{"id":4170,"date":"2024-09-18T15:58:53","date_gmt":"2024-09-18T15:58:53","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=4170"},"modified":"2024-11-21T17:44:38","modified_gmt":"2024-11-21T17:44:38","slug":"quadratic-functions-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/quadratic-functions-get-stronger\/","title":{"raw":"Quadratic Functions: Get Stronger","rendered":"Quadratic Functions: Get Stronger"},"content":{"raw":"

Introduction to Quadratic Functions and Parabolas<\/span><\/h2>\r\n

For the following exercises, rewrite the quadratic functions in standard form and give the vertex.<\/p>\r\n\r\n

    \r\n \t
  1. [latex]g(x)=x^2+2x\u22123[\/latex]<\/li>\r\n \t
  2. [latex]f(x)=x^2+5x\u22122[\/latex]<\/li>\r\n \t
  3. [latex]k(x)=3x^2\u22126x\u22129[\/latex]<\/li>\r\n \t
  4. [latex]f(x)=3x^2-5x-1[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.\r\n
      \r\n \t
    1. [latex]f(x)=2x^2\u221210x+4[\/latex]<\/li>\r\n \t
    2. [latex]f(x)=4x^2+x\u22121[\/latex]<\/li>\r\n \t
    3. [latex]f(x)=\\frac{1}{2}x^2+3x+1[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, determine the domain and range of the quadratic function.\r\n
        \r\n \t
      1. [latex]f(x)=(x\u22123)^2+2[\/latex]<\/li>\r\n \t
      2. [latex]f(x)=x^2+6x+4[\/latex]<\/li>\r\n \t
      3. [latex]k(x)=3x^2\u22126x\u22129[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, use the vertex [latex](h,k)[\/latex] and a point on the graph [latex](x,y)[\/latex] to find the general form of the equation of the quadratic function.\r\n
          \r\n \t
        1. [latex](h,k)=(\u22122,\u22121),(x,y)=(\u22124,3)[\/latex]<\/li>\r\n \t
        2. [latex](h,k)=(2,3),(x,y)=(5,12)[\/latex]<\/li>\r\n \t
        3. [latex](h,k)=(3,2),(x,y)=(10,1)[\/latex]<\/li>\r\n \t
        4. [latex](h,k)=(1,0),(x,y)=(0,1)[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.\r\n
            \r\n \t
          1. [latex]f(x)=x^2\u22126x\u22121[\/latex]<\/li>\r\n \t
          2. [latex]f(x)=x^2\u22127x+3[\/latex]<\/li>\r\n \t
          3. [latex]f(x)=4x^2\u221212x\u22123[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, write the equation for the graphed quadratic function.\r\n
              \r\n \t
            1. \"Graph<\/li>\r\n \t
            2. \"Graph<\/li>\r\n \t
            3. \"Graph<\/li>\r\n<\/ol>\r\n

              Complex Numbers and Operations<\/span><\/h2>\r\nFor the following exercises, plot each number in the complex plane.\r\n
                \r\n \t
              1. [latex]4[\/latex]<\/li>\r\n \t
              2. [latex]\u20133i[\/latex]<\/li>\r\n \t
              3. [latex]\u20132+3i[\/latex]<\/li>\r\n \t
              4. [latex]2 + i[\/latex]<\/li>\r\n \t
              5. [latex]-2[\/latex]<\/span><\/li>\r\n \t
              6. [latex]4i[\/latex]<\/span><\/li>\r\n \t
              7. [latex]1+2i[\/latex]<\/span><\/li>\r\n \t
              8. [latex]-1-i[\/latex]<\/span><\/li>\r\n<\/ol>\r\nFor the following exercises, solve.\r\n
                  \r\n \t
                1. [latex](2+3i) + (3-4i)[\/latex]<\/li>\r\n \t
                2. [latex](3-5i) - (-2-i)[\/latex]<\/li>\r\n \t
                3. [latex](1-i) + (2+4i)[\/latex]<\/li>\r\n \t
                4. [latex](\u00a0-2-3i) - (4-2i)[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, multiply the expressions.\r\n
                    \r\n \t
                  1. latex]3(2+4i)[\/latex]<\/li>\r\n \t
                  2. [latex](2i)(-1-5i)[\/latex]<\/li>\r\n \t
                  3. [latex](2-4i)(1+3i)[\/latex]<\/li>\r\n \t
                  4. [latex]2(-1+3i)[\/latex]<\/li>\r\n \t
                  5. [latex](3i)(2-6i)[\/latex]<\/li>\r\n \t
                  6. [latex](1-i)(2+5i)[\/latex]<\/li>\r\n<\/ol>\r\n \r\n
                      \r\n \t
                    1. Plot the number [latex]2+3i[\/latex]. Does multiplying by [latex]1-i[\/latex] move the point closer to or further from the origin? Does it rotate the point, and if so which direction?<\/li>\r\n \t
                    2. Plot the number [latex]2+3i[\/latex]. Does multiplying by [latex]0.75+0.5i[\/latex] move the point closer to or further from the origin? Does it rotate the point, and if so which direction.<\/li>\r\n<\/ol>\r\n

