Introduction To Parabola

Graphing of a Parabola  : Let us understand  how to Graph Parabola . For graphing parabola we will first draw rough sketches of parabola and various terms associated to them are given below:
1. Equation: y^2 = 4ax, vertex = (0, 0), focus = (a, 0), Latustrectum = 4a, Directrix: x = -a.
2. Equation: y^2 = -4ax, vertex = (0, 0), focus = (-a, 0), Latustrectum = 4a, Directrix: x = a.
3. Equation: x^2 = 4ay, vertex = (0, 0), focus = (0, a), Latustrectum = 4a, Directrix: y = -a.
4. Equation: x^2 = -4ay, vertex = (0, 0), focus = (0, -a), Latustrectum = 4a, Directrix: y = a.

Sketching of curves represented by y = ax^2 + bx + c. For graphing a parabola equation y = ax^2 + bx + c always represents Vertex of Parabola (-b/2a, -D/4a) and axis x = -b/2a.

The parabola graph opens upwards or downward according as a > 0 or < 0. It meets x-axis at (alpha, 0) and (beta, 0), where alpha and beta are the roots of the equation ax^2 + bx + c = 0.

If the roots of this equation are not real, then the parabola does not cross x-axis. In order to draw rough sketch of the parabolas given by the equations of the form y = ax^2 + bx + c, we may follow the following algorithm.

Algorithm: 
Step 1: Obtain the equation and observe the sign of the coefficient of x^2 in it.
Step 2: Put y = 0 in the given equation and get the values of x. Let the values be alpha and beta.
Step 3: Mark the points A(alpha, 0) and B(beta, 0) on x-axis.
Step 4: Draw a parabola passing through points A and B having its vertex on x = -b/2a = (alpha + beta)/2 and opening upward and downward according as the coefficient of x^2 in the given equation is positive or negative.

In the above algorithm, if the values of alpha and beta are imaginary, then the equation y = ax^2 + bx + c represents a parabola having vertex at (-b/2a, -D/4a) and opens upward or downward according as a > 0 or a < 0.

Sketching of curves represented by x = ay^2 + by + c. For a graphing a parabola the equation x = ay^2 + by + c also represents a Parabola Vertex at (-D/4a, -b/2a) axis y = -b/2a and so the parabola graph opens leftward or rightward according as a < 0 or > 0.

It crosses y –axis at (0, alpha) and (0, beta), where alpha and beta are the roots of the equation ay^2 + by + c = 0.
If alpha and beta are not real, then the parabola does not cross y-axis and it opens rightward if a > 0 and leftward if a < 0.
In order to draw a rough sketch of the parabolas given by the equations of the form x = ay^2 + by + c, we may follow the following algorithm.

Algorithm:
Step 1: Obtain the equation and observe the sign of the coefficient of y^2 in it.
Step 2: Put x = 0 in the given equation and get the values of y. Let the values be alpha and beta.
Step 3: Mark the points A(0, alpha) and B(0, beta) on y-axis.
Step 4: Draw a parabola passing through points A and B having its vertex on y = -b/2a = (alpha + beta)/2 and opening upward and downward according as a > 0 or a < 0.