https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:quadratic-forms-features/v/interpret-quadratic-models-factored-form 2022-09-10T01:04:29.754011735Z Interpret quadratic models: Factored form Given a quadratic function that models a relationship, we can rewrite the function to reveal certain properties of the relationship. Factored form helps us identify the x-intercepts or zeros of the function. Sal Khan video https://cdn.kastatic.org/ka_thumbnails_cache/93772e35-6d9b-4087-b53e-9b0f03fbd865_1280_720_base.png Interpret quadratic models: Factored form Given a quadratic function that models a relationship, we can rewrite the function to reveal certain properties of the relationship. Factored form helps us identify the x-intercepts or zeros of the function. https://cdn.kastatic.org/ka-youtube-converted/ITBRCLusU1s.mp4/ITBRCLusU1s.mp4 240 Quadratic equations and functions https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:quadratic-forms-features/v/interpret-quadratic-models 2022-09-10T01:04:29.754011735Z Interpret quadratic models: Vertex form Given a quadratic function that models a relationship, we can rewrite the function to reveal certain properties of the relationship. Sal Khan video https://cdn.kastatic.org/ka_thumbnails_cache/e268af1e-4465-43ff-ab14-bddce8f28f96_1280_720_base.png Interpret quadratic models: Vertex form Given a quadratic function that models a relationship, we can rewrite the function to reveal certain properties of the relationship. https://cdn.kastatic.org/ka-youtube-converted/JdWEdb90V1s.mp4/JdWEdb90V1s.mp4 353 Quadratic equations and functions https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:quadratic-forms-features/e/interpreting-quadratic-models 2025-12-04T14:30:43.376793863Z Interpret quadratic models Given a quadratic function that models a relationship, rewrite the function to reveal properties of the relationship. Jeff Dodds exercise https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:vertex-form/v/vertex-form-intro 2021-06-30T17:32:38Z Vertex form introduction One of the common forms for quadratic functions is called vertex form, because it highlights the coordinates of the vertex of the function's graph. video https://cdn.kastatic.org/googleusercontent/cAD4buJDDstUZwLyYSMxCqMRrR83QC7L8kTAL4xJTNUII989_xtGR6jOEw8WqTg3vLhiM8M0MHtTylbLG7fAQIRh Vertex form introduction One of the common forms for quadratic functions is called vertex form, because it highlights the coordinates of the vertex of the function's graph. https://cdn.kastatic.org/ka-youtube-converted/_QqhuLixNEk.mp4/_QqhuLixNEk.mp4 373 Quadratic equations and functions https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:vertex-form/v/graphing-a-parabola-in-vertex-form 2021-06-30T17:32:36Z Graphing quadratics: vertex form Learn how to graph any quadratic function that is given in vertex form. Here, Sal graphs y=-2(x-2)²+5. Sal Khan video https://cdn.kastatic.org/googleusercontent/VwgW4fiqJMlUTcwMlUxaYRWsZjDL0rEuiNuqf2Yj0c5xiOwU71CUEpTdMdk3BlI29wHfhJgFDJGlg9c4U5EmRYVLDQ Graphing quadratics: vertex form Learn how to graph any quadratic function that is given in vertex form. Here, Sal graphs y=-2(x-2)²+5. https://cdn.kastatic.org/ka-youtube-converted/7QMoNY6FzvM.mp4/7QMoNY6FzvM.mp4 196 Quadratic equations and functions https://www.khanacademy.org/math/algebra-home/alg-quadratics/vertex-form-alg/e/graphing-quadratics-in-vertex-form-warmup 2026-04-07T22:23:57.656416987Z Warmup: graphing quadratics in vertex form Make sure you know how to graph quadratic functions that are given in vertex form f(x)=k(x-a)²+b, as a warmup to our exercise. Tomer Gal exercise https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:vertex-form/e/graphing_parabolas_1 2026-03-24T23:41:07.080866137Z Graph quadratics in vertex form Graph quadratic functions that are given in the vertex form a(x+b)²+c. For example, graph y=-2(x-2)²+5. Sid Patel exercise https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:intro-parabolas/v/parabolas-intro 2021-05-14T17:03:12Z Parabolas intro Graphs of quadratic functions all have the same shape which we call "parabola." All parabolas have shared characteristics. For example, they are all symmetric about a line that passes through their vertex. This video covers this and other basic facts about parabolas. video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Parabolas intro Graphs of quadratic functions all have the same shape which we call "parabola." All parabolas have shared characteristics. For example, they are all symmetric about a line that passes through their vertex. This video covers this and other basic facts about