21. [latex]\\dfrac{3\\pi }{2}[\/latex]<\/li>\r\n \t
    22. [latex]6\\pi {a}^{2}[\/latex]<\/li>\r\n \t
    23. [latex]2\\pi ab[\/latex]<\/li>\r\n \t
    24. [latex]\\dfrac{1}{3}\\left(2\\sqrt{2}-1\\right)[\/latex]<\/li>\r\n \t
    25. [latex]7.075[\/latex]<\/li>\r\n \t
    26. [latex]6a[\/latex]<\/li>\r\n \t
    27. [latex]\\dfrac{2\\pi \\left(247\\sqrt{13}+64\\right)}{1215}[\/latex]<\/li>\r\n \t
    28. [latex]59.101[\/latex]<\/li>\r\n \t
    29. [latex]\\dfrac{8\\pi }{3}\\left(17\\sqrt{17}-1\\right)[\/latex]<\/li>\r\n<\/ol>","rendered":"

      Fundamentals of Parametric Equations<\/span><\/h2>\n
        \n
      1. \"A
        \norientation: bottom to top<\/li>\n
      2. \"A
        \norientation: left to right<\/li>\n
      3. \"Half<\/li>\n
      4. \"A<\/li>\n
      5. \"A<\/li>\n
      6. \"An<\/li>\n
      7. \"An<\/li>\n
      8. \"A
        \nAsymptotes are [latex]y=x[\/latex] and [latex]y=\\text{-}x[\/latex]<\/li>\n
      9. \"A<\/li>\n
      10. \"A<\/li>\n
      11. [latex]y=\\dfrac{\\sqrt{x+1}}{2}[\/latex]; domain: [latex]x\\in \\left[1,\\infty \\right)[\/latex].<\/li>\n
      12. [latex]\\dfrac{{x}^{2}}{16}+\\dfrac{{y}^{2}}{9}=1[\/latex]; domain [latex]x\\in \\left[-4,4\\right][\/latex].<\/li>\n
      13. [latex]y=3x+2[\/latex]; domain: all real numbers.<\/li>\n
      14. [latex]{\\left(x - 1\\right)}^{2}+{\\left(y - 3\\right)}^{2}=1[\/latex]; domain: [latex]x\\in \\left[0,2\\right][\/latex].<\/li>\n
      15. [latex]y=\\sqrt{{x}^{2}-1}[\/latex]; domain: [latex]x\\in \\left[-\\infty,1\\right][\/latex].<\/li>\n
      16. [latex]{y}^{2}=\\dfrac{1-x}{2}[\/latex]; domain: [latex]x\\in \\left[2,\\infty \\right)\\cup \\left(\\text{-}\\infty ,-2\\right][\/latex].<\/li>\n
      17. [latex]y=\\text{ln}x[\/latex]; domain: [latex]x\\in \\left(1,\\infty \\right)[\/latex].<\/li>\n
      18. [latex]y=\\text{ln}x[\/latex]; domain: [latex]x\\in \\left(0,\\infty \\right)[\/latex].<\/li>\n
      19. [latex]{x}^{2}+{y}^{2}=4[\/latex]; domain: [latex]x\\in \\left[-2,2\\right][\/latex].<\/li>\n
      20. line<\/li>\n
      21. parabola<\/li>\n
      22. circle<\/li>\n
      23. ellipse<\/li>\n
      24. hyperbola<\/li>\n
      25. \"A
        \nThe equations represent a cycloid.<\/li>\n
      26. \"A<\/li>\n<\/ol>\n

        Calculus with Parametric Curves<\/span><\/h2>\n
          \n
        1. [latex]0[\/latex]<\/li>\n
        2. [latex]\\dfrac{-3}{5}[\/latex]<\/li>\n
        3. [latex]\\text{Slope}=0[\/latex]; [latex]y=8[\/latex].<\/li>\n
        4. Slope is undefined; [latex]x=2[\/latex].<\/li>\n
        5. [latex]\\tan{t}=\\left(-2\\right)[\/latex] [latex]\\left(\\dfrac{4}{\\sqrt{5}},\\dfrac{-8}{\\sqrt{5}}\\right),\\left(\\dfrac{4}{\\sqrt{5}},\\dfrac{-8}{\\sqrt{5}}\\right)[\/latex].<\/li>\n
        6. No points possible; undefined expression.<\/li>\n
        7. [latex]y=\\text{-}\\left(\\dfrac{4}{e}\\right)x+5[\/latex]<\/li>\n
        8. [latex]y=-2x + 3[\/latex]<\/li>\n
        9. [latex]\\dfrac{\\pi }{4},\\dfrac{5\\pi }{4},\\dfrac{3\\pi }{4},\\dfrac{7\\pi }{4}[\/latex]<\/li>\n
        10. [latex]\\dfrac{dy}{dx}=\\text{-}\\tan\\left(t\\right)[\/latex]<\/li>\n
        11. [latex]\\dfrac{dy}{dx}=\\dfrac{3}{4}[\/latex] and [latex]\\dfrac{{d}^{2}y}{d{x}^{2}}=0[\/latex], so the curve is neither concave up nor concave down at [latex]t=3[\/latex]. Therefore the graph is linear and has a constant slope but no concavity.<\/li>\n
        12. [latex]\\dfrac{dy}{dx}=4,\\dfrac{{d}^{2}y}{d{x}^{2}}=-6\\sqrt{3}[\/latex]; the curve is concave down at [latex]\\theta =\\dfrac{\\pi }{6}[\/latex].<\/li>\n
        13. No horizontal tangents. Vertical tangents at [latex]\\left(1,0\\right),\\left(-1,0\\right)[\/latex].<\/li>\n
        14. [latex]\\text{-}{\\sec}^{3}\\left(\\pi t\\right)[\/latex]<\/li>\n
        15. Horizontal [latex]\\left(0,-9\\right)[\/latex]; vertical [latex]\\left(\\pm2,-6\\right)[\/latex].<\/li>\n
        16. [latex]1[\/latex]<\/li>\n
        17. [latex]0[\/latex]<\/li>\n
        18. [latex]4[\/latex]<\/li>\n
        19. Concave up on [latex]t>0[\/latex].<\/li>\n
        20. [latex]\\dfrac{e\\frac{1}{2}-1}{2}[\/latex]<\/li>\n
        21. [latex]\\dfrac{3\\pi }{2}[\/latex]<\/li>\n
        22. [latex]6\\pi {a}^{2}[\/latex]<\/li>\n
        23. [latex]2\\pi ab[\/latex]<\/li>\n
        24. [latex]\\dfrac{1}{3}\\left(2\\sqrt{2}-1\\right)[\/latex]<\/li>\n
        25. [latex]7.075[\/latex]<\/li>\n
        26. [latex]6a[\/latex]<\/li>\n
        27. [latex]\\dfrac{2\\pi \\left(247\\sqrt{13}+64\\right)}{1215}[\/latex]<\/li>\n
        28. [latex]59.101[\/latex]<\/li>\n
        29. [latex]\\dfrac{8\\pi }{3}\\left(17\\sqrt{17}-1\\right)[\/latex]<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":11,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":109,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/185"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":1,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/185\/revisions"}],"predecessor-version":[{"id":196,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/185\/revisions\/196"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/parts\/109"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/185\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=185"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=185"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=185"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=185"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}