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What is the focus of the parabola y=8x2−4y = 8x^2 - 4?\newlineSimplify any fractions.\newline(______,______)

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Q. What is the focus of the parabola y=8x2−4y = 8x^2 - 4?\newlineSimplify any fractions.\newline(______,______)
  1. Identify Parabola Type: We got y=8x2−4y = 8x^2 - 4, which is a vertical parabola.\newlineFind the vertex form by completing the square if needed.\newlineBut here, no need to complete the square since xx is already squared and there's no xx term.\newlineSo, the vertex form is y=8(x−0)2−4y = 8(x - 0)^2 - 4, where a=8a = 8, h=0h = 0, and k=−4k = -4.
  2. Find Vertex Form: Now, let's find the value of pp using the formula p=14ap = \frac{1}{4a}. So, p=14×8p = \frac{1}{4\times 8}. p=132p = \frac{1}{32}.
  3. Calculate p Value: The focus of a vertical parabola is (h,k+p)(h, k + p). We already know h=0h = 0, k=−4k = -4, and p=132p = \frac{1}{32}. So, the focus is (0,−4+132)(0, -4 + \frac{1}{32}).
  4. Determine Focus Coordinates: Now, let's simplify −4+132-4 + \frac{1}{32}.−4-4 is the same as −12832-\frac{128}{32}.So, −12832+132=−12732-\frac{128}{32} + \frac{1}{32} = -\frac{127}{32}.The focus is (0,−12732)(0, -\frac{127}{32}).

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