Exercises on the Equation of a Parabola

Q: 🔍 How do you find a parabola's focus and directrix from just the equation?

<p class="font-claude-response-body break-words whitespace-normal" dir="ltr">The equation tells you everything. <strong>Look at which variable is squared</strong>—if y is squared, the parabola opens left or right; if x is squared, it opens up or down. The number multiplying the non-squared variable tells you the parameter p. The vertex is always at the center of the parabola, the focus lies p/2 units away from the vertex in the opening direction, and the directrix is a line p/2 units away on the opposite side.</p>

A parabola is an infinite collection of points that are equidistant from a fixed line and a fixed point in the plane. The definition above is also known as the locus of a parabola.

The parabola contains characteristic elements such as the directrix and the focus, which are the line and point, both fixed, mentioned in the definition. It also has a vertex, which is the point closest to the directrix and focus.

In the following problems and exercises, we work with two modalities: we obtain the elements of the parabola from knowing its equation, and if we know the elements of the parabola, we obtain its equation.

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Obtaining the Elements of the Parabola

Score:

Given the parabola $\text{[math]}$ , calculate its vertex, focus, and directrix.

Solution

The parameter is:

$\text{[math]}$

This is a reduced equation, so the vertex is at the origin:

$\text{[math]}$

The squared term in the equation is $\text{[math]}$ , so the axis of the parabola coincides with the x-axis. Moreover, the parabola is on the positive side of the x-axis because the coefficient accompanying the non-squared term (in this case $\text{[math]}$ ) is 8, which is positive:

$\text{[math]}$

The graphical representation of the parabola $\text{[math]}$ is

representacion gráfica de la parábola

Given the parabola $\text{[math]}$ , calculate its vertex, focus, and directrix.

Solution

The parameter is:

$\text{[math]}$

This is a reduced equation, so the vertex is at the origin:

$\text{[math]}$

The squared term in the equation is $\text{[math]}$ , so the axis of the parabola coincides with the x-axis. Moreover, the parabola is on the negative side of the x-axis because the coefficient accompanying the non-squared term (in this case $\text{[math]}$ ) is −8, which is negative:

$\text{[math]}$

The graphical representation of the parabola $\text{[math]}$ is

ecuación reducida de la parábola representación gráfica

Given the parabola $\text{[math]}$ , calculate its vertex, focus, and directrix.

Solution

The parameter is:

$\text{[math]}$

This is a reduced equation, so the vertex is at the origin:

$\text{[math]}$

The squared term in the equation is $\text{[math]}$ , so the axis of the parabola coincides with the y-axis. Moreover, the parabola is on the positive side of the y-axis because the coefficient accompanying the non-squared term (in this case $\text{[math]}$ ) is 8, which is positive:

$\text{[math]}$

The graphical representation of the parabola $\text{[math]}$ is

ecuaciones de la parabola representación gráfica

Given the parabola $\text{[math]}$ , calculate its vertex, focus, and directrix.

Solution

The parameter is:

$\text{[math]}$

This is a reduced equation, so the vertex is at the origin:

$\text{[math]}$

The squared term in the equation is $\text{[math]}$ , so the axis of the parabola coincides with the y-axis. Moreover, the parabola is on the negative side of the y-axis because the coefficient accompanying the non-squared term (in this case $\text{[math]}$ ) is −8, which is negative:

$\text{[math]}$

The graphical representation of the parabola $\text{[math]}$ is

elementos de las parabolas representación gráfica

Given the parabola $\text{[math]}$ , calculate its vertex, focus, and directrix.

Solution

The parameter is:

$\text{[math]}$

This is not a reduced equation, so the vertex is at:

$\text{[math]}$

The squared term in the equation is $\text{[math]}$ , so the axis of the parabola is parallel to the x-axis. Moreover, the coefficient accompanying the non-squared term (in this case $\text{[math]}$ ) is 8, which is positive, so the focus is to the right of the vertex:

$\text{[math]}$

The graphical representation of the parabola $\text{[math]}$ is

parabola con eje paralelo al eje OX representación gráfica

Given the parabola $\text{[math]}$ , calculate its vertex, focus, and directrix.

