I was given the equation (y-2)^2 - x^2 =1 and asked to locate the center, vertices,and asymptotes. ifound the center to be (0,2) and the verticies are(0,1)and (0,3). but I'm not sure how to find the asymptotes. Could you please help me find them and explain the process. Thanks for your time
Sure! To find the asymptotes of the given equation, we first need to understand the general form of a hyperbola equation. The standard form is:
(x-h)^2 / a^2 - (y-k)^2 / b^2 = 1
Where (h, k) represents the center of the hyperbola, and a and b are positive real numbers.
In your equation, (y-2)^2 - x^2 = 1, we can rewrite it by comparing it to the standard form:
(x-0)^2 / 1^2 - (y-2)^2 / (√1)^2 = 1
From this, we can determine that the center of the hyperbola is (h, k) = (0, 2).
Now, let's find the asymptotes. The asymptotes of a hyperbola are lines that the curve approaches but never touches. For a vertical hyperbola (like in this case), the equations of the asymptotes are given by:
y = k ± (b/a) * (x-h)
Using the values we have, the equation of the asymptotes will be:
y = 2 ± (1/1) * (x-0)
Simplifying this, we get:
y = 2 ± x
Therefore, the equations of the asymptotes are y = 2 + x and y = 2 - x.
I hope this helps! Let me know if you have any further questions.