A parabola is a U-shaped mirror symmetric curve which has different characteristics and applications in mathematics and physics. A parabola is described through the quadratic functions y = x2 in most of the two dimensional diagrams. Students of mathematics are often given problems on parabola and in many assignments parabola forms the part of the assignments. In physics too parabola plays a significant role in mechanics. Some assignments seem so hard that it becomes tough for the students of mathematics to solve them within the given deadlines.
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To form a parabola, one needs to start with a line and a point off to one side. This line is called “Directrix” and the point is called the “Focus“. The parabola is a curve where the points on the curve are equidistant from the directrix and focus.
Any geometrical structure of parabola will reveal that it is a U-shaped geometric image. There is a sharp turning point in a parabola from which the graphical figure changes the direction abruptly. There may be two types of turning points –
The axis of symmetry
If a parabola has a turning point (h,k), then the line of symmetry is given by x = h.
The turning point gives us information on the location of the line of symmetry along with the highest point or lowest point of the parabola.
The point on the “Axis of symmetry” which is exactly the midway between the focus and Directrix is called the vertex.
Quadratic forms of parabola
Apart from y = x2, parabolas can also be of the following forms –
y=ax2+bx+c This is the general form of y interception.
y=a(x+d) (x+e), This is the x intercepted form with x as the primary factor.
y=a(x+k)2+v This is known as the turning point form.
Each of these quadratic forms of parabola signifies some specific characteristics of the most popular and useful geometric figure.
It is evident from all these quadratic functions that a parabola has only one y intercept whereas the x intercepts may vary from zero to two.
Like hyperbola, ellipse, and circle, parabola is a special conic section with the above mentioned characteristics.
If a real life situation is considered, if any light or sound ray is inflowing the parabola, which is parallel to the axis of symmetry, and hitting the inner surface of the parabolic “bowl“, it will be reflected back to the focus.
Bionic ears and radio telescopes etc. are the real life examples of parabolic dishes. With the help of their parabolic property, a signal is received with a maximum concentration onto a receiver. In general, a parabola is differentiated from other conic sections, in the following aspects –
The assignments on parabola inevitably requires the following –
In some assignments, real life problems may be given which may need some more formulas to solve successfully. Again, many problems on parabola may be solved in more than one ways. The examiner may ask the students to follow any particular process to solve those equations.
In any case, problems on parabola are not easy to solve. The students are expected to show the highest levels of acumen and knowledge on that subject to solve the equations and assignments.
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