--- title: Accurate Function Graphing - Explained Simply With Example description: "Accurate graphing of functions uses the derivative: intercepts, stationary points, and rising or falling parts." url: https://astra-ai.co/videos/derivatives/stationary-points-increasing-decreasing-functions/accurate-graphing-of-functions/ locale: en type: video-lesson keywords: - graphing of functions - stationary points - increasing and decreasing - accurate sketch published: 2026-10-01T10:06:16.513Z updated: 2026-10-04T22:02:47.181Z alternates: bg: https://astra-ai.co/bg/videos/derivati/statsionarni-tochki/tochno-grafika-na-funktsiite/ br: https://astra-ai.co/br/videos/derivadas/capitulo-matematica-038/video-matematica-081/ cs: https://astra-ai.co/cs/videos/derivaty/stacionarni-body/presne-zakreslovani-grafu-funkci/ da: https://astra-ai.co/da/videos/differentialregning/stationaere-punkter-og-voksende-aftagende-funktioner/pr-cis-graftegning-af-funktioner/ de: 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https://astra-ai.co/pl/videos/pochodne/punkty-stacjonarne-funkcje-rosnace-malejace/dokladne-rysowanie-wykresow-funkcji/ pt: https://astra-ai.co/pt/videos/derivadas/pontos-estacionarios/graficos-precisos-de-funcoes/ pt-TL: https://astra-ai.co/pt-TL/videos/derivadas/capitulo-de-video-032/licao-de-video-081/ ro: https://astra-ai.co/ro/videos/derivate/puncte-stationare-functii-crescatoare-descrescatoare/reprezentarea-grafica-precisa-a-functiilor/ sk: https://astra-ai.co/sk/videos/derivacie/kapitola-032/lekcia-081/ sl: https://astra-ai.co/sl/videos/odvodi/stacionarne-tocke/natancno-risanje-grafov-funkcij/ sr: https://astra-ai.co/sr/videos/izvodi/poglavlje-032/lekcija-081/ sv: https://astra-ai.co/sv/videos/derivator/stationara-punkter-vaxande-och-avtagande-funktioner/noggrann-grafritning-av-funktioner/ th: https://astra-ai.co/th/videos/anuphan/bot-38-chut-ning-fangchan-phoem-lot/botrian-081/ tr: https://astra-ai.co/tr/videos/turevler/bolum-032/ders-081/ uk: https://astra-ai.co/uk/videos/uk-topic-13/uk-chapter-38/uk-lesson-cbfb3af6/ vi: https://astra-ai.co/vi/videos/vi-topic-13/vi-chapter-38/ve-o-thi-ham-so-chinh-xac/ zh-HK: https://astra-ai.co/zh-HK/videos/shuxue-kecheng-013/shuxue-zhang-038/shuxue-kecheng-081/ zh-TW: https://astra-ai.co/zh-TW/videos/derivatives/chapter-3c240780/lesson-cbfb3af6/ --- # Accurate Graphing of Functions > Accurate graphing of functions uses intercepts and the derivative. This lesson marks stationary points and where the graph rises or falls. ## INTRODUCTION TO GRAPHING FUNCTIONS Accurately graphing a function means not only sketching its approximate shape but also analyzing the function's most important features. These include zeros (x-intercepts), stationary points, intervals of increase and decrease, possible extrema, asymptotes, concavity, and inflection points. Each of these features contributes to the correct shape of the graph. ## STEPS FOR GRAPHING A FUNCTION To graph a function correctly, a specific sequence of steps is followed: - **DETERMINE THE DOMAIN OF THE FUNCTION:** Check for which values of x the function is defined (e.g., division by zero is not allowed, the square root of a negative number is not defined in the real number system, etc.). - **FIND THE ZEROS (X-INTERCEPTS) OF THE FUNCTION:** Solve the equation f(x) = 0. These values represent the points where the graph intersects the x-axis. - **FIND THE FIRST