https://www.khanacademy.org/math/cambridge-o-level-mathematics-4024/x3f4c73d082a798eb:algebra-and-graphs/x3f4c73d082a798eb:sketching-curves/v/sat-math-h9-easier 2021-02-09T20:46:47Z Graphing linear equations — Basic example Watch Sal work through a basic Graphing linear equations problem. video https://cdn.kastatic.org/googleusercontent/G7d57yXqy8N5Wte1Hv_ZqVG-dqP_R84PzfgIwoDnKW0vlMuqxRGbdMil9cNeytTQHzriINfhyfWfZBmtfsYdwQCXInzEVW4j1FQ9 Graphing linear equations — Basic example Watch Sal work through a basic Graphing linear equations problem. https://cdn.kastatic.org/ka-youtube-converted/PesqhjGfQvY.mp4/PesqhjGfQvY.mp4 158 Sketching curves https://www.khanacademy.org/math/cambridge-o-level-mathematics-4024/x3f4c73d082a798eb:algebra-and-graphs/x3f4c73d082a798eb:sketching-curves/a/v2-sat-lesson-graphs-of-linear-equations-and-functions 2022-10-18T02:23:40.2192308Z Graphs of linear equations and functions | Lesson A guide to graphs of linear equations and functions on the digital SAT article https://www.khanacademy.org/math/cambridge-o-level-mathematics-4024/x3f4c73d082a798eb:algebra-and-graphs/x3f4c73d082a798eb:sketching-curves/v/sat-math-h9-harder 2021-02-10T21:38:00Z Graphing linear equations — Harder example Watch Sal work through a harder Graphing linear equations problem. video https://cdn.kastatic.org/googleusercontent/823gpGwygTJWF3_7gSvBv5V5cbIZoABrL4AkVQsQkARNxNjewe6GLWMbZITLqOSFcBV7xzHHXtF5xobT_InjxvcMo2yuhWsZZBfbzSg Graphing linear equations — Harder example Watch Sal work through a harder Graphing linear equations problem. https://cdn.kastatic.org/ka-youtube-converted/xlxQ_kUREMU.mp4/xlxQ_kUREMU.mp4 207 Sketching curves https://www.khanacademy.org/math/cambridge-o-level-mathematics-4024/x3f4c73d082a798eb:algebra-and-graphs/x3f4c73d082a798eb:sketching-curves/e/v2-graphs-of-linear-equations-and-functions-1 2023-12-04T17:55:05.230178085Z Graphs of linear equations and functions: foundations Graphs of linear equations and functions: foundations Duo Yang 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 Sketching curves 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: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 Sketching curves 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 Sketching curves 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-home/alg-quadratics/alg-features-of-quadratic-functions/e/warmup-features-of-quadratic-functions 2021-04-29T07:54:31Z Warmup: Features of quadratic functions Have some quick review of finding features of quadratic functions given in different forms. Tomer Gal 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: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/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 Sketching curves 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 Sketching curves 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/algebra2/x2ec2f6f830c9fb89:poly-graphs/x2ec2f6f830c9fb89:poly-end-behavior/v/polynomial-end-behavior 2021-06-30T17:32:35Z Intro to end behavior of polynomials Sal explains what "end behavior" is and what affects the end behavior of polynomial functions. Sal Khan video https://cdn.kastatic.org/googleusercontent/kJCEVazTkyR1RTZeo7Pnup1OJ3-kjdFMf_fB4Y1NamfEKLUxriMCU6fTz4-8R8LRe3B7w9JbTZtrFqI9Unt9-scDgA Intro to end behavior of polynomials Sal explains what "end behavior" is and what affects the end behavior of polynomial functions. https://cdn.kastatic.org/ka-youtube-converted/tZKzaF28sOk.mp4/tZKzaF28sOk.mp4 490 Sketching curves https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-graphs/x2ec2f6f830c9fb89:poly-end-behavior/a/end-behavior-of-polynomials 2023-05-03T21:01:13.225276573Z End behavior of polynomials Learn what the end behavior of a polynomial is, and how we can find it from the polynomial's equation. article https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-graphs/x2ec2f6f830c9fb89:poly-end-behavior/e/determine-the-end-behavior-of-polynomials 2023-05-07T21:16:17.274254006Z End behavior of polynomials Analyze polynomial functions to determine how they behave as the input variable increases to positive infinity or decreases to negative infinity. Tomer Gal exercise https://www.khanacademy.org/math/algebra-2-tx/xf7d2a662bc9ffb25:cubic-and-cube-root-functions-and-equations/xf7d2a662bc9ffb25:cubic-functions/v/transforming-a-cubic-function 2025-09-11T05:00:47.574338337Z Graphing cubic function transformations Sal demonstrates the relationship between changes to the equation of the parent function x^3 and transformations of its original graph. Sal Khan video https://cdn.kastatic.org/ka_thumbnails_cache/2b0854e2-6199-4015-bd1b-cda89ff22515_1280_720_base.png Graphing cubic function transformations Sal demonstrates the relationship between changes to the