                      Application of Quadratic Functions<\/span><\/h2>\r\n
                        \r\n \t
                      1. Suppose that the price per unit in dollars of a cell phone production is modeled by [latex]p=$45-0.0125x[\/latex], where [latex]x[\/latex] is in thousands of phones produced, and the revenue represented by thousands of dollars is [latex]R=x\\cdot p[\/latex]. Find the production level that will maximize revenue.<\/li>\r\n \t
                      2. A ball is thrown in the air from the top of a building. Its height, in meters above ground, as a function of time, in seconds, is given by [latex]h(t)=-4.9t^2+24t+8[\/latex]. How long does it take to reach maximum height?<\/li>\r\n \t
                      3. A farmer finds that if she plants [latex]75[\/latex] trees per acre, each tree will yield [latex]20[\/latex] bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by [latex]3[\/latex] bushels. How many trees should she plant per acre to maximize her harvest?<\/li>\r\n<\/ol>","rendered":"

                        Introduction to Quadratic Functions and Parabolas<\/span><\/h2>\n

                        For the following exercises, rewrite the quadratic functions in standard form and give the vertex.<\/p>\n

                          \n
                        1. [latex]g(x)=x^2+2x\u22123[\/latex]<\/li>\n
                        2. [latex]f(x)=x^2+5x\u22122[\/latex]<\/li>\n
                        3. [latex]k(x)=3x^2\u22126x\u22129[\/latex]<\/li>\n
                        4. [latex]f(x)=3x^2-5x-1[\/latex]<\/li>\n<\/ol>\n

                          For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.<\/p>\n

                            \n
                          1. [latex]f(x)=2x^2\u221210x+4[\/latex]<\/li>\n
                          2. [latex]f(x)=4x^2+x\u22121[\/latex]<\/li>\n
                          3. [latex]f(x)=\\frac{1}{2}x^2+3x+1[\/latex]<\/li>\n<\/ol>\n

                            For the following exercises, determine the domain and range of the quadratic function.<\/p>\n

                              \n
                            1. [latex]f(x)=(x\u22123)^2+2[\/latex]<\/li>\n
                            2. [latex]f(x)=x^2+6x+4[\/latex]<\/li>\n
                            3. [latex]k(x)=3x^2\u22126x\u22129[\/latex]<\/li>\n<\/ol>\n

                              For the following exercises, use the vertex [latex](h,k)[\/latex] and a point on the graph [latex](x,y)[\/latex] to find the general form of the equation of the quadratic function.<\/p>\n

                                \n
                              1. [latex](h,k)=(\u22122,\u22121),(x,y)=(\u22124,3)[\/latex]<\/li>\n
                              2. [latex](h,k)=(2,3),(x,y)=(5,12)[\/latex]<\/li>\n
                              3. [latex](h,k)=(3,2),(x,y)=(10,1)[\/latex]<\/li>\n
                              4. [latex](h,k)=(1,0),(x,y)=(0,1)[\/latex]<\/li>\n<\/ol>\n

