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**Unformatted text preview: **and to make good use of this fact, we set Bx + H = a
and solve. We ﬁrst subtract the H (causing the horizontal shift) and then divide by B . If B 100 Relations and Functions 1
is a positive number, this induces only a horizontal scaling by a factor of B . If B < 0, then
we have a factor of −1 in play, and dividing by it induces a reﬂection about the y -axis. So we
have x = a−H as the input to g which corresponds to the input x = a to f . We now evaluate
B
g a−H = Af B · a−H + H + K = Af (a) + K = Ab + K . We notice that the output from f is
B
B
ﬁrst multiplied by A. As with the constant B , if A > 0, this induces only a vertical scaling. If
A < 0, then the −1 induces a reﬂection across the x-axis. Finally, we add K to the result, which is
our vertical shift. A less precise, but more intuitive way to paraphrase Theorem 1.7 is to think of
the quantity Bx + H is the ‘inside’ of the function f . What’s happening inside f aﬀects the inputs
or x-coordinate...

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