Graphing the Quadratic Equation

Graphing of Quadratic Equation

Introduction to graphs

Graphs, make the mathematics visual. Without graphs, mathematics is just numbers, symbols and alphabets. Which is very difficult to analyse intuitively. Graphs make it simple to visualise any geometrical problem. Graphs do not exist solely. It exists as a complementary of another branches of mathematics, like, algebra, trigonometry, geometry, statistics, probability. The relation of graphs and algebra is so strong. Any algebraic expression (or function) becomes very realistic and visual when we plot it’s graph.

When you’re reading this, it is expected that you are well aware about graphing of an equation. To revise, we know that in a co-ordinate plane, the horizontal line is known as \(x\)-axis and the vertical line is \(y\)-axis. From our knowledge of functions, we know that, in general, a function is written as \(y=f(x)\) where \(x\) is domain and \(y\) is range of function.

A graph basically works as:

1. We define the domain of a function.

2. We define the range of the function corresponding to every point of domain.

3. We plot every point as \((x,y)\) which is \((x_1,f(x_1))\), \((x_2,f(x_2))\), \((x_3,f(x_3))\),........ for all \(x\) in domain.

4. All the points makes a curve or line when plotted together. And hence we get a graph of an equation.

This is how a graph works. But, in reality, it is nearly impossible to do like this. The reason is, the domains, generally, exist in intervals. Like \((a,b)\), \([a,b)\), \((a,b]\) or \([a,b]\), where \(a\) and \(b\) are the end points of domain. Between two end points, there exist infinitely many points (values of \(x\)). And for those values of \(x\), it is obvious there will be many, many values of \(f(x)\). So, by plotting individual points like \((x_1,f(x_1))\), \((x_2,f(x_2))\), \((x_3,f(x_3))\), .... is not possible. Therefore we just find out some points which satisfy the equation, observe the pattern of graph, and by doing so we get a graph.

This will be more clear when we’ll draw the graph of a quadratic equation.

Brief overview of quadratic equations

Quadratic equation, in general, expressed as \(y=ax^2+bx+c\) where \(a, b, c\) are the real numbers. \(x\) is variable of input. \(y\) is variable of output. Means by changing the \(x\), we get different values of \(y\).

Example of quadratic equations are \(y=2x^2+5x+8\), \(y=-7x^2\), \(y=\frac{1}{2}x^2+9x=0\), etc.

Method to draw the graph of a quadratic equation

You can see that as we’ll change \(x\) we will get different value of \(y\).

Let us consider the most simple example to build our understanding is \(y=x^2\). Here input is \(x\) and output is \(y\). If \(x\) is \(0\), \(y\) will be \(0\). If \(x\) is \(1.5\), \(y\) will be \(2.25\). If \(x\) is \(6\), \(y\) is \(36\), and so on. Let’s write it systematically in a table for better presentation.

Here we took random values of \(x\), now let us take the sequential values for convenience in drawing and understanding the pattern of quadratic equations’ graph. We’ll consider integers from \(-4\) to \(4\) and will observe it's graph.

From above table, we can obtain the following points on co-ordinate plane: \((-4,16)\), \((-3,9)\), \((-2,4)\), \((-1,1)\), \((0,0)\), \((1,1)\), \((2,4)\), \((3,9)\), \((4,16)\).

Just observe the pattern in which the co-ordinate points are appearing in the following graph.

When we'll join the points through a curve (the basic knowledge of graph), we'll get the graph as: 

Now let us draw it properly for clear view:

This is how a graph of a quadratic equation looks like when plotted in co-ordinate plane. The U-shaped pattern what is forming here is called 'parabola'. Parabola is pattern of graph of second degree equations. It can either be U-shaped or ∩-shaped.

Some examples of graphs of quadratic equations: 

You can see different types of parabolas. These different shapes are just a result of different quadratic equation because of variation in coefficients \(a,b,c\).

As we’ve discussed above that the graph of a quadratic equation can either be U-shaped or ∩-shaped, but this notation is called “concave upward” or “concave downward” respectively.

Characteristics of a quadratic graph

Now we will learn to some characteristics of a quadratic graph.

See, if in the quadratic equation, \(a\), means the coefficient of \(x^2\) is a positive then quadratic equation will be concave upwards.

The minimum or maximum point of a quadratic equation is \((\dfrac{-b}{2a},\dfrac{-D}{4a})\). 

Example. Let’s take the equation: \(x^2-5x+6=0\). According to this equation, graph must be concave upwards and the minimum point must be \((\frac{-(-5)}{2·1},\frac{-[(-5)^2-4·1·6]}{4·1})\)=\((\frac{5}{2},\frac{-1}{4})\) which is \((2.5,-0.25)\). Now let’s see the graph. 

Now the question arises that when we will say \((\frac{-b}{2a},\frac{-D}{4a})\) minimum and when maximum?

If \(a\) is positive then \((\frac{-b}{2a},\frac{-D}{4a})\) will be the minimum point, and when \(a\) is negative then, maximum point will be \((\frac{-b}{2a},\frac{-D}{4a})\). An important thing is to considered is, by saying minimum and maximum point, we mean that the minimum and maximum value of \(y\) at any value of \(x\).