Solution

The parameter is:

$\text{[math]}$

This is not a reduced equation, so the vertex is at:

$\text{[math]}$

The squared term in the equation is $\text{[math]}$ , so the axis of the parabola is parallel to the y-axis. Moreover, the coefficient accompanying the non-squared term (in this case $\text{[math]}$ ) is 8, which is positive, so the focus is above the vertex:

$\text{[math]}$

The graphical representation of the parabola $\text{[math]}$ is

parábolas representación gráfica

Determine, in reduced form, the equations of the following parabolas, indicating the value of the parameter, the coordinates of the focus, and the equation of the directrix:

$\text{[math]}$
$\text{[math]}$
$\text{[math]}$

Solution

1 $\text{[math]}$

We solve for the term $\text{[math]}$ :

$\text{[math]}$

The equation of the parabola: $\text{[math]}$

The parameter is:

$\text{[math]}$

This is a reduced equation, so the vertex is at the origin:

$\text{[math]}$

The squared term in the equation is $\text{[math]}$ , so the axis of the parabola coincides with the x-axis. Moreover, the coefficient accompanying the non-squared term (in this case $\text{[math]}$ ) is 2, which is positive, so the focus is to the right of the vertex:

$\text{[math]}$

The graphical representation of the parabola $\text{[math]}$ is

parábolas acostadas representación gráfica y^2=2x

2 $\text{[math]}$

We solve for the term $\text{[math]}$ :

$\text{[math]}$

The parameter is:

$\text{[math]}$

This is a reduced equation, so the vertex is at the origin:

$\text{[math]}$

The squared term in the equation is $\text{[math]}$ , so the axis of the parabola coincides with the x-axis. Moreover, the coefficient accompanying the non-squared term (in this case $\text{[math]}$ ) is $\text{[math]}$ , which is negative, so the focus is to the left of the vertex:

$\text{[math]}$

The graphical representation of the parabola $\text{[math]}$ is

ecuacion reducida de la parabola representación gráfica

3 $\text{[math]}$

We solve for the term $\text{[math]}$ :

$\text{[math]}$

The parameter is:

$\text{[math]}$

This is a reduced equation, so the vertex is at the origin:

$\text{[math]}$

The squared term in the equation is $\text{[math]}$ , so the axis of the parabola coincides with the y-axis. Moreover, the coefficient accompanying the non-squared term (in this case $\text{[math]}$ ) is $\text{[math]}$ , which is negative, so the focus is below the vertex:

$\text{[math]}$

The graphical representation of the parabola $\text{[math]}$ is

parabola hacia abajo representación gráfica

Calculate the coordinates of the vertex and focus, and the equations of the directrices of the parabolas:

- $\text{[math]}$

- $\text{[math]}$

- $\text{[math]}$

Solution

1 $\text{[math]}$

vertice y foco de la parabola representación gráfica

We complete the square:

$\text{[math]}$

We simplify:

$\text{[math]}$

We solve:

$\text{[math]}$

The parameter is:

$\text{[math]}$

2 $\text{[math]}$

directriz de la parabola representación gráfica

We complete the square:

$\text{[math]}$

We simplify:

$\text{[math]}$

We solve:

$\text{[math]}$

Therefore:

$\text{[math]}$

The parameter is:

$\text{[math]}$

The squared term in the equation is $\text{[math]}$ , so the axis of the parabola is parallel to the y-axis. Moreover, the coefficient accompanying the non-squared term (in this case $\text{[math]}$ ) is 6, which is positive, so the focus is above the vertex:

$\text{[math]}$

3 $\text{[math]}$

vertice de la parabola representación gráfica

We complete the square:

$\text{[math]}$

We simplify:

$\text{[math]}$

We solve:

$\text{[math]}$

The parameter is:

