The definition of a hyperbola is the set of all points in a plane, the DIFFERENCE of whose distance from two fixed points (foci) is constant. The definition is similar to an ellipse, but instead of adding the distances from a point on the hyperbola to each of the two foci, we subtract those distances. For any point that you pick on the hyperbola, when you take the two distances from that point to each of the foci and subtract them, you will get the same value. The general format of the equation for a hyperbola is similar to that of an ellipse. The difference is that you have a subtraction sign between the x squared and y squared terms. The general form of the equation of a hyperbola centered at (0,0) is below. Unlike an ellipse, where the orientation was determined based on whether the bigger denominator was under the x or the y, with a hyperbola, the orientation is determined based on which coordinate is being subtracted, the x or the y. Image courtesy of www.mathwarehouse.com/hyperbola/graph-equation-of-a-hyperbola.php For the hyperbola with the horizontal transverse axis (y^2 is being subtracted), the basic properties are as follows: Vertices are at (a,0) and (-a,0) Equations of asymptotes are y= - b/a and y = b/a (slope m= (change in y) / (change in x) Note In this case the b is under the y and the a is under the x which is why we use b/a (y/x). Foci are at (c,0) and (-c,0) where c is defined as c^2 = a^2 + b^2. For a parabola with a vertical transverse axis (x^2 is being subtracted), you are pretty much swapping the coordinates. Vertices are at (0,a) and (0,-a) Equations of asymptotes are y= - a/b and y = a/b (slope m= (change in y) / (change in x) Note In this case the a is under the y and the b is under the x which is why we use a/b (y/x). Foci are at (0,c) and (0,-c) where c is defined as c^2 = a^2 + b^2. If you have a horizontal and/or vertical shifts, the format of the equations looks as follows: Note that to find the asymptotes for a hyperbola with a shifted center, use the point slope formula with (h,k) as your point, and the slope still being +/- ( a/b) or (b/a) depending on the orientation of the hyperbola. Included at the end of this post are some more notes on hyperbolas with examples worked out, the homework answers worked out and some good websites with videos on hyperbolas. Videos - basics of a hyperbola www.youtube.com/watch?v=i6vM82SNAUk Part I www.youtube.com/watch?v=6Xahrwp6LkI Part II Graphing hyperbolas centered at the origin. www.youtube.com/watch?v=W0IXOdsna9A&feature=youtu.be www.youtube.com/watch?v=vT-gnaAG51o&feature=youtu.be Graphing a hyperbola not centered at the origin. www.youtube.com/watch?v=_ssQfu8N6XE&feature=youtu.be www.youtube.com/watch?v=0qAB_5v_2G4&feature=youtu.be Finding the equation of a hyperbola given center, focus, vertex www.youtube.com/watch?v=ZwqNou_Z7oI&feature=youtu.be Not a video site, but goes into nice detail on how the calculations are derived. www.ck12.org/book/ck-12-algebra-ii-with-trigonometry-concepts/section/10.7/ www.ck12.org/book/ck-12-algebra-ii-with-trigonometry-concepts/section/10.8/ www.ck12.org/book/ck-12-algebra-ii-with-trigonometry-concepts/section/10.9/ Homework Assignment: pages 768-69: 2, 4, 6, 16, 18, 20, 28 pages 781-82: 10, 12 Due Tuesday - 4/28/20 There will be an assessment on this unit near the end of next week.
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