Lastly, without fancy mounting, let's take a look at a single parabola, in black, at right. .A parabola is a reversing curve with the equation:s

X is customarily horizontal, Y is vertical -- so they are on this page. What the graph gives us is a way to look at all the outcomes of the equation, its "function", over a given range of inputs -- here -3 to 2..

Here, 'a' and 'b' are both 1, 'c' is 0. The function becomes.Y = x2 + x . The curve starts high, dips low and then returns to high, illustrating the tug-o-war between positive and negative values that typically takes place near the origin, (0,0). .In math, a negative times a negative is a positive, meaning that all "real" values of x squared, (or x times x), will be positive numbers. .Since the square is the most powerful part of this function, ultimately it will carry the curve north on both sides. But near 0, squares become very weak -- (squares of fractions are smaller fractions) -- and this allows the normally overpowered first degree component: 'b' times x, (which is negative when x is), to come out and play. .Without changing the shape of the function, (and this is unusual), it has pulled it left and downward -- slightly into negative territory -- along a line shown in cyan. .. This line is the function ..... y = one half x. .It is what I call a finding function, since it has "found" the relative minimum of the parabola. .Math understands itself well.
Now let's take a look at the blue line. .This is the 0 line, so called not because it lies at 0, but because here 'a' equals 0. That makes its function: Y = b times x. .It's kind of strange to think that there would be a line in the center of a stream of parbolas. But that's part of the interconnectedness of math. .On its own terms, it is about form and flow. .The black parabola will come down and touch its zero line at one point. .It will not cross. .As we increase 'a' it rises from and creeps along it, towards the origin. .As we reduce 'a', the function slides down the zero line, and opens so that it seems to be sleeping on it. .If 'a' becomes negative, the parabola flips across the zero line and behaves as an exact mirror image of its former self, growing down instead of up, moving left when it would have moved right, and always along that line. .This motion has everything to do with modifying 'a', the squared coefficient, while leaving 'b', the linear coefficient alone. .If we do change 'b' the zero line changes. .If we change only 'b', it is not a line.