$\text{[math]}$

The squared term in the equation is $\text{[math]}$ , so the axis of the parabola is parallel to the y-axis. Moreover, the coefficient accompanying the non-squared term (in this case $\text{[math]}$ ) is 1, which is positive, so the focus is above the vertex:

$\text{[math]}$

Obtaining the Equation of the Parabola

Score:

Determine the equations of the parabolas that have:

Directrix $\text{[math]}$ , focus $\text{[math]}$
Directrix $\text{[math]}$ , vertex $\text{[math]}$
Directrix $\text{[math]}$ , focus $\text{[math]}$
Directrix $\text{[math]}$ , focus $\text{[math]}$
Focus $\text{[math]}$ , vertex $\text{[math]}$
Focus $\text{[math]}$ , vertex $\text{[math]}$
Focus $\text{[math]}$ , vertex
Focus $\text{[math]}$ , vertex $\text{[math]}$

Solution

1 Directrix $\text{[math]}$ , focus $\text{[math]}$

obtener ecuacion de una parabola representación gráfica

First we calculate the distance between the focus and the directrix to obtain the parameter $\text{[math]}$ :

$\text{[math]}$

Since the focus is on the x-axis, the directrix is parallel to the y-axis, and they are equidistant from the origin, this is a reduced equation:

$\text{[math]}$

Since the axis coincides with the x-axis and the focus is to the right of the vertex, the equation is:

$\text{[math]}$

2 Directrix $\text{[math]}$ , vertex $\text{[math]}$

obtener la ecuacion de parabolas representación gráfica

First we calculate the distance between the vertex and the directrix to obtain $\text{[math]}$ :

$\text{[math]}$

We note that the vertex is at the origin and the directrix is parallel to the x-axis, so this is a reduced equation.

Since the axis coincides with the y-axis and the focus is below the vertex, the equation is:

$\text{[math]}$

3 Directrix $\text{[math]}$ , focus $\text{[math]}$

ecuacion parabolica representación gráfica

First we calculate the distance between the focus and the directrix to obtain the parameter $\text{[math]}$ :

$\text{[math]}$

We note that the directrix is parallel to the x-axis, the focus is on the y-axis, and they are equidistant from the origin, so this is a reduced equation.

Since the focus is above the directrix, the equation is:

$\text{[math]}$

4 Directrix $\text{[math]}$ , focus $\text{[math]}$

parabola representación gráfica

First we calculate the distance between the focus and the directrix to obtain the parameter $\text{[math]}$ :

$\text{[math]}$

We note that the directrix is parallel to the y-axis, the focus is on the x-axis, and they are equidistant from the origin, so this is a reduced equation.

Since the focus is to the left of the directrix, the equation is:

$\text{[math]}$

5 Focus $\text{[math]}$ , vertex $\text{[math]}$

parabolas representación gráfica de foco 2,0 y vertice 0,0

First we calculate the distance between the focus and the vertex to obtain the parameter $\text{[math]}$ :

$\text{[math]}$

We note that the directrix is parallel to the y-axis, the focus is on the x-axis, and they are equidistant from the origin, so this is a reduced equation.

Since the focus is to the right of the directrix, the equation is:

$\text{[math]}$

6 Focus $\text{[math]}$ , vertex $\text{[math]}$

obtener la ecuacion de una parabola ejercicios representación gráfica foco 3,2 y vértice 5,2

First we calculate the distance between the focus and the vertex to obtain $\text{[math]}$ :

$\text{[math]}$

We note that the directrix is parallel to the y-axis.

Since the focus is to the left of the vertex, the equation is:

$\text{[math]}$

7 Focus $\text{[math]}$ , vertex

ecuaciones parabolicas representación gráfica

First we calculate the distance between the focus and the vertex to obtain $\text{[math]}$ :

$\text{[math]}$

We note that the directrix is $\text{[math]}$ and is parallel to the x-axis.

