Question

What is the equation of the following graph?

Answer 1

@jim_thompson5910

Answer 2

This is a hyperbola.


The center is located at the center of the rectangle


So average the x coordinates that make up the length to get (-4+6)/2 = 2/2 = 1

and average the y coordinates that make up the height to get (0+4)/2 = 4/2 = 2


So the center of the rectangle is (1,2). This means that the center of the hyperbola is (1,2)


The distance from the center to either vertex will be the length of half of the transverse axis. This distance is 5 (since 6 - 1 = 5). Because the hyperbola opens left/right, we know that it's in the form (x-h)^2/a^2-(y-k)^2/b^2=1


The value of 'a' is the length of half of the transverse axis. So a = 5


The value of 'b' is the distance from the vertex to either adjacent corner of the rectangle. In this case, that distance is 2. So b = 2


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In short, we found that the center was (1,2), so (h,k) = (1,2) giving us h = 1 and k = 2


We also know that a = 5 and b = 2


So the equation


(x-h)^2/a^2-(y-k)^2/b^2=1


becomes


(x-1)^2/(5^2)-(y-2)^2/(2^2)=1


which turns into


(x-1)^2/25-(y-2)^2/4=1



So the answer is \[\Large \frac{(x-1)^2}{25}-\frac{(y-2)^2}{4}=1\]

Answer 3

Thank you soooo much! =)

Answer 4

you're welcome, anytime