Unformatted text preview: e formulas of
f (x) and m(x), we have m(x) = f (x + 3) − 2. We have 3 being added to an input, indicating
a horizontal shift, and 2 being subtracted from an output, indicating a vertical shift. We
leave it to the reader to verify that, in this particular case, the order in which we perform
these transformations is immaterial; we will arrive at the same graph regardless as to which
transformation we apply ﬁrst.2 We follow the convention ‘inputs ﬁrst’,3 and to that end we
ﬁrst tackle the horizontal shift. Letting m1 (x) = f (x + 3) denote this intermediate step,
Theorem 1.3 tells us that the graph of y = m1 (x) is the graph of f shifted to the left 3 units.
Hence, we subtract 3 from each of the xcoordinates of the points on the graph of f .
y
(1, 2) y
(4, 2)
2 (−2, 1) (1, 1)
(−3, 0) (0, 0)
−3 −2 −1
−1 2
1 1
1 2 3 4 −3 −2 −1
−1 x 1 2 3 4 x −2 −2 shift left 3 units y = f (x) = √ −− − − − −→
−−−−−−
x subtract 3 from each xcoordinate y = m1 (x) = f (x + 3) = √
x+3 2
We...
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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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