Stitz-Zeager_College_Algebra_e-book

# What is the marbles maximum height above the ground

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Unformatted text preview: e multiply the output by a, we multiply the y -coordinates on the graph of g by a, so the point (0, 0) remains (0, 0) and remains the vertex. Next, we deﬁne g2 (x) = g1 (x − h) = a(x − h)2 . This induces a horizontal shift right or left h units3 moves the vertex, in either case, to (h, 0). Finally, f (x) = g2 (x) + k = a(x − h)2 + k which eﬀects a vertical shift up or down k units4 resulting in the vertex moving from (h, 0) to (h, k ). To verify the vertex formula for a quadratic function in general form, we complete the square to convert the general form into the standard form.5 f (x) = ax2 + bx + c b = a x2 + x + c a b b2 = a x2 + x + 2 + c − a a 4a 2 b 4ac − b2 = a x+ + 2a 4a b2 4a2 complete the square factor; get a common denominator Comparing this last expression with the standard form, we identify (x − h) as x + 2ba so that b 2 2 h = − . Instead of memorizing the value k = 4ac−b , we see that f − 2ba = 4ac−b . As such, 4a 4a 2a 2 Just a scaling if a > 0. If a < 0, there is a reﬂection involved. Right if h > 0, left if h &l...
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