parabolas. https://cdn.kastatic.org/ka-youtube-converted/BGz3pkoGPag.mp4/BGz3pkoGPag.mp4 494 Quadratic equations and functions https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:intro-parabolas/e/parabolas-intro 2026-04-15T14:40:40.016885978Z Parabolas intro Make sure you understand the basic features of parabolas: vertex, axis of symmetry, intercepts, parabolas that "open up" or "open down." Tomer Gal exercise https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:quadratic-forms-features/v/finding-features-of-quadratic-functions 2022-10-06T23:01:00.745424633Z Finding features of quadratic functions Sal finds the zeros, the vertex, & the line of symmetry of quadratic functions given in vertex form, factored form, & standard form. video https://cdn.kastatic.org/ka_thumbnails_cache/20ca5e7a-7f75-41ef-8bf4-7e86045363fa_1280_720_base.png Finding features of quadratic functions Sal finds the zeros, the vertex, & the line of symmetry of quadratic functions given in vertex form, factored form, & standard form. https://cdn.kastatic.org/ka-youtube-converted/V-BAdpvejTw.mp4/V-BAdpvejTw.mp4 503 Quadratic equations and functions https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:quadratic-forms-features/v/rewriting-a-quadratic-function-to-find-roots-and-vertex 2021-06-30T17:32:36Z Forms & features of quadratic functions Different forms of quadratic functions reveal different features of those functions. Here, Sal rewrites f(x)=x²-5x+6 in factored form to reveal its zeros and in vertex form to reveal its vertex. Sal Khan video https://cdn.kastatic.org/googleusercontent/IYWi7RoLLOMjWtLJvwRZz3CEG3Bma-p6nwFbm1bb8Ss8LqTcIrbzl4eyR9bkXrWuuliJKoYMAKpXggGwpvRUgVD- Forms & features of quadratic functions Different forms of quadratic functions reveal different features of those functions. Here, Sal rewrites f(x)=x²-5x+6 in factored form to reveal its zeros and in vertex form to reveal its vertex. https://cdn.kastatic.org/ka-youtube-converted/4Bc5-HRop5Y.mp4/4Bc5-HRop5Y.mp4 509 Quadratic equations and functions https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:quadratic-forms-features/v/strategy-with-forms-of-quadratics 2022-10-06T23:01:00.745424633Z Worked examples: Forms & features of quadratic functions Choosing whether factored form, vertex form or standard form of a quadratic is best in different situations. video https://cdn.kastatic.org/ka_thumbnails_cache/e725d6d8-0971-44b2-82c9-a4901522cedb_1280_720_base.png Worked examples: Forms & features of quadratic functions Choosing whether factored form, vertex form or standard form of a quadratic is best in different situations. https://cdn.kastatic.org/ka-youtube-converted/eRbgHCaWQQE.mp4/eRbgHCaWQQE.mp4 402 Quadratic equations and functions https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:quadratic-forms-features/v/quadratic-functions-2 2021-06-30T17:32:40Z Vertex & axis of symmetry of a parabola Sal rewrites a quadratic equation in vertex form and shows how it reveals the vertex of the corresponding parabola. Sal Khan video https://cdn.kastatic.org/googleusercontent/tRp--KEO-LXgBvdYKjp_StBVMt1J8kcCUBpoOog6VLUwnN7iMVHEW1QrDJqj3VnLZ_3xFVtVpZ4I5Ulrir3CY3w Vertex & axis of symmetry of a parabola Sal rewrites a quadratic equation in vertex form and shows how it reveals the vertex of the corresponding parabola. https://cdn.kastatic.org/ka-youtube-converted/dfoXtodyiIA.mp4/dfoXtodyiIA.mp4 442 Quadratic equations and functions https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:quadratic-forms-features/e/rewriting-expressions-to-reveal-information 2026-04-08T00:16:02.937952483Z Features of quadratic functions Find the zeroes ("roots") of quadratic functions and the vertex of their parabolas. Functions are given in standard, vertex, and factored form. Tomer Gal exercise https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:quadratic-forms-features/e/identifying-and-using-quadratic-forms 2023-12-12T00:32:03.629852833Z Features of quadratic functions: strategy Identify the form of a quadratic function that immediately reveals a given feature of that function. Features in question are the y-intercept of the graph, the zeroes ("roots") of the function, and the vertex of the parabola. Tomer Gal exercise https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:standard-form-quadratic/v/ex3-completing-the-square 2021-06-30T17:32:40Z Finding the vertex of a parabola in standard form Sal rewrites the equation y=-5x^2-20x+15 in vertex form (by completing the square) in order to identify the vertex of the corresponding parabola. Sal Khan video