**The parabola in coordinates**

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In general, the parabola is the set of points (***x, y*) which satisfy a quadratic equation: *y* = *ax*^{2}* + bx + c* (*a* 0). The simplest example is where *a* = 1 and *b* = *c* = 0. This gives the equation *y = x*^{2}, the graph of which is illustrated here.

** ****Investigate**** On squared or graph paper, reproduce the illustrated graph by plotting a number of points (***x, y*) where *y = x*^{2}, and joining them up with a smooth curve. Alternatively, produce the curve with the help of a graphics calculator.

**Now do the same with the curve ***y = x*^{2} +* c* for *c* = ... 2, 1, –1, –2, ... . What effect does the value of *c* have on the graph? Where does the vertex lie?

**Finally, look at the graph with equation ***y = x*^{2} + 2*x*. Where does the vertex lie? How would you describe the graph with equation *y = x*^{2} + 4*x*? *y = x*^{2} + *bx*?