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What is the focus of the parabola y=6x2−3y = 6x^2 - 3?\newlineSimplify any fractions.\newline(______,______)

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Q. What is the focus of the parabola y=6x2−3y = 6x^2 - 3?\newlineSimplify any fractions.\newline(______,______)
  1. Identify Parabola Form: y=6x2−3y = 6x^2 - 3 is a vertical parabola cuz it's in the form y=ax2+bx+cy = ax^2 + bx + c.
  2. Find Vertex Form: The vertex form of a parabola is y=a(x−h)2+ky = a(x - h)^2 + k, so we need to find aa, hh, and kk.
  3. Determine Parameters: Comparing y=6x2−3y = 6x^2 - 3 to y=a(x−h)2+ky = a(x - h)^2 + k, we get a=6a = 6, h=0h = 0, and k=−3k = -3.
  4. Calculate p Value: Now we find pp using the formula p=14ap = \frac{1}{4a}. So p=14×6=124p = \frac{1}{4\times 6} = \frac{1}{24}.
  5. Locate Focus: The focus of a vertical parabola is (h,k+p)(h, k + p). So we plug in h=0h = 0, k=−3k = -3, and p=124p = \frac{1}{24}.
  6. Simplify Coordinates: The focus is (0,−3+1/24)(0, -3 + 1/24). We simplify −3+1/24-3 + 1/24 to get −72/24+1/24-72/24 + 1/24.
  7. Simplify Coordinates: The focus is (0, -3 + rac{1}{24}). We simplify -3 + rac{1}{24} to get - rac{72}{24} + rac{1}{24}.After simplifying, we get - rac{71}{24}. So the focus is (0, - rac{71}{24}).

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