The equation of a vertical hyperbola centered at (h, k) is (y-k)^2a^2- (x-h)^2b^2=1. The vertices are (h,k ± a). How can you find the foci and the asymptotes?
We will find the desired information and use it to draw the graph of the hyperbola.
Vertices, Foci, and Asymptotes
Let's start by recalling the equation of hyperbolas centered at (h, k).
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Horizontal & & Vertical
Hyperbola & & Hyperbola
(x-h)^2/a^2-(y-k)^2/b^2=1 & & (y-k)^2/a^2-(x-h)^2/b^2=1
Consider the given equation.
(y+6)^2/20-(x-1)^2/25=1 ⇕ (y+6)^2/(sqrt(20))^2-(x-1)^2/5^2=1
From the above formula we can see that the equation represents a vertical hyperbola. Next, let's review the main characteristics of this type of hyperbola.
Using this information we can identify that the center of the hyperbola is (1, - 6), and the vertices are ( 1, - 6- sqrt(20)) = ( 1, - 6- 2sqrt(5) ) and (1, - 6+ sqrt(20) )=(1, - 6+ 2sqrt(5) ). By substituting h=1, k=-6, a=sqrt(20)=2sqrt(5), and b=5 into the equation of the asymptotes, we can have two equations for the asymptotes.
y-( -6)=± 2sqrt(5)/5(x-1) ⇕ y+6=± 2sqrt(5)/5(x-1)
Now let's calculate c, the absolute value of the nonzero coordinate of the foci. To do so, we will substitute a=sqrt(20) and b=5 into c^2=a^2+b^2.
Note that when solving the above equation, we only needed to consider the principal root because c is a positive number. The foci of the hyperbola are (1, -6 ± 3sqrt(5)).
Graph
To graph the function, let's summarize all of the information that we have found.
