Vertical and Horizontal Translations

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Vertical and Horizontal Translations
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(2, 6)
(1, 3)
(2, 4)
(1, 1)
(0, 2)
(0, 0)
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(2, 4)
(0, 0)
(1, 1)
(2, 1)
(1, -2)
(0, -3)
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Vertical Shifts

• cgt0
• The graph of g(x) f(x) c is the same as the
  graph of f(x) but shifted UP by c.
• The graph of g(x) f(x) - c is the same as the
  graph of f(x) but shifted DOWN by c.

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• Use the graph of
• to obtain the graphs of

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HORIZONTAL TRANSLATIONS
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Horizontal Shifts

• The graph of is the
  graph of shifted h units to the left.
• The graph of is the
  graph of
• shifted h units to the right.

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The graph below is obtained by translating the
graph of . Find a formula for the
function graphed.
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The graph below is obtained by translating the
graph of . Find a formula for the
function graphed.
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The graph below is obtained by translating the
graph of . Find a formula for the
function graphed.
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Reflections about the x-Axis and the y-Axis

• The graph of y - f(x) is obtained by
  reflecting the graph of f(x) about the x-axis.
• The graph of y f(-x) is obtained by reflecting
  the graph of f(x) about the y-axis.

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Vertical Compression

• When the right side of a function yf(x) is
  multiplied by a positive number a, the graph of
  the new function yaf(x) is obtained by
  multiplying each y-coordinate on the graph of
  yf(x) by a. The new graph is a vertically
  compressed (if 0 lt a lt 1) or a vertically
  stretched (if a gt 1) version of the graph of
  yf(x).

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Horizontal Compression
If the argument x of a function yf(x) is
multiplied by a positive number a, the graph of
the new function yf(ax) is obtained by
multiplying each x-coordinate of yf(x) by
1/a. A horizontal compression results if agt1,
and a horizontal stretch occurs if 0 lt a lt 1.
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yx²-1 y(2x)²-1 y((1/3)x)²-1 (1,0) ((1/2),0)
(3,0) (-1,0) (((-1)/2),0) (-3,0) (0,-1) (0,-1)
(0,-1)
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