What is the equation of the quadratic graph with a focus of (3, -1) and a directrix of y = 1?

Question

What is the equation of the quadratic graph with a focus of (3, -1) and a directrix of y = 1?

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RuslanHeatt 6 months 2021-07-29T20:33:44+00:00 1 Answers 7 views 0

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    2021-07-29T20:35:32+00:00

    Answer:

    The equation of the quadratic graph is y =  -\frac{1}{4} (x – 3)²

    Step-by-step explanation:

    The standard form of the equation of the quadratic graph is


    (x – h)² = 4p(y – k)
    , where

    • The vertex of the parabola is (h, k)
    • The focus is (h, k + p)
    • The directrix is at y = k – p  

    ∵ The focus is (3, -1)

    ∵ The focus is (h, k + p)

    → Compare them

    h = 3

    k + p = -1 ⇒ (1)

    ∵ The directrix is at y = 1

    ∵ The directrix is at y = k – p

    → Compare them

    k – p = 1 ⇒ (2)

    → Add equations (1) and (2) to eliminate p

    ∵ (k + k) + (p – p) = (-1 + 1)

    ∴ 2k + 0 = 0

    ∴ 2k = 0

    → Divide both sides by 2

    K = 0

    → Substitute the value of k in equation (1) to find p

    ∵ 0 + p = -1

    p = -1

    → Substitute the values of h, k, and p in the form of the equation above

    (x – 3)² = 4(-1)(y – 0)

    ∴ (x – 3)² = -4(y)

    (x – 3)² = -4y

    → Divide both sides by -4

    -\frac{1}{4} (x – 3)² = y

    → Switch the two sides

    y =  -\frac{1}{4} (x – 3)²

    The equation of the quadratic graph is y =  -\frac{1}{4} (x – 3)²

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Giải phương trình 1 ẩn: x + 2 - 2(x + 1) = -x . Hỏi x = ? ( )