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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)
• 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