Since the focus is above the vertex, the equation is:

$\text{[math]}$

8 Focus $\text{[math]}$ , vertex $\text{[math]}$

La parabola representación gráfica de foco 3,4 y vertice 1,4

First we calculate the distance between the focus and the vertex to obtain the parameter $\text{[math]}$ :

$\text{[math]}$

We note that the directrix is parallel to the y-axis.

Since the focus is to the right of the vertex, the equation is:

$\text{[math]}$

Find the equation of the parabola whose vertex coincides with the origin of coordinates and passes through the point $\text{[math]}$ , with axis on the x-axis.

Solution

Since the vertex is at the origin and the axis coincides with the x-axis of the plane, its equation is in reduced form, specifically:

$\text{[math]}$

It passes through the point (3, 4), so its coordinates satisfy the previous equation:

$\text{[math]}$

We divide by 3:

$\text{[math]}$

Thus, the equation is:

$\text{[math]}$

Write the equation of the parabola with axis parallel to the y-axis, vertex on the x-axis, and passing through the points $\text{[math]}$ and $\text{[math]}$ .

Solution

From the problem we know that:

$\text{[math]}$

Since the curve passes through points A and B, their coordinates must satisfy the equation of the parabola:

$\text{[math]}$

We take the first equation and multiply it by 4:

$\text{[math]}$

We subtract the second equation (or add its negative):

$\text{[math]}$

And thus obtain:

$\text{[math]}$

We simplify by dividing everything by 3:

$\text{[math]}$

The two solutions for $\text{[math]}$ give us two distinct parabola equations:

$\text{[math]}$

Determine the equation of the parabola that has directrix $\text{[math]}$ and focus at the origin.

Solution

$\text{[math]}$

We square both sides to eliminate the square root:

$\text{[math]}$

Find the equation of the parabola with vertical axis passing through the points: $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ .

Solution

The equation must be of the form:

$\text{[math]}$

If it passes through points A, B, and C, their coordinates satisfy the previous equation:

$\text{[math]}$

Solving the system of 3 unknowns we obtain:

$\text{[math]}$

Finally:

$\text{[math]}$

Determine the equation of the parabola that has directrix $\text{[math]}$ and focus at point $\text{[math]}$ .

Solution

$\text{[math]}$

Finally, the parabola equation is:

$\text{[math]}$

Calculate the relative position of the line $\text{[math]}$ with respect to the parabola $\text{[math]}$ .

Solution

Interseccion de una recta y una parabola representación gráfica

$\text{[math]}$

We expand:

$\text{[math]}$

We solve:

$\text{[math]}$

Points of intersection:

$\text{[math]}$

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Agostina Babbo

Agostina Babbo is an English and Italian to Spanish translator and writer, specializing in product localization, legal content for tech, and team sports—particularly handball and e-sports. With a degree in Public Translation from the University of Buenos Aires and a Master's in Translation and New Technologies from ISTRAD/Universidad de Madrid, she brings both linguistic expertise and technical insight to her work.

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Frequently asked questions

🔍 How do you find a parabola's focus and directrix from just the equation?

The equation tells you everything. Look at which variable is squared—if y is squared, the parabola opens left or right; if x is squared, it opens up or down. The number multiplying the non-squared variable tells you the parameter p. The vertex is always at the center of the parabola, the focus lies p/2 units away from the vertex in the opening direction, and the directrix is a line p/2 units away on the opposite side.

✏️ When you know the focus and directrix, why is finding the parabola's equation so straightforward?

By definition, every point on a parabola is equidistant from the focus and directrix. So you use the distance formula: distance from a point to the focus equals distance to the directrix. Square both sides to eliminate the square roots, simplify, and you have your equation. The math flows directly from the definition.

Chapters

Obtaining the Elements of the Parabola
Obtaining the Equation of the Parabola

Parabola Equation - Problems and Exercises

Obtaining the Elements of the Parabola

Obtaining the Equation of the Parabola

Exercises on the Equation of an Ellipse

Circle Equation Exercises I

Elliptical Exercises: Definition, Practical Questions, & More!

Exercises on Parabolas