https://cdn.kastatic.org/googleusercontent/wigSts0--ntP7W8-W5pW-wvBMuV6nwqB7t0bYBb0q7mTKETAJtsDEsZMGzGvwVqvw7zYQxHm1k9BRxBT8e0iTMtd Finding the vertex of a parabola in standard form Sal rewrites the equation y=-5x^2-20x+15 in vertex form (by completing the square) in order to identify the vertex of the corresponding parabola. https://cdn.kastatic.org/ka-youtube-converted/IbI-l7mbKO4.mp4/IbI-l7mbKO4.mp4 340 Quadratic equations and functions https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:graphing-and-comparing-quadratics/v/comparing-features-of-quadratic-functions 2022-10-06T23:01:00.745424633Z Comparing features of quadratic functions Sal compares the y-intercepts, the zeros, and the concavity of quadratic functions given graphically and algebraically. video https://cdn.kastatic.org/ka_thumbnails_cache/b5917311-4af9-4a3f-8165-3ef2fd30b9a1_1280_720_base.png Comparing features of quadratic functions Sal compares the y-intercepts, the zeros, and the concavity of quadratic functions given graphically and algebraically. https://cdn.kastatic.org/ka-youtube-converted/UdLYAqN0gNY.mp4/UdLYAqN0gNY.mp4 251 Quadratic equations and functions https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:graphing-and-comparing-quadratics/v/comparing-features-of-functions-2-example-2 2020-11-23T05:22:12Z Comparing maximum points of quadratic functions Given several quadratic functions represented in different forms, Sal finds the one with the lowest maximum value. Sal Khan video https://cdn.kastatic.org/googleusercontent/4Hr043EISIY55-vRJUtLpGFco8Lsc8F9hF3Ve2Tw9cNk5EWCyfOVIwFsalnf_q4hBfYs3f5Ra1SbyCNbh-zfI9S3 Comparing maximum points of quadratic functions Given several quadratic functions represented in different forms, Sal finds the one with the lowest maximum value. https://cdn.kastatic.org/ka-youtube-converted/4Bx06GFyhUA.mp4/4Bx06GFyhUA.mp4 239 Quadratic equations and functions https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:graphing-and-comparing-quadratics/e/compare-properties-quadratic-functions 2023-12-12T00:32:03.629852833Z Compare quadratic functions Compare the properties of two quadratic functions, each represented in a different way. Sulinya Ramanan exercise https://www.khanacademy.org/math/algebra-1-illustrative-math/x6418b49dfbc9d0c9:quadratic-equations-part1/x6418b49dfbc9d0c9:rewriting-quadratic-expressions-in-vertex-form/v/rewriting-a-quadratic-function-from-vertex-form-to-standard-form 2024-06-10T22:09:52.259723494Z Rewriting a quadratic function from vertex form to standard form Given a quadratic function in vertex form, rewrite it in standard form. Sal Khan video https://cdn.kastatic.org/ka_thumbnails_cache/cd2fcc1d-5126-45a3-923b-347f589a9a23_1280_720_base.png Rewriting a quadratic function from vertex form to standard form Given a quadratic function in vertex form, rewrite it in standard form. https://cdn.kastatic.org/ka-youtube-converted/NbNfSIG65Qc.mp4/NbNfSIG65Qc.mp4 170 Quadratic equations and functions https://www.khanacademy.org/math/algebra-1-illustrative-math/x6418b49dfbc9d0c9:quadratic-equations-part1/x6418b49dfbc9d0c9:rewriting-quadratic-expressions-in-vertex-form/e/rewrite-a-quadratic-function-from-vertex-form-to-standard-form 2024-10-01T22:06:52.489588313Z Rewrite a quadratic function from vertex form to standard form Given a quadratic function in vertex form, rewrite it in standard form. Nick Kokkinis exercise https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:factored-form-quadratics/v/graphing-quadratics-in-factored-form 2021-06-30T17:32:37Z Graphing quadratics in factored form An example for a quadratic function in factored form is y=½(x-6)(x+2). We can analyze this form to find the x-intercepts of the graph, as well as find the vertex. video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Graphing quadratics in factored form An example for a quadratic function in factored form is y=½(x-6)(x+2). We can analyze this form to find the x-intercepts of the graph, as well as find the vertex. https://cdn.kastatic.org/ka-youtube-converted/EV57jv7JKCs.mp4/EV57jv7JKCs.mp4 363 Quadratic equations and functions https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:quadratic-forms-features/e/graphing_parabolas_2 2026-03-24T23:41:07.080866137Z Graph parabolas in all forms Graph quadratic functions given in any form. Sid Patel exercise https://www.khanacademy.org/math/algebra-home/alg-quadratics/factored-form-alg/e/graphing-quadratics-in-factored-form-warmup 2026-04-08T00:16:02.937952483Z Warmup: graphing quadratics in factored form Make sure you know how to graph quadratic functions that are given in the factored form f(x)=(x+a)(x+b), as a warmup to our exercise. Tomer Gal exercise