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05. graph of quadratic expression/ polynomial

IN THIS CHAPTER:



What type of graph does a quadratic equation have?

To answer this question, first of all let me show you the what we mean by graph of any function , starting from a simple linear function( linear function means a function in degree 1 i.e. the highest power is 1)

Let me define a linear function , like

y = 2x +1

What we mean by the graph is for each value of x, we get a particular value of y and we put the points (x,y) on the graph paper, and join the points.


In the above example, we get the graph of the function very easily as it is a linear function.
 Putting x=0, we get y=1.
Putting x=1, we get y=3.

 Now i join the points (0,1) and (1,3) to get a line and extend it. You can see that all other points satisfying the equation lies on the line.

 

 

So now that you know what a graph is, can you guess what the graph of a quadratic equation must look like?

For example, what is the graph of the function
f(x)= x2 +2x +3   0r y= x2 +2x +3

So if you don't know what is the shape of the graph you will try to find it out by plotting some points (x,y) which satisfy the above equation and join them.

 

WRONG GRAPH
The points which satisfy the equation have been joined by straight lines. This is wrong.
They have to be joined by appropriate curved lines.

The correct graph is-


 

CORRECT GRAPH

We observe that the graph of a quadratic equation is a parabola.

So now your obvious question is- why such a difference between the graph of linear equation and quadratic equation. Why aren't the points joined by a straight line?

Note that i still stick to the same definition of graph. If you plot a large number of points and then join them you will notice that it is approaching the form of the correct graph. But why the difference???

This is because of the slope of the graph. The slope of a linear function is constant, whereas the slope for the quadratic function is not constant, the slope itself changes as a linear function of x.

Calculating slope :

For linear function y= ax +b
Slope= dy/dx= a =constant

For quadratic function y = ax2 + bx + c
Slope = dy/dx = 2ax + b , which is a linear function of x, and thus is continuously changing with the change in x.

Hence such a nature of the graph.

WHEN a<0:

When a in the expression ax2 + bx + c is less than 0, then the graph of the function is upside down.

For example, the  graph of the function y = -2x2 + 4x + 5   is

 


VERTEX:

Now that you know that the graph of the quadratic function is a parabola,  you can observe that there is a lowest point after which the parabola changes direction.
In case the parabola is upside down( when a <0, the parabola is upside down), there is a highest point.

How to find the vertex ?

You can observe that the slope at the vertex is 0.
dy/dx= 0

or, 2ax +b =0

or, x = -b/2a

Now d2y/dx2 = 2a

Clearly, when a is positive,  d2y/dx2  is positive.

So by the principle of maxima and minima, there is a minima, i.e. a lowest point, as i told you. Just proved this using calculus.

When a <0, d2y/dx2  <0

So, there is a maxima, or a highest point, as you can see the graph in the above example.

So now we know the x- coordinate of the vertex.
Putting x= -b/2a in  y = ax2 + bx + c, we get the y coordinate of the vertex.

y = a(-b/2a)2 + b(-b/2a) + c

or, y= (-b2 +4ac)/4a

or, y= -D/4a    , where D = b2 -4ac

Thus, the coordinate of the vertex is (-b/2a, -D/4a)

IMPORTANT POINTS TO KEEP IN MIND WHEN PLOTTING THE GRAPH:

  • Graph of quadratic function is a parabola.
  •   See the sign of a. If a is positive, parabola will be facing upwards.
            If a is negative, parabola will be upside down.
  •   Find the vertex. It can be found out as explained above.
  •   The parabola will be symmetric about the vertex.
  •   Find the roots of the quadratic equation. The roots are the points where the graph intersects the X- axis, or points at which y=0.
  •   If the roots are real and distinct, we can get in total three points, the vertex and the two roots. So we can easily join them to form the parabola, as shown below:


  •   If the roots are real and equal, you can easily find out that the vertex and the two points of intersection with X- axis all three comes out to be same point.
For example, graph of function y =x2 +2x +1 is.
All the three points comes out to be (-1,0)



  •   When the roots are complex, we cannot find the real roots.
 So , in this case,when a>0, the vertex will be above X-axis. When a<0, vertex will  be  below   X-axis.

 See the complete analysis

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