find the vertex and focus of the parabola: y² - 4y + 12x - 8 = 0 vertex = (?, ) focus = (, )

Explanation:

Step1: Complete the square for the (y) terms

Given (y^{2}-4y + 12x-8=0). Group the (y) terms: ((y^{2}-4y)=-12x + 8). Complete the square for (y^{2}-4y): (y^{2}-4y=(y - 2)^{2}-4). So, ((y - 2)^{2}-4=-12x + 8), then ((y - 2)^{2}=-12x+12). Factor out (- 12) on the right - hand side: ((y - 2)^{2}=-12(x - 1)).

Step2: Identify the vertex

For a parabola of the form ((y - k)^{2}=4p(x - h)), the vertex is ((h,k)). Comparing ((y - 2)^{2}=-12(x - 1)) with ((y - k)^{2}=4p(x - h)), we have (h = 1) and (k = 2). So the vertex is ((1,2)).

Step3: Find the value of (p)

Since ((y - k)^{2}=4p(x - h)) and ((y - 2)^{2}=-12(x - 1)), then (4p=-12), so (p=-3).

Step4: Find the focus

The focus of the parabola ((y - k)^{2}=4p(x - h)) is ((h + p,k)). Substitute (h = 1), (k = 2), and (p=-3) into ((h + p,k)). (h + p=1+(-3)=-2). So the focus is ((-2,2)).

Answer:

Vertex (=(1,2)) Focus (=(-2,2))