Graphing Hyperbolas

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In this section, we continue to learn about conic sections. Here, we will discuss how to graph a hyperbola that is centered at the origin. The hyperbola is defined as the set of all points in a plane in which the absolute value of the difference of the distances from two fixed points is constant. These fixed points are referred to as foci. The easiest scenario occurs when the hyperbola is centered at the origin. When this occurs, we can graph our hyperbola using a few easy steps. 1) Identify if the hyperbola is horizontal or vertical: $$horizontal\hspace{.25em}hyperbola:\hspace{.25em}\frac{x^2}{a^2}- \frac{y^2}{b^2}=1$$ $$vertical\hspace{.25em}hyperbola:\hspace{.25em}\frac{y^2}{a^2}- \frac{x^2}{b^2}=1$$ Take note of how the minus sign switches around between a horizontal and vertical hyperbola. 2) Find and plot the intercepts, for a horizontal hyperbola, we will have x-intercepts of (a,0) and (-a,0) and no y-intercepts. In the case of a vertical hyperbola, we will have y-intercepts of (0,a) and (0,-a) and no x-intercepts. 3) Find the fundamental rectangle: we plot the points (a,b), (-a,b),(-a,-b), and (a,-b) (horizontal) or (b,a),(-b,a),(b,-a),(-b,-a) (vertical). 4) We sketch the asymptotes, these are found by graphing the equations:
a) Horizontal: $$y=\pm \frac{b}{a}x$$ b) Vertical: $$y=\pm \frac{a}{b}x$$ 5) Sketch the graph of the hyperbola: we draw each branch through the intercepts and approaching the asymptotes.
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