FUNKCJA HOMOGRAFICZNA 🔥 Wszystko, co musisz wiedzieć: dziedzina, asymptoty, wykres i wzór | Liceum

Section: Functions Topic: Homographic Function [There's a MINI-QUIZ in the description—test yourself after the video! 🎯] "HOMOGRAPHIC FUNCTION 🔥 Everything you need to know: domain, asymptotes, graph, and formula | High School" ——— Want to finally understand the homographic function without the chaos—know where the asymptotes come from, how to graph it, and how to read the most important properties in just a few seconds? In this video, you'll get a complete outline of how to work with the homographic function: from the general formula, through the shifted form, to problems with graphing and finding the function's formula. 🔹 General form: f(x) = (ax + b) / (cx + d), where c ≠ 0 and ad − bc ≠ 0. 🔹 Domain: all real numbers except the one that makes the denominator zero, i.e., x different from −d/c. 🔹 Asymptotes: • vertical: x = −d/c, • horizontal: y = a/c. 🔹 Set of values: all real numbers except a/c. 🔹 Zeros: find ax + b = 0 from the equation, provided a ≠ 0. 🔹 Y-intercept: substitute x = 0 if 0 is in the domain. 🔹 Shifted form: f(x) = A / (x − p) + q — thanks to this, you can immediately see the asymptotes and the center of symmetry of the graph. 🔹 The center of symmetry of the graph has coordinates S(p, q), and the entire graph is a shifted hyperbola. 🔹 When the coefficient A is positive, the branches are arranged as in the graph of y = 1/x, and when A is negative, the opposite is true. 🔹 Monotonicity: • when ad − bc is positive, the function is increasing on every interval of its domain, • when ad − bc is negative, the function is decreasing on every interval of its domain. 🔹 How to quickly determine a formula from asymptotes and a point: start with the form A / (x − p) + q, then substitute the coordinates of the point. 🔹 Common pitfalls: missing a number excluded from the domain, confusing vertical and horizontal asymptotes, finding the zero in the denominator instead of the numerator, incorrect signs after transformations. Finally—a quick challenge: try to read the asymptotes, center of symmetry, and formula of the function from the graph itself. 😎 Who is this video for? 🔹 For high school students who want to understand the homographic function from scratch. 🔹 For those preparing for a function exam. 🔹 For anyone who wants to confidently analyze graphs and formulas before their final exams. ——— MINI-QUIZ (Homographic Function) 1. For the function f(x) = (2x + 3) / (x − 4), state the domain and asymptotes. 2. For the function g(x) = (3x − 6) / (x + 1), determine the zero and y-intercept. 3. Write the function h(x) = (5x + 7) / (x − 2) as A / (x − p) + q and specify the center of symmetry of the graph. 4. Determine whether the function k(x) = (x + 3) / (2x − 1) is increasing or decreasing on each interval of its domain. 5. Determine the formula of the homographic function with asymptotes x = 3 and y = −2 that passes through the point (4, 1). ——— If you want more videos like this, give us a thumbs up and subscribe to the channel—we'll tackle every section together! Leave a comment and say "cheat sheet" if you'd like a short list of formulas for printing. #math #highschool #homographicfunction #functions #algebra #matura LINK TO SUPPORT THE CHANNEL:    / @matwujek   Follow me on: @matwujek 🖥️ https://matwujek.pl 👾 DC:   / discord   🎶 Twitter:   / matwujek   📸 Instagram:   / matwujek   Fb:   / matwujek   📧 @: [email protected] ——— MINI-QUIZ ANSWERS 1. Domain: all real numbers except 4. Asymptotes: x = 4 and y = 2. 2. Zero: 3x − 6 = 0, so x = 2. Y-intercept: g(0) = −6, so the point is (0, −6). 3. 5x + 7 = 5(x − 2) + 17, so h(x) = 5 + 17 / (x − 2). Therefore, A = 17, p = 2, q = 5, and the center of symmetry is S(2, 5). 4. For this function, ad − bc = 1 · (−1) − 3 · 2 = −7, which is negative. The function is therefore decreasing on every interval of its domain, i.e., for x less than 1/2 and for x greater than 1/2. 5. We look for a formula of the form f(x) = A / (x − 3) − 2. We substitute the point (4, 1): 1 = A / (4 − 3) − 2, so 1 = A − 2, hence A = 3. Finally: f(x) = 3 / (x − 3) − 2.