Quadratic Equations and Graphs

Definition of quadratic equation : An equation in which one or more of the terms is squared but raised to no higher power, having the general form ax2 + bx + c = 0, where a, b, and c are constants.

Quadratic Equations and Graphs:

The general form of a quadratic equation is

ax2 + bx + c =0

Where a, b and c are real numbers and a`!=` 0

It is called so because it has a second degree term.

The values of the variable involves in a quadratic equation are called its solution.

The following method are usually used to solve a quadratic equation.

completing square method
using the quadratic formula
Factorization method


In general, the shape of the graph of a quadratic equations a parabola.

The steps we follow to sketch the graph of the quadratic equation are:

1. Check if a > 0 or a < 0

If a > 0, then the parabola is u-shaped ie it opens upwards

If a < 0, then the parabola is  n-shaped ie it opens downwards

2. Find Vertex

The x-coordinate of the vertex is –b/2a

Substitute the value of x in the function to have the y coordinate.

Thus we have the (x, y) coordinates of the vertex.

3. Find y-intercept

The coordinates of the y-intercept can be found by substituting x = 0.

4. Find x-intercept

The coordinates of the x-intercept can be found by substituting y = 0 and solving the quadratic equation

5. Finding other coordinates

Substitute any possible value for x to get the corresponding value of y or vice versa

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Quadratic Equations and Graphs-example

Graph of the function y =  2x2 - 8x + 6

Here a= 2 b=-8 c=6

1. Since a > 0, then the parabola is u-shaped i.e. it opens upwards

2. The x-coordinate of the vertex is –b/2a = 8/(2*2) = 2

Substituting the value of x in the function y = 2x2 - 8x + 6 to have the y coordinate.

We get y = 2*22 - 8*2 + 6 = -2

So the vertex is (2,-2)

3. The coordinates of the y-intercept can be found by substituting x = 0.

We get y = 2*02 - 8*0+ 6 =6

So the coordinates of the y-intercept is (0,6)

4. The coordinates of the x-intercept can be found by substituting y = 0 and solving   the quadratic equation

We get

2x2 - 8x + 6=0

2(x2 - 4x + 3)=0

2(x - 1)(x - 3) = 0

So x = 1, or x = 3.

So the coordinates of the x-intercept is (1,0)  and (3,0)

5. Finding some other coordinates by substituting any possible value for x

We can now plot the graph.

Quadratic Equations and Graphs-graphs