Answer:
Solution:
[tex]\frac{\left(y-1\right)^2}{9}-\frac{\left(x+4\right)^2}{4}=1[/tex]
Step-by-step explanation:
We have the general equation for our up - down facing hyperbola as (y - k)² / b²- (x - h)² / a² = 1. Let's start listing the properties of this graph -
• Taking a look at the graph we see that the center point of our hyperbola here is (- 4, 1). Therefore (h, k) = (- 4,1).
• Another thing here is our key value a. This is the semi distance from the center to one of the vertices. Here it will be the distance from points (- 4,1) and (- 4,4) or 3 unit difference. Therefore a = 3.
• That leaves us with our asymptotes. Now remember that it will be in the form y = ± b / a. We already know a = 3, so we have to find b. Looking at this graph I can say that another point besides (- 4,1) that lies on the " dotted line " is (- 2, - 2). Calculating the slope of the dotted line would be as follows,
Given: (- 4,1) and (- 2, - 2)
Slope = - 2 + 4 / - 2 - 1 = 2 / 3
We have the equation y = 2 / 3x. Therefore b = 2. Let's substitute to receive our equation...
h = - 4, k = 1, b = 2, a = 3
(y - 1)² / (3)²- (x + 4)² / (2)² = 1
(y - 1)² / 9 - (x + 4)² / 4 = 1 ~ your solution is the last one, option d
