# Precalculus : Find the Focus and the Directrix of a Parabola

## Example Questions

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### Example Question #1 : Find The Focus And The Directrix Of A Parabola

Find the focus and the directrix of the following parabola: .

Focus:
Directrix:

Focus:

Directrix:

Focus:

Directrix:

Focus:

Directrix:

Focus:
Directrix:

Explanation:

To find the focus from the equation of a parabola, first set the equation to resemble the form  where  represents any numerical value.

For our problem, it is already in this form.

Therefore,

.

Solve for  then .

The focus for this parabola is given by .

So,  is the focus of the parabola.

The directrix is represented as .

Therefore, the directrix for this problem is .

### Example Question #2 : Find The Focus And The Directrix Of A Parabola

Find the directerix of the parabola with the following equation:

Explanation:

Recall the standard form of the equation of a vertical parabola:

, where  is the vertex of the parabola and  gives the focal length.

When , the parabola will open up.

When , the parabola will open down.

For the parabola in question, the vertex is  and . This parabola will open up. Because the parabola will open up, the directerix will be located  units down from the vertex. The equation for the directerix is then .

### Example Question #3 : Find The Focus And The Directrix Of A Parabola

Find the directerix for the parabola with the following equation:

Explanation:

Recall the standard form of the equation of a vertical parabola:

, where  is the vertex of the parabola and  gives the focal length.

When , the parabola will open up.

When , the parabola will open down.

For the parabola in question, the vertex is  and . This parabola will open up. Because the parabola will open up, the directerix will be located  unit down from the vertex. The equation for the directerix is then .

### Example Question #4 : Find The Focus And The Directrix Of A Parabola

Find the directerix of the parabola with the following equation:

Explanation:

Recall the standard form of the equation of a vertical parabola:

, where  is the vertex of the parabola and  gives the focal length.

When , the parabola will open up.

When , the parabola will open down.

Start by putting the equation in th estandard form of the equation of a vertical parabola.

Isolate the  terms to one side.

Complete the square for the  terms. Remember to add the same amount on both sides!

Factor out both sides of the equation to get the standard form of a vertical parabola.

For the parabola in question, the vertex is  and . This parabola will open down. Because the parabola will open down, the directerix will be located  units above the vertex. The equation for the directerix is then .

### Example Question #5 : Find The Focus And The Directrix Of A Parabola

Find the focus of the parabola with the following equation:

Explanation:

Recall the standard form of the equation of a vertical parabola:

, where  is the vertex of the parabola and  gives the focal length.

When , the parabola will open up.

When , the parabola will open down.

Start by putting the equation into the standard form.

Isolate the  terms on one side.

Complete the square. Remember to add teh same amount on both sides!

Factor both sides of the equation to get the standard equation for the parabola.

For the parabola in question, the vertex is  and . This parabola will open up. Because the parabola will open up, the focus will be located  unit up from the vertex. The focus is then located at .

### Example Question #6 : Find The Focus And The Directrix Of A Parabola

Find the focus of the parabola with the following equation:

Explanation:

Recall the standard form of the equation of a vertical parabola:

, where  is the vertex of the parabola and  gives the focal length.

When , the parabola will open up.

When , the parabola will open down.

For the parabola in question, the vertex is  and . This parabola will open down. Because the parabola will open down, the focus will be located  units down from the vertex. The focus is then located at .

### Example Question #7 : Find The Focus And The Directrix Of A Parabola

Find the focus of the parabola with the following equation:

Explanation:

Recall the standard form of the equation of a vertical parabola:

, where  is the vertex of the parabola and  gives the focal length.

When , the parabola will open up.

When , the parabola will open down.

For the parabola in question, the vertex is  and . This parabola will open down. Because the parabola will open down, the focus will be located  units down from the vertex. The focus is then located at

### Example Question #8 : Find The Focus And The Directrix Of A Parabola

Find the directerix of the parabola with the following equation:

Explanation:

Recall the standard form of the equation of a horizontal parabola:

, where  is the vertex of the parabola and  is the focal length.

When , the parabola opens to the right.

When , the parabola opens to the left.

For the given parabola, the vertex is  and . This means the parabola is opening to the left and that the directerix will be located  unit to the right of the vertex. The directerix is then .

### Example Question #9 : Find The Focus And The Directrix Of A Parabola

Find the directerix of the parabola with the following equation:

Explanation:

Recall the standard form of the equation of a horizontal parabola:

, where  is the vertex of the parabola and  is the focal length.

When , the parabola opens to the right.

When , the parabola opens to the left.

For the given parabola, the vertex is  and . This means the parabola is opening to the left and that the directerix will be located  units to the right of the vertex. The directerix is then .

### Example Question #10 : Find The Focus And The Directrix Of A Parabola

Find the directerix of the parabola with the following equation:

Explanation:

Recall the standard form of the equation of a horizontal parabola:

, where  is the vertex of the parabola and  is the focal length.

When , the parabola opens to the right.

When , the parabola opens to the left.

Put the given equation into the standard form. Start by isolating the  terms to one side.

Complete the square. Remember to add the same amount to both sides of the equation.

Factor both sides of the equation to get the standard form of the equation of a horizontal parabola.

For the given parabola, the vertex is  and . This means the parabola is opening to the right and that the directerix will be located  units to the left  of the vertex. The directerix is then .

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