# Ways to Draw Graph of the Quadratic Function

Graph of the function y=ax^2+bx+c, where a!=0 is parabola. To draw it three methods are used, that will be illustrated in following example.

Example . Draw graph of the function y=-0.5x^2-x+4.

First way: finding of coordinates (x_0,y_0) of vertex of parabola using following formulas: x_0=-b/(2a); y_0=(4ac-b^2)/(4a).

Here a=-0.5,b=-1,c=4. Therefore, x_0=-(-1)/(2*(-0.5))=-1, y_0=(4*(-0.5)*4-1)/(4*(-0.5))=9/2. So, (-1;9/2) is vertex of parabola. To graph, let's find a couple more points on parabola: for example, (0;4),(1;5/2),(2;0). Draw vertex of parabola, obtained points and points, that are symmetric to them about axis of parabola. Now, connect all points - we obtained graph of the parabola (see figure).

Second way: drawing parabola using points whose y-coordinate equals free member of quadratic function.

Let's find points of graph, whose y-coordinate equals free member of quadratic cfunction. For this we need to solve equation -0.5x^2-x+4=4, i.e. 0.5x^2+x=0. This equations has two roots: x_1=0,x_2=-2.

So, we found two points of graph: A(0;4) and B(-2;4).

Since points A and B lie on graph and have same y-coordinate then they are symmetric about axis of symmetry of parabola, thus axis passes through middle of segment AB perpendicular to it. x-coordinate of point A equals 0, x-coordinate of point B equals -2, therefore, x-coordinate of middle is (0+(-2))/2=-1, thus x=-1 is axis of symmetry of parabola. Now, y(-1)=-0.5*(-1)^2-(-1)+4=9/2. Therefore, point C(-1,9/2) is vertex of parabola (i.e. the only point of parabola that lies on its axis of symmetry). Finally, we draw points A,B,C; connect them with line and obtain graph of the parabola (see figure).

Third way: drawing parabola using roots of quadratic function.

Equation -0.5x^2-x+4=0 has two roots: x_1=-4 and x_2=2. Therefore, we know two points of parabola: D(-4;0) and E(2;0). Since axis of symmetry is perpendicular to segment DE and passes through middle of it, then x-coordinate of middle is (-4+2)/2=-1. So, x=-1 is line of symmetry. Therefore, vertex of parabola is point C(-1;9/2). Using three points D,E and C we draw graph (see figure).