IXL - Write equations of hyperbolas in standard form (Precalculus practice)

The graph shows a hyperbola. Write its equation in standard form.

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question

The graph shows a hyperbola. Write its equation in standard form.

key idea

A hyperbola that has a horizontal orientation opens right and left. A hyperbola that has a vertical orientation opens up and down.

The equation of a hyperbola that has a horizontal orientation in standard form is:

(

–

)

–

(

–

)

The center is

(

)

the vertices are

(

)

and the co-vertices are

(

)

Note that the vertices are on the hyperbola, but the co-vertices are not.

The transverse axis is the line segment with endpoints at the vertices and has length

The conjugate axis is the line segment with endpoints at the co-vertices and has length

(

)

(

–

)

(

)

(

–

)

(

)

solution

To write the equation in standard form, find the center

(

)

Then, determine the orientation, and find the values of

and

Find the center.

The asymptotes intersect at the center of the hyperbola.

The center is

(

)

Determine the orientation, and find

and

The hyperbola opens right and left, so it has a horizontal orientation. This means the transverse axis is horizontal and the conjugate axis is vertical.

To help visualize the transverse and conjugate axes, draw the rectangle whose length equals that of the transverse axis and whose diagonals are the asymptotes of the hyperbola. The width of this rectangle is the length of the conjugate axis.

The length of the transverse axis is the distance between the two vertices, which is 6. Since

is half the length of the transverse axis,

The length of the conjugate axis is the distance between the two co-vertices, which is 4. Since

is half the length of the conjugate axis,

Write the equation in standard form.

The equation of the hyperbola in standard form is:

(

–

)

–

(

–

)

Hyperbola with a horizontal orientation

(

–

)

–

(

–

)

Plug in

and

Plug in

and

–

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Not feeling ready yet? These can help:

U.10Find properties of hyperbolas

GG.6Write equations of hyperbolas in standard form from graphs

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