Quadratic equations and parabolas
As you know any equation can be represented by a graph. The graphs of different equations have different shapes.
But there is an interesting fact about the graph of a quadratic equation.
The graph of a quadratic function is always a parabola.
Let us find out how that has been concluded.
Establishing the relation between quadratic equations and parabolas
The general form of a quadratic equation is, y = ax2 + bx + c
Let us do some algebraic operations with this equation.
y = ax2 + bx + c
or, y – c = a[x2 + (b/a)x]
= a[x2 + (b/a)x + (b/2a)2 – (b/2a)2]
or, y = a[x2 + (b/a)x + (b/2a)2]– (b2/4a) + c
= a[x + (b/2a)]2 – [(b2/4a) – c ]
= a[x + (b/2a)]2 – [(b2 – 4ac)/4a]
This is in the form of the equation, y = a(x – h )2 + k, which is nothing but the standard form of the equation of a parabola.
Thus, the graph of a quadratic function is a parabola.
Conclusions from the relation between quadratic equations and parabolas
y = a[x + (b/2a)]2 – [(b2 – 4ac)/4a], is derived from the general form of a quadratic equation.
y = a(x – h )2 + k, is the standard form of the equation of a parabola where (h, k) are the coordinates of the vertex.
By comparing the two identities, the vertex of the parabola is [(-b/2a), (b2 – 4ac)/4a]
In the equation, y = a(x – h )2 + k, if a is positive, the value of y increases with increase in x. In other words, the parabola is concave up. By the same argument, the parabola is concave down if a is negative.
A parabola can have only one vertex and that is the lowest point or the highest point depending upon whether it is concave up or concave down. Therefore a quadratic equation can have only a minimum value or a maximum value depending upon the sign of coefficient of a.
The maximum or minimum value of a parabola is at the vertex point and hence the maximum or minimum value of a quadratic equation is when x = (-b/2a) and the value is, (b2 – 4ac)/4a.
I like to share this Solving Quadratic Equations by Graphing with you all through my blog.
But there is an interesting fact about the graph of a quadratic equation.
The graph of a quadratic function is always a parabola.
Let us find out how that has been concluded.
Establishing the relation between quadratic equations and parabolas
The general form of a quadratic equation is, y = ax2 + bx + c
Let us do some algebraic operations with this equation.
y = ax2 + bx + c
or, y – c = a[x2 + (b/a)x]
= a[x2 + (b/a)x + (b/2a)2 – (b/2a)2]
or, y = a[x2 + (b/a)x + (b/2a)2]– (b2/4a) + c
= a[x + (b/2a)]2 – [(b2/4a) – c ]
= a[x + (b/2a)]2 – [(b2 – 4ac)/4a]
This is in the form of the equation, y = a(x – h )2 + k, which is nothing but the standard form of the equation of a parabola.
Thus, the graph of a quadratic function is a parabola.
Conclusions from the relation between quadratic equations and parabolas
y = a[x + (b/2a)]2 – [(b2 – 4ac)/4a], is derived from the general form of a quadratic equation.
y = a(x – h )2 + k, is the standard form of the equation of a parabola where (h, k) are the coordinates of the vertex.
By comparing the two identities, the vertex of the parabola is [(-b/2a), (b2 – 4ac)/4a]
In the equation, y = a(x – h )2 + k, if a is positive, the value of y increases with increase in x. In other words, the parabola is concave up. By the same argument, the parabola is concave down if a is negative.
A parabola can have only one vertex and that is the lowest point or the highest point depending upon whether it is concave up or concave down. Therefore a quadratic equation can have only a minimum value or a maximum value depending upon the sign of coefficient of a.
The maximum or minimum value of a parabola is at the vertex point and hence the maximum or minimum value of a quadratic equation is when x = (-b/2a) and the value is, (b2 – 4ac)/4a.
I like to share this Solving Quadratic Equations by Graphing with you all through my blog.