# Precalculus : Determine the equation of a parabola and graph a parabola

## Example Questions

### Example Question #25 : Parabolas

Rewrite  in standard form.

Explanation:

The standard form of a parabola is .

Factorize the right side of , and simplify.

### Example Question #11 : Determine The Equation Of A Parabola And Graph A Parabola

If the vertex of the parabola is , and the y-intercept is , find the equation of the parabola, if possible.

Explanation:

First, write the equation of the parabola in standard form.

Determine the values of the coefficients.  The value of the y-intercept is 4, which means that .

Write the vertex formula.

The given point of the vertex is , which indicates that:

Substitute the value of the point and  into the standard form.

Substitute this value into  to determine the value of .

Substitute the values of  coefficients into the standard form of the parabola.

### Example Question #27 : Parabolas

Find the standard form of the equation of the following parabola:

Explanation:

Recall the standard equation of a horizontal parabola:

, where  is the vertex and  is the focal length.

Start by isolating the  terms.

Complete the square on the left. Make sure to add the same amount to both sides of the equation!

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

### Example Question #28 : Parabolas

Find the standard form of equation of the following parabola:

Explanation:

Recall the standard equation of a horizontal parabola:

, where  is the vertex and  is the focal length.

Start by isolating the  terms.

Complete the square on the left. Make sure to add the same amount to both sides of the equation!

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

### Example Question #29 : Parabolas

Find the standard form of the equation for the following parabola:

Explanation:

Recall the standard equation of a horizontal parabola:

, where  is the vertex and  is the focal length.

Start by isolating the  terms.

Complete the square on the left. Make sure to add the same amount to both sides of the equation!

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

### Example Question #30 : Parabolas

Find the standard form of the equation for the following parabola:

Explanation:

Recall the standard equation of a vertical parabola:

, where  is the vertex and  is the focal length.

Start by isolating the  terms.

Complete the square on the left. Make sure to add the same amount to both sides of the equation!

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

### Example Question #31 : Parabolas

Find the standard form of the equation for the following parabola:

Explanation:

Recall the standard equation of a vertical parabola:

, where  is the vertex and  is the focal length.

Start by isolating the  terms.

Complete the square on the left. Make sure to add the same amount to both sides of the equation!

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

### Example Question #32 : Parabolas

Find the standard form of the equation for the following parabola:

Explanation:

Recall the standard equation of a vertical parabola:

, where  is the vertex and  is the focal length.

Start by isolating the  terms.

Complete the square on the left. Make sure to add the same amount to both sides of the equation!

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

### Example Question #33 : Parabolas

Find the standard form of the equation of a parabola with the following equation:

Explanation:

Recall the standard equation of a vertical parabola:

, where  is the vertex and  is the focal length.

Start by isolating the  terms.

Complete the square on the left. Make sure to add the same amount to both sides of the equation!

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

### Example Question #34 : Parabolas

Find the standard form of the equation of a parabola that has its focus at  and its directerix at .

Explanation:

First, we need to figure out if this is a vertical parabola or a horizontal parabola. Since we need to move down  units from the focus to the directerix, we know that this is a vertical parabola.

Recall the standard equation of a vertical parabola:

, where  is the vertex and  is the focal length.

Remember that the distance from the focus to the directerix is . Then, .

With this information, we can then figure out the vertex of the parabola. Since this is a vertical parabola, the -coordinate of the focus and the vertex are the same. To find the -coordinate of the vertex, just move down  units from the focus. The vertex is then .

Since the focus is above the vertex, we know that .

Now, from the vertex, we know that  and .

Now, plug in all the information to write the standard equation for this parabola: