Quadratic function y=f(x)=ax^{2}+bx+c always represent a parabola. If the co-efficient of x^{2} is positive , the parabola will be concave up and negative , the parabola will be concave down. For manually sketch the quadratic function graph, we will find two points for which y=f(x)=0 that means the parabola passes through this two points and the concavity of parabola depend on the co-efficient of x^{2} . As for example y=f(x)=x^{2}-5x+4=(x-1)(x-4). If x=1 or x=4, then y will be zero.That means the parabola passes through (1, 0) and (4, 0). Since the co-efficient of x^{2} is 1 which is positive, so the parabola will be concave up. Geogebra Free graphing calculator for math graph drawing website www.graph2d.com can draw any qudratic function. You can type x**2-5*x+4 in the input box and then press drawing button to get the graph. You can also try to draw with the function f(x)=x^{2}, f(x)=-x^{2}, f(x)=x^{2}+5x, f(x)=x^{2}-3x, f(x)=x^{2}+2x+5 etc.

If we find only one point for which y=0, then the parabola touches x-axis on that point. As for example, y=f(x)=2(x-2) ^{2}, there is only one point (2, 0) for which y=0, so the parabola touches x-axis at (2, 0) which is critical point. Since the co-efficient of x^{2} is 2 which is positive, so the parabola will be concave up. Free graphing calculator for math drawing website www.graph2d.com can draw any qudratic function. You can type 2*(x-2)**2 in the input box and then press drawing button to get the graph.

If we can not find any point for which y=0, then we will try to get a term of whole square. As for example, y=f(x)=x^{2}-4x+7=(x-2) ^{2}+3. When x=2, then the (x-2) ^{2}=0 and hence y=3. So the point (2, 3) is a critical point. Since the co-efficient of x^{2} is 1 which is positive, so the parabola will be concave up. Free graphing calculator for math drawing website www.graph2d.com can draw any qudratic function. You can type x**2-4*x+7 in the input box and then press drawing button to get the graph.
For learn about Implicit function of Mathematics and graphing, visit: Graphing Implicit function of Math.

A polynomial function of degree two is called quadratic function. The graph of quadratic function is always represent a parabola which can be drawn by quadratic function graph maker,browsing link: Graphing calculator of Math