DERIVATIVE OF THE FUNCTION AND SOLVE f'(x) = 0:** These are candidates for local maxima, minima, or stationary inflection points. - **ANALYZE INCREASING AND DECREASING INTERVALS:** Determine where the function is increasing (f'(x) > 0) and where it is decreasing (f'(x) < 0) based on the sign of the first derivative. - **CHECK FOR LOCAL EXTREMA:** By observing the change in the sign of the derivative around stationary points, we can determine if it is a maximum, a minimum, or just a flat point (saddle/stationary inflection point). - **DETERMINE CONCAVITY AND INFLECTION POINTS:** Calculate the second derivative, f''(x). If f''(x) > 0, the graph is concave upwards (convex). If f''(x) < 0, it is concave downwards. Points where the concavity changes are inflection points (typically where f''(x) = 0 or is undefined, and changes sign). - **CHECK FOR ASYMPTOTES (IF ANY):** If the function contains fractions or logarithmic expressions, check the behavior for very large or very small values of x, and where denominators are zero. - **CALCULATE THE FUNCTION'S VALUES FOR SOME X-VALUES:** Substitute specific values of x to obtain points through which the graph passes (e.g., x = 0, x = 1, x = -1 ...). ## IMPORTANT ELEMENTS OF THE GRAPH - **Y-INTERCEPT:** Determined by calculating f(0). - **X-INTERCEPTS (ZEROS):** Determined from the equation f(x) = 0. - **STATIONARY POINTS AND EXTREMA:** Found using the first derivative. - **CONCAVITY AND INFLECTION POINTS:** Found using the second derivative. - **ASYMPTOTES:** Horizontal, vertical, or slant (if the function has them). - **INCREASING/DECREASING INTERVALS:** Determined by the sign of the first derivative. ## EXAMPLE Let the function be f(x) = x³ - 3x. - **DOMAIN:** The entire set of real numbers. - **ZEROS:** Solve x³ - 3x = 0 → x(x² - 3) = 0 → x(x - √3)(x + √3) = 0. Zeros are x = 0, x = √3, x = -√3. - **FIRST DERIVATIVE:** f'(x) = 3x² - 3. Solve f'(x) = 0 → 3(x² - 1) = 0 → x = ±1. These are stationary points. - **SIGN OF THE FIRST DERIVATIVE:** - - For x < -1 (e.g., x = -2), f'(-2) = 3(-2)² - 3 = 12 - 3 = 9 > 0. The function is **increasing**. - For -1 < x < 1 (e.g., x = 0), f'(0) = 3(0)² - 3 = -3 < 0. The function is **decreasing**. - For x > 1 (e.g., x = 2), f'(2) = 3(2)² - 3 = 12 - 3 = 9 > 0. The function is **increasing**. - **LOCAL EXTREMA:** - - At x = -1, the function changes from increasing to decreasing, so there is a local **maximum**. f(-1) = (-1)³ - 3(-1) = -1 + 3 = 2. Point: (-1, 2). - At x = 1, the function changes from decreasing to increasing, so there is a local **minimum**. f(1) = (1)³ - 3(1) = 1 - 3 = -2. Point: (1, -2). - **SECOND DERIVATIVE:** f''(x) = 6x. Set f''(x) = 0 → 6x = 0 → x = 0. - - For x < 0, f''(x) < 0 (concave downwards). - For x > 0, f''(x) > 0 (concave upwards). The point (0, f(0)) = (0, 0) is an **inflection point** where concavity changes. - The function has **no asymptotes**. - **OTHER POINTS:** We already have f(0) = 0. Let's check f(2) = 2³ - 3(2) = 8 - 6 = 2. Point (2, 2). ## CONCLUSION Accurately graphing functions is based on an analytical investigation of their properties. By examining domains, derivatives, changes in direction, concavity, and other key features in the correct order, a complete picture of the function's behavior across its entire domain can be created. This is not just a drawing exercise but a structured approach to a deeper understanding of functions.