Problems based on Cartesian Equation of Parabola

A parabola is an open plane curve formed when a right circular cone meets a plane parallel to one of its sides. All parabola’s points are at  equal distance from a fixed line. A parabola is a graphical portrayal of a second-degree equation or a quadratic equation. The projectile motion of a body along a parabolic curved path, footbridges in the shape of a parabola, reflective telescopes, and antennae are all examples of parabolas. 

Properties of a Parabola

There are six properties of a Parabola:

1. Vertex- Vertex is found in the curve’s centre. It can either be at the origin or anywhere in the Cartesian plane.

2. Concavity- The orientation of the parabolic curve is the concavity of a parabola. The curve may open to the left or right or upward or downward.

3. Focus- A parabolic curve’s axis of symmetry is the focal point. It is a distance of ‘a’ units from the parabola’s vertex.

4. Axis of Symmetry- The imaginary line including the vertex, focus, and midway of the directrix is called the axis of symmetry. It is the imaginary line that divides the parabola into equal individual shares that are mirror images of one another.

5. Directrix- A parabola’s directrix is the line that runs parallel towards both axes. The directrix is ‘a’ units from the vertex and ‘2a’ units from focus away from the vertex.

6. Latus rectum- The latus rectum is a segment that passes through the focus of a parabolic curve and is perpendicular to the axis of symmetry. 

Graph of Parabola

A parabola’s(y^2=4ax) focus is ‘a’ units distant from the vertex and is directly on the right or left side, depending on whether it opens to the right or left. The focus of a parabola, on the other hand, is precisely above or below the vertex if it expands upward or downward.

The axis of symmetry is either the x-axis or parallel to the x-axis if the parabola opens to the right or left. The axis of symmetry is either the y-axis or parallel to the y-axis if the parabola opens upward or downward.

Let us see how to graph a parabola step by step:

1. Determine the parabolic equation’s concavity. The directions of the curve’s opening may be found in the table above. It may open to the left or right, as well as upward or downward.

2. Determine the parabola’s vertex. The vertex might be  (0, 0) or (1, 1),(h, k).

3. Determine the parabola’s focus.

4. Evaluate the latus rectum’s coordinate.

5. Find the parabolic curve’s directrix. The directrix is at the same distance from the vertex as the focus but in the opposite direction.

6. Draw a curve connecting the vertex and the coordinates of the latus rectum to graph the parabola. Then, to finish, label all of the parabola’s major points.

Cartesian Equation of Parabola

An analytic equation for the parabola may be constructed if the parabola is put in the Cartesian plane with the specified Cartesian coordinate system O(x,y), with the vertex V at the origin O and the focus F on the positive portion of the coordinate axis y.

A parabola’s general equation is y = a(x-h)^2 + k or x = a(y-k)^2 +h,

 where (h,k) is the vertex.

A normal parabola’s standard equation is y^2 = 4ax.

The Cartesian coordinate plane, with the axes  x and y, is used in our equation. We have two types of standard equations that alter depending on which way our parabola is pointing. 

Parabola Calculator

A parabola calculator is a tool that lets you input data about a parabola and obtain a variety of results.

Although technically, any calculator that provides information about a parabola might be deemed a parabola calculator, these are frequently available online. Many advanced graphing calculators contain parabola applications, allowing them to turn into a parabola calculator while maintaining other features.

Conclusion

To conclude, we will say that a Cartesian plane is a graph that has two axes, one for the x-axis and one for the y-axis. These two axes are parallel to one another. The origin (O) is located at the precise middle of the graph, where the two axes intersect. The numbers on the x-axis to the right of zero are positive, whereas the numbers to the left of zero are negative. Numbers below zero on the y-axis are negative, whereas those above zero are positive