Quadratic equation is a polynomial equation of degree 2 and this equation can be written in standard form given by-
ax2 + bx + c = 0
Where a,b and c can have any value but a cannot be equal to 0.
For an example: 5x2 + 3x + 4 = 0
This makes this equation a quadratic.
If we will replace 0 with y, then this quadratic function is formed which further can be written as
y = ax2 + bx + c,
And the graph of this equation is parabola. But to draw the graph more accurately, we have to convert this equation to the following equation
y = a(x-h)2 +k
This equation is also called vertex form of quadratic equation.
- h = -b/2a,
- k = f(h)
In other words we can calculate h = -b/2a and k by calculating the whole equation for x=h.
Representation of quadratic equation
The points on graph intersects the x axis will be the solution to the equation, ax2 + bx + c = 0. If the polynomial ax2 +bx + c can be factorized to (x-p) (x-q) is also called zero product property that if (x-p) (x-q) = 0 either (x-p) = 0 or (x-q) = 0. Then p and q are the solutions to the equation ax2 + bx +c =0 and therefore the xx -intercepts.
Since the xx -coordinate of the vertex of a parabola is exactly the midpoint of the xx -intercepts, the xx -coordinate of the vertex will be P+Q.
We can use the xx -coordinate of the vertex to find the yy -coordinate.
Now we have the vertex and 22 other points on the parabola (namely, the xx -intercepts). One can use multiple points in order to graph the points. The resultant would be an ellipse.
Some rules to remember
- Huge value of a shrink the curve inwards
- Small value of a expand the curve outwards
- Negative value of a will flip it upside down
Let us look at the example –
Example 1: Draw the graph of the following quadratic equation given by-
Solution: Let us note down –
- a = 2,
- b = -12 and
- c = 16
As we know that the value of a is positive so therefore the graph is upwards and the value of a is 2 so therefore it is shirked.
Value of a can be calculated as –
h = -b/2a = -(-12)/4 = 3
and k = f(3) = 2(3)2 –(12)(3)+16= 18-36 + 16 = -2
Example 2: Graph the function y = x2-8x+12 using factorizing.
Solution : Let us note down –
- a = 1,
- b = -8 and
- c = 12
Now, as we know that the value of a is positive so therefore the graph is upwards and the value of a is 1 so therefore it is shirked.
You can calculate the value of a as –
h = -b/2a = -(-8)/1 = 8
and k = f(8) = 1(8)2 –(8)(8)+12 = 64 -64 +12 = 12