Practice Problems on Parabola

A quadratic function is a polynomial with a degree of two. A quadratic function contains only one unknown. Such quadratic functions can be represented with the help of graphs. Various quadratic equations lead to various shapes on a graph. One of those shapes is a parabola. 

A parabola is a plane curve. The approximate shape of a parabola is like the letter U, and it is mirror-symmetric. Parabola equation examples show us what should be the coefficient of the x-axis and y-axis for a parabola. Parabola belongs to the family of conic sections and has interesting properties.

Parabola

As discussed, a parabola is a plane curve. A parabola is shaped like the letter u, and it is symmetric in nature. A parabola can be defined as a locus of points in a plane in which the points are equidistant from the focus and the directrix. A parabola is also formed when a right circular conical section intersects with the plane, which is parallel to another plane.

This other plane is tangential to the conical surface. 

The axis of the symmetry of a parabola is a line that divides the parabola into two symmetric parts. This line passes through the focus of the parabola and is perpendicular to the directrix. The vertex of the parabola is a point at which the axis of symmetry intersects with the parabola. A sharp curve can be observed at this point. 

The Properties of a Parabola

There are various properties of the parabola that define a parabola. Let us take a look at those properties. 

• Normal:

Each parabola has a normal. A normal parabola is a line that passes through the point of contact and is perpendicular to the tangent of the parabola.

The Normal also passes through the focus of the parabola. Let’s take one of the equation examples.

Suppose the equation of a parabola is

y²= 4ax

and the point of contact is (x1, y1) along with the slope of -y1/2a.

Then the equation of the normal for that parabola will be

 (y – y1) = (-y1 / 2a) × (x – x1)

• Tangent:

A tangent is a line that touches the parabola at a point. Suppose the point of contact is (x1, y1) and the equation of the parabola is,

y²= 4ax

then, the equation of the tangent will be

yy1 = 2a( x + x1)

• Pole and the Polar:

Suppose a point is lying outside the parabola. Then the locus made by the points of intersection by the tangents, drawn at the end of the chords, which are drawn from this particular point, is called the polar. If the coordinates of the polar x1 and y1 and the equation of the parabola,

y²= 4ac

then, the polar will have the equation,

yy1 = 2x (x + x1)

Parabola Equation Examples

The parabola equation examples are

y= a(x – h)² + k

x = a(y – k)² + h

Here (h, k) denotes the vertex. This is the general parabola equation example. The standard equation of the parabola is y²= 4ax. The parabola with the equation,

y= a(x – h)² + k is a regular parabola, while the parabola with the equation,

x = a(y – k)² + h is a sideways parabola. 

In the equation of the parabola, if the term y² appears, then the axis of symmetry of the parabola is along the x-axis, and if the equation contains a term with x², then the axis of symmetry of the parabola is along the y axis. In the parabola equation examples, suppose x has a positive coefficient. The parabola will open to the right, and if it has a negative coefficient, the parabola will open to the left.

Parabola Graph Equation

We can draw a parabola graph based on the Parabola graph equation.

Suppose the parabola has the equation

y = 3x² – 6x + 5

Which means a = 3, b = -6, and c = -5.

The graph of this parabola will open up as the a is positive.

Since h = -b / 2a, the value of h will become 1, and since k = f(h), the value of k will become 2, hence the vertex will be (1,2).  The axis of symmetry that is X will be equal to 1, and the directrix that is y will be equal to zero.

Conclusion

A quadratic equation can be graphically represented with the help of graphs. The graphs of the quadratic equation take various shapes. A parabola is one such shape; it is a plane curve. The equations,

y = a(x – h)² + k

x = a(y – k)² + h

are the general forms of the equation of the parabola.

A parabola has a point (a, 0) which is the focus of the parabola. The eccentricity of a parabola will be equal to 1. The latest rectum of a parabola is equal to 4a. A parabola also had a focal chord and focal distance.