Graphing a quadratic equation/function:

A quadratic equation's graph has the shape of a parabola. It is a U shaped structure that can be upright or inverted.

The main characteristics of a parabola are:

1. Vertex: The vertex is the apex point of a parabola. It is the point where the axis of symmetry of the parabola intersects the parabola.
2. Axis of symmetry: The axis of symmetry is a line that can be drawn through a parabola in such a way that the part of the parabola on one side of it is the mirror image of the part on the other side.

To graph a quadratic equation or a quadratic function, we first need to get the coordinates of the vertex of its parabola, and then we need to get a few points on either side of the axis of symmetry.

There are three steps in graphing a quadratic equation/function:

Step 1: Get the coordinates of the vertex

In order to get the coordinates of the vertex of a parabola, either of the two ways can be followed:

1. If the equation/function is given in the standard form:

Let the vertex be (h, k).

For any quadratic equation in the standard form ax^2 + bx + c, value of 'h' can be obtained by

h = -b / (2a)

Obtain 'k' by putting x = h in the above equation. That is, calculate the value of the given quadratic expression at x = h. This is the value of 'k'. Hence

k = a(h)^2 + b(h) + c

Finally write the point as (h, k).

An example: For the quadratic equation

f(x) = x^2 + 6x + 5,

the value of 'h' is

h = -6 / (2*1) = -6/2 = -3

And the value of k is

k = (-3)^2 + 6(-3) + 5

k = 9 - 18 + 5

k = -4

Therefore coordinates of its vertex are (-3 , -4)

2. If the equation/function is in the vertex form

For any quadratic equation/function given in the vertex form,

a(x - h)^2 + k = 0,

the coordinates of the vertex of its graph are given by

Vertex = (h, k)

For example, for the equation

(x + 3)^2 - 4,

the coordinates of its vertex are:

Vertex = (-3, -4)

Learn more about

• How to convert a quadratic equation/function to the standard form?
• How to convert a quadratic equation/function to the vertex form?

Step 2: Get the coordinates of a few points

Now after getting the coordinates of the vertex, we need to get a few more points that lie on the parabola.

For this, take at least one value of 'x' bigger than that of the x-coordinate of the vertex, and at least one value of 'x' smaller than that of the x-coordinate of the vertex. Plug in both these values in the quadratic equation/function in order to obtain the corresponding values. These corresponding pairs of x and y values form the points lying on the parabola written in the form (x, y).

Example: Find the points of the quadratic function

f(x) = x^2 + 6x + 5

We know that the vertex of this function is (-3, -4) as we calculated that above.

Taking a value of x lesser than -3, we get

For x = -4, y = (-4)^2 + 6(-4) + 5 = -3

Therefore a point is (-4, -3)

Taking a value of x greater than -3, we get

For x = 0, y = (0)^2 + 6(0) + 5 = 5
Therefore the point is (0, 5)

The last step:

Graph the vertex and the two points obtained above, and join them with the help of a free hand curve. So we get the following graph for the function

f(x) = x^2 + 6x + 5