IXL - Write equations of parabolas in vertex form using the focus and directrix (Geometry practice)

Write the equation in vertex form for the parabola with focus

(

–

)

and directrix

–

Simplify any fractions.

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question

Write the equation in vertex form for the parabola with focus

(

)

and directrix

–

Simplify any fractions.

key idea

A parabola is the set of points in a plane that are the same distance from a fixed line and a fixed point that is not on the line. The fixed line lies outside the parabola and is called the

directrix.

The fixed point lies inside the parabola and is called the

focus.

The vertex form of a parabola that opens up or down is

(

–

)

The vertex is

(

)

the parabola opens up. If

the parabola opens down.

solution

To write the equation in vertex form, write expressions for the distance from a point on the parabola to the focus and the directrix. Set these equal and solve.

Let

(

)

be a point on the parabola.

Write an expression for the distance between

and the focus.

Use the distance formula to write an expression for the distance between

(

)

and the focus

(

)

(

–

)

(

–

)

(

–

)

Write an expression for the distance between

and the directrix.

The directrix

–

is a horizontal line. The distance between a point and a horizontal line is the length of the vertical line segment from the point to the line.

The vertical line segment from

(

)

–

intersects the directrix at

(

–

)

Use the distance formula to write an expression for the distance between

(

)

and

(

–

)

(

–

)

(

–

)

(

)

(

)

Set the expressions equal and solve.

(

–

)

(

)

(

–

)

(

)

–

The set of all points equidistant from

(

)

and

–

is described by the equation

Looking for more? Try these:

Lesson: Parabolas

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Teacher tools

Work it out

Not feeling ready yet? These can help:

Z.4Write equations of parabolas in vertex form from graphs

Z.2Find the focus or directrix of a parabola

Lesson: Parabolas

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