                                For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.<\/p>\n

                                  \n
                                1. [latex]f(x)=x^2\u22126x\u22121[\/latex]<\/li>\n
                                2. [latex]f(x)=x^2\u22127x+3[\/latex]<\/li>\n
                                3. [latex]f(x)=4x^2\u221212x\u22123[\/latex]<\/li>\n<\/ol>\n

                                  For the following exercises, write the equation for the graphed quadratic function.<\/p>\n

                                    \n
                                  1. \"Graph<\/li>\n
                                  2. \"Graph<\/li>\n
                                  3. \"Graph<\/li>\n<\/ol>\n

                                    Complex Numbers and Operations<\/span><\/h2>\n

                                    For the following exercises, plot each number in the complex plane.<\/p>\n

                                      \n
                                    1. [latex]4[\/latex]<\/li>\n
                                    2. [latex]\u20133i[\/latex]<\/li>\n
                                    3. [latex]\u20132+3i[\/latex]<\/li>\n
                                    4. [latex]2 + i[\/latex]<\/li>\n
                                    5. [latex]-2[\/latex]<\/span><\/li>\n
                                    6. [latex]4i[\/latex]<\/span><\/li>\n
                                    7. [latex]1+2i[\/latex]<\/span><\/li>\n
                                    8. [latex]-1-i[\/latex]<\/span><\/li>\n<\/ol>\n

                                      For the following exercises, solve.<\/p>\n

                                        \n
                                      1. [latex](2+3i) + (3-4i)[\/latex]<\/li>\n
                                      2. [latex](3-5i) - (-2-i)[\/latex]<\/li>\n
                                      3. [latex](1-i) + (2+4i)[\/latex]<\/li>\n
                                      4. [latex](\u00a0-2-3i) - (4-2i)[\/latex]<\/li>\n<\/ol>\n

                                        For the following exercises, multiply the expressions.<\/p>\n

                                          \n
                                        1. latex]3(2+4i)[\/latex]<\/li>\n
                                        2. [latex](2i)(-1-5i)[\/latex]<\/li>\n
                                        3. [latex](2-4i)(1+3i)[\/latex]<\/li>\n
                                        4. [latex]2(-1+3i)[\/latex]<\/li>\n
                                        5. [latex](3i)(2-6i)[\/latex]<\/li>\n
                                        6. [latex](1-i)(2+5i)[\/latex]<\/li>\n<\/ol>\n

                                           <\/p>\n

                                            \n
                                          1. Plot the number [latex]2+3i[\/latex]. Does multiplying by [latex]1-i[\/latex] move the point closer to or further from the origin? Does it rotate the point, and if so which direction?<\/li>\n
                                          2. Plot the number [latex]2+3i[\/latex]. Does multiplying by [latex]0.75+0.5i[\/latex] move the point closer to or further from the origin? Does it rotate the point, and if so which direction.<\/li>\n<\/ol>\n

                                            Application of Quadratic Functions<\/span><\/h2>\n
                                              \n
                                            1. Suppose that the price per unit in dollars of a cell phone production is modeled by [latex]p=$45-0.0125x[\/latex], where [latex]x[\/latex] is in thousands of phones produced, and the revenue represented by thousands of dollars is [latex]R=x\\cdot p[\/latex]. Find the production level that will maximize revenue.<\/li>\n
                                            2. A ball is thrown in the air from the top of a building. Its height, in meters above ground, as a function of time, in seconds, is given by [latex]h(t)=-4.9t^2+24t+8[\/latex]. How long does it take to reach maximum height?<\/li>\n
                                            3. A farmer finds that if she plants [latex]75[\/latex] trees per acre, each tree will yield [latex]20[\/latex] bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by [latex]3[\/latex] bushels. How many trees should she plant per acre to maximize her harvest?<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":23,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":185,"module-header":"practice","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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4170"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4170\/revisions"}],"predecessor-version":[{"id":5955,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4170\/revisions\/5955"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/185"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4170\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=4170"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=4170"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=4170"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=4170"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}