Find the equations for the asymptotes of a hyperbola

Learning Objectives Define what an asymptote of a hyperbola is and its relationship to the hyperbola's branches. Identify the center (h, k), and the values of 'a' and 'b' from the standard equation of a hyperbola. State the formulas for the asymptotes of both horizontal and vertical hyperbolas centered at (h, k). Derive the equations of the asymptotes for a hyperbola given its standard form equation. Differentiate between the asymptote equations for a horizontal hyperbola and a vertical hyperbola based on the slope. Use the center and slopes to accurately describe the lines that guide the hyperbola's graph. Ever wonder what invisible lines guide the path of a comet as it slingshots around the sun? ☄️ These guiding lines are the asymptotes of a h...

TermDefinitionExample HyperbolaA conic section consisting of two separate, mirror-image branches. It is defined as the set of all points where the difference of the distances from two fixed points (the foci) is constant.The graph of the equation `x^2/9 - y^2/4 = 1` is a hyperbola that opens to the left and right. AsymptoteA line that a curve approaches but never touches as it heads towards infinity. For a hyperbola, the two asymptotes form an 'X' shape that the branches get closer and closer to.For the hyperbola `x^2/9 - y^2/4 = 1`, the lines `y = (2/3)x` and `y = -(2/3)x` are its asymptotes. Center (h, k)The midpoint between the two vertices of the hyperbola. It is the point where the two asymptotes intersect.In the equation `(x-1)^2/16 - (y+2)^2/9 = 1`, the center is at the po...

Asymptote Equations for a Horizontal Hyperbola For `(x-h)^2/a^2 - (y-k)^2/b^2 = 1`, the asymptotes are `y - k = ±(b/a)(x - h)`. Use this formula when the x-term is positive, meaning the hyperbola opens left and right. The slope of the asymptotes is `±(b/a)`, which can be remembered as `±(y-radius / x-radius)` of the central box. Asymptote Equations for a Vertical Hyperbola For `(y-k)^2/a^2 - (x-h)^2/b^2 = 1`, the asymptotes are `y - k = ±(a/b)(x - h)`. Use this formula when the y-term is positive, meaning the hyperbola opens up and down. The slope of the asymptotes is `±(a/b)`. Note the change in the slope formula compared to the horizontal case.