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Unformatted text preview: = h, we get y = k , so (h, k ) is on the graph. If x = h, then x − h = 0 and so (x − h)2 is a positive
number. If a > 0, then a(x − h)2 is positive, and so y = a(x − h)2 + k is always a number larger
than k . That means that when a > 0, (h, k ) is the lowest point on the graph and thus the parabola
must open upwards, making (h, k ) the vertex. A similar argument shows that if a < 0, (h, k ) is the
highest point on the graph, so the parabola opens downwards, and (h, k ) is also the vertex in this
case. Alternatively, we can apply the machinery in Section 1.8. The vertex of the parabola y = x2
is easily seen to be the origin, (0, 0). We leave it to the reader to convince oneself that if we apply
any of the transformations in Section 1.8 (shifts, reﬂections, and/or scalings) to y = x2 , the vertex
of the resulting parabola will always be the point the graph corresponding to (0, 0). To obtain the
formula f (x) = a(x − h)2 + k , we start with g (x) = x2 and ﬁrst deﬁne g1 (x) = ag (x) = ax2 . This
is results in a vertical scaling and/or reﬂection.2 Since w...
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