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 the origin.
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Horizontal & & Vertical
Hyperbola & & Hyperbola
x^2/a^2-y^2/b^2=1 & & y^2/a^2-x^2/b^2=1
Consider the given equation.
y^2/18-x^2/20=1 ⇔ y^2/( sqrt(18))^2-x^2/( sqrt(20))^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 vertices are (0,± sqrt(18)) = (0,± 3sqrt(2)). Let's substitute a=3sqrt(2) and b=sqrt(20)=2sqrt(5) into the formula for the asymptotes and obtain their equations.
The asymptotes are y=± 3sqrt(10)10x. Now let's calculate c, the absolute value of the nonzero coordinate of the foci. To do so we will substitute a=3sqrt(2) and b=2sqrt(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 (0,± sqrt(38)).
Graph
To graph the function let's summarize all of the information that we have found.
