We locate the points listed above on a graph paper and draw the graph.

Is the graph of y = x² – 3x – 4 a straight line? No, it is like a shaped curve. It is intersecting the X-axis at two points.

In fact, for any quadratic polynomial ax²+ bx + c, a not equal to 0, the graph of the corresponding equation y = ax² + bx + c (a not equal to 0) either opens upwards like ∪ or opens downwards like ∩ . This depends on whether a > 0 or a < 0. (The shape of these curves are called parabolas.)

From the table, we observe that -1 and 4 are zeroes of the quadratic polynomial. From the graph, we see that -1 and 4 are also X coordinates of points of intersection of the parabola with the X-axis. Zeroes of the quadratic polynomial x²– 3x – 4 are the x-coordinates of the points where the graph of y = x– 3x – 4 intersects the X-axis. For the polynomial P(x) = y = x²– 3x – 4; P(-1)=0, its graph is intersecting the X-axis at (-1, 0). Also P(4)=0 its graph is intersecting the X-axis at (4, 0). In general for polynomial P(x) if P(a)=0, its graph intersects X-axis at (a, 0).

This is true for any quadratic polynomial, i.e., the zeroes of a quadratic polynomial ax²+ bx + c, (a not equal to 0) are precisely the x-coordinates of the points where the parabola representing y = ax²+ bx + c (a not equal to 0) intersects the X-axis.