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Graphing Quadratic Equations – Math Tricks

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.

Where:

  • 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.
                                                                    2

        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-

f(x)= 2x2-12x+16

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