https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:standard-form-quadratic/v/graphing-a-parabola-using-roots-and-vertex 2021-06-30T17:32:35Z Graphing quadratics: standard form Learn how to graph any quadratic function that is given in standard form. Here, Sal graphs y=5x²-20x+15. Sal Khan video https://cdn.kastatic.org/googleusercontent/cRCnKKAjKKn3-iOi2QgaDA6zvAPnzjaC2vxXEMkYLqSqmBOmYKhQS4-c8xgz_MWVN15PIZlWRUOunkPSX74kvrAviw Graphing quadratics: standard form Learn how to graph any quadratic function that is given in standard form. Here, Sal graphs y=5x²-20x+15. https://cdn.kastatic.org/ka-youtube-converted/MQtsRYPx3v0.mp4/MQtsRYPx3v0.mp4 281 Quadratic equations and functions https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:standard-form-quadratic/e/graphing_parabolas_0.5 2026-03-24T23:41:07.080866137Z Graph quadratics in standard form Graph quadratic functions given in the standard form ax²+bx+c. For example, graph y=5x²-20x+15. Sid Patel exercise https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:factored-form-quadratics/e/graph-quadratic-functions-in-factored-form 2026-05-01T17:50:49.79434997Z Graph quadratics in factored form Graph quadratic functions that are given in factored form. For example, graph y=(x-4)(x+2). Sulinya Ramanan exercise https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:quadratic-forms-features/a/graphing-quadratics-review 2025-12-08T18:50:48.544869696Z Graphing quadratics review The graph of a quadratic function is a parabola, which is a "u"-shaped curve. In this article, we review how to graph quadratic functions. article https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:transform-quadratic-functions/v/shifting-and-scaling-parabolas 2020-12-28T16:56:56Z Intro to parabola transformations Sal discusses how we can shift and scale the graph of a parabola to obtain any other parabola, and how this affects the equation of the parabola. Sal Khan video https://cdn.kastatic.org/googleusercontent/ospzQQp01E5WLNQf0eRF5tfoGKLcxkp4iLMdevZPRhsOwt9jBImlY9EmQbB_tQVp3Eat_1xTEh_zuxc6x9QmUlU Intro to parabola transformations Sal discusses how we can shift and scale the graph of a parabola to obtain any other parabola, and how this affects the equation of the parabola. https://cdn.kastatic.org/ka-youtube-converted/99v51U3HSCU.mp4/99v51U3HSCU.mp4 481 Quadratic equations and functions https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:transform-quadratic-functions/v/example-translating-parabola 2020-12-28T16:56:57Z Shifting parabolas The graph of y=(x-k)²+h is the resulting of shifting (or translating) the graph of y=x², k units to the right and h units up. For example, y=(x-3)²-4 is the result of shifting y=x² 3 units to the right and -4 units up, which is the same as 4 units down. video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Shifting parabolas The graph of y=(x-k)²+h is the resulting of shifting (or translating) the graph of y=x², k units to the right and h units up. For example, y=(x-3)²-4 is the result of shifting y=x² 3 units to the right and -4 units up, which is the same as 4 units down. https://cdn.kastatic.org/ka-youtube-converted/ZmVOR6n_fzY.mp4/ZmVOR6n_fzY.mp4 282 Quadratic equations and functions https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:transform-quadratic-functions/e/shift-parabolas 2023-12-12T00:32:03.629852833Z Shift parabolas We can shift a parabola by moving it up, down, left, or right. Practice how this is expressed graphically and algebraically. Tomer Gal exercise https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:transform-quadratic-functions/v/scaling-and-reflecting-parabolas 2020-12-28T16:56:57Z Scaling & reflecting parabolas The graph of y=k⋅x² is the graph of y=x² scaled by a factor of |k|. If k<0, it's also reflected (or "flipped") across the x-axis. In this worked example, we find the equation of a parabola from its graph. video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Scaling & reflecting parabolas The graph of y=k⋅x² is the graph of y=x² scaled by a factor of |k|. If k<0, it's also reflected (or "flipped") across the x-axis. In this worked example, we find the equation of a parabola from its graph. https://cdn.kastatic.org/ka-youtube-converted/AMsbZLxd3lE.mp4/AMsbZLxd3lE.mp4 285 Quadratic equations and functions https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:transform-quadratic-functions/e/stretch-and-shrink-parabolas 2026-05-12T22:10:47.015864082Z Scale & reflect parabolas We can make parabolas wider or narrower, and we can reflect them over the $x$-axis. Practice how this is done, graphically and algebraically. Tomer Gal exercise