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Anonymous
Anonymous asked in Science & MathematicsMathematics · 6 years ago

How do you graph a parabola with only the vertex and directrix given?

I've already plotted the coordinates of the vertex and the directrix. I just don't completely understand how I could get the focus.

1) Vertex : (8, 3)

Directrix : y = 7

2) Vertex : (0, 3)

Directrix : x = 4

2 Answers

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  • 6 years ago
    Favorite Answer

    look here for a diagram

    http://img513.imageshack.us/img513/1720/unb5.png

    1)

    The focus F is on the symmetry axes of the parabola x = 8

    V(8,3) is 4 far from the directrix (7 - 3 = 4)

    FV = 4

    remember that F is "inside" the parabola

    thus

    F(8, -1)

    you can find the equation directly

    P(x,y) is a point of the parabola

    PF = PH (where H is the projection of P on the directrix)

    PF² = PH²

    (x - 8)² + (y + 1)² = (y - 7)²

    y = - x^2/16 + x - 1

    2)

    V(0,3)

    x = 4

    symmetry axes y = 3

    F(-4,3)

    equation

    (x + 4)² + (y - 3)² = (x - 4)²

    x = - y^2/16 + (3/8)y - 9/16

    Attachment image
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  • 6 years ago

    I presume that you know that the equation of a parabola whose vertex is at the origin, and whose axis is the positive x-axis is

    y² = 4ax

    Since the definition of a parabola is that any point on the parabola is the same distance from the focus and the directrix, it follows that the focus is the same distance from the vertex (but on the opposite side) as is the directrix. The parameter a in the equation is the distance from the vertex to the directrix (on one side), and to the focus on the other side. That is, if the equation is y² = 4x, then the directrix is at y = -1, and the focus is at (1, 0).

    1) Since the directrix is y = 7, and the vertex is at x = 8, it means that the focus must be at x = 9. The directrix is vertical, therefore the axis is horizontal, therefore the focus is at (9, 3)

    2) Similaly, for this one, the directrix being to the right of the vertex means that the parobola opens to the left, so the focus must be at x = -4, that is, the point (-4, 3).

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