equation of the parent function x^3 and transformations of its original graph. https://cdn.kastatic.org/ka-youtube-converted/gpaRJMGFmyI.mp4/gpaRJMGFmyI.mp4 183 Sketching curves https://www.khanacademy.org/math/algebra-2-tx/xf7d2a662bc9ffb25:cubic-and-cube-root-functions-and-equations/xf7d2a662bc9ffb25:cubic-functions/e/graph-cubic-function-transformations 2025-07-01T01:15:48.63946125Z Graph cubic function transformations analyze the effect on the graph of 𝑓(𝑥) = 𝑥³ when 𝑓(𝑥) is replaced by 𝑎𝑓(𝑥), 𝑓(𝑏𝑥), 𝑓(𝑥 − 𝑐), and 𝑓(𝑥) + 𝑑 for specific positive and negative real values of 𝑎, 𝑏, 𝑐, and 𝑑 Nate Kinast exercise https://www.khanacademy.org/math/math3/x5549cc1686316ba5:rationals/x5549cc1686316ba5:rational-graphs/v/horizontal-vertical-asymptotes 2021-06-29T20:27:55Z Graphing rational functions 1 Sal graphs f(x)=(2x+10)/(5x-15). Sal Khan video https://cdn.kastatic.org/googleusercontent/RQTfWiQddnto4liAxw2i0S7h1dmlGpPdYX-ehKArcSxri0vMh_OrpD-ugcIdfPEDOfL2ALGzCWxO0p4jsN7-QQON Graphing rational functions 1 Sal graphs f(x)=(2x+10)/(5x-15). https://cdn.kastatic.org/ka-youtube-converted/fvC0dm2wzIo.mp4/fvC0dm2wzIo.mp4 664 Sketching curves https://www.khanacademy.org/math/math3/x5549cc1686316ba5:rationals/x5549cc1686316ba5:rational-graphs/v/another-rational-function-graph-example 2021-06-29T20:27:56Z Graphing rational functions 2 Sal graphs y=(2x)/(x+1). Sal Khan video https://cdn.kastatic.org/googleusercontent/Q9hVO0uY8QGuoa4qVNKoWZAR_xcssRHbllEYMFR1ScG7bCzfjRvgEFeGxa1TfKlZDIqzakz2HCZpQ95_hbwINoK- Graphing rational functions 2 Sal graphs y=(2x)/(x+1). https://cdn.kastatic.org/ka-youtube-converted/ReEMqdZEEX0.mp4/ReEMqdZEEX0.mp4 500 Sketching curves https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:exponential-growth-decay/x2f8bb11595b61c86:exponential-vs-linear-growth/v/exponential-growth-functions 2022-11-16T04:22:43.7414897Z Intro to exponential functions An exponential function represents the relationship between an input and output, where we use repeated multiplication on an initial value to get the output for any given input. Exponential functions can grow or decay very quickly. Exponential functions are often used to model things in the real world, such as populations, radioactive materials, and compound interest. Sal Khan video https://cdn.kastatic.org/googleusercontent/MOdCeqFGlDqk69XEhUqpMPoJA42wLRt7GwZjEAthvS_AxHPH0ieVuA4DqeI3ImMs5u7K5aCZdV6MCqKxXrs-d4g Intro to exponential functions An exponential function represents the relationship between an input and output, where we use repeated multiplication on an initial value to get the output for any given input. Exponential functions can grow or decay very quickly. Exponential functions are often used to model things in the real world, such as populations, radioactive materials, and compound interest. https://cdn.kastatic.org/ka-youtube-converted/6WMZ7J0wwMI.mp4/6WMZ7J0wwMI.mp4 461 Sketching curves https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:exponential-growth-decay/x2f8bb11595b61c86:graphs-of-exponential-growth/v/graphing-exponential-functions 2023-05-24T15:30:27.32390848Z Exponential function graph We can graph an exponential function, like y=5ˣ, by picking a few inputs (x-values) and finding their corresponding outputs (y-values). We'll see that an exponential function has a horizontal asymptote in one direction and rapidly changes in the other direction. Sal Khan video https://cdn.kastatic.org/googleusercontent/ntaWYGvlJPukwyhP7oC72DTO8x4xLVpcF5KAFo38Bh93MyaHdExf2ElxhokBqnWf3H-UmjGETTe2s0_Kv9rod86J1A Exponential function graph We can graph an exponential function, like y=5ˣ, by picking a few inputs (x-values) and finding their corresponding outputs (y-values). We'll see that an exponential function has a horizontal asymptote in one direction and rapidly changes in the other direction. https://cdn.kastatic.org/ka-youtube-converted/9SOSfRNCQZQ.mp4/9SOSfRNCQZQ.mp4 332 Sketching curves https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:exponential-growth-decay/x2f8bb11595b61c86:graphs-of-exponential-growth/v/graphs-of-exponential-growth 2021-02-16T05:02:26Z Graphs of exponential growth Identifying which graph represents a given exponential function. video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Graphs of exponential growth Identifying which graph represents a given exponential function. https://cdn.kastatic.org/ka-youtube-converted/6rX2VNybXEE.mp4/6rX2VNybXEE.mp4 261 Sketching curves https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:exponential-growth-decay/x2f8bb11595b61c86:graphs-of-exponential-growth/e/graphing-exponential-growth-intro 2025-03-03T19:47:54.930712342Z Graphs of exponential growth Start exploring the relationship between exponential functions and their graphs. Tomer Gal exercise