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Theorem 2
The next theorem shows that the graph of y = ax2 + bx + c is exactly like the graph of y = ax2, except that its vertex is at the point (x0, y0) where the curve has slope zero. The focus and directrix are still at a distance of Ľa above and below the vertex. THEOREM 2 The graph of the equation y = ax2 + bx + c (where a ≠ 0) is a vertical parabola. Its vertex is at the point (x0, y0) where the curve has slope zero, the focus is F(x0, y0 + Ľa), and the directrix is y = y0 - Ľa. PROOF We first compute x0. The curve y = ax2 + bx + c has slope dy/dx = 2ax + b. The slope is zero when 2ax + b = 0, x = -b/2a. Thus x0 = -b/2a. Let p be the parabola with focus F(x0, y0 + Ľa) and directrix y = y0 - Ľa. Put X = x - x0 and Y = y - y0. In terms of X and Y, the focus and directrix are at (X, Y) = (0,Ľa), Y = -4a. By Theorem 1, p has the equation Y = aX2, or y - y0 = a(x - x0)2, y = ax2 - 2ax0x + (ax02 + y0). Substituting -b/2a for x0, we have y = ax2 + bx + (b2/4a + y0). This shows that the parabola p and the curve y = ax2 + bx + c differ at most by a constant. Moreover, the point (x0, y0) lies on the curve. (x0, y0) is also the vertex of the parabola p, where (X, Y) = (0, 0). Therefore the curve and the parabola are the same.
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Last Update: 2006-11-05