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translations

# translations - Translations of graphs Shifting a graph...

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Translations of graphs Shifting a graph vertically up or down . Example 1 : The graph of the equation = y x 2 is a parabola opening upwards with the y axis as an axis of symmetry and having its lowest point at the origin. Given a table of values for the equation = y x 2 , it is straightforward to construct a table of values for the equation = y + x 2 1. We simply need to add 1 to each of the y values. ( See the first three rows in the table below.) Similarly, we can obtain values for the equation = y + x 2 2 by adding 2 to each of the y values given for = y x 2 . y values for the equation = y - x 2 2 can be obtained by subracting 2 from each of the y values otained for = y x 2 . The following table gives y values corresponding to values of x between - 3 and 3 for the equations = y x 2 , = y + x 2 1, = y + x 2 2, = y - x 2 2 and = y - x 2 4. x | - 3 - 5 2 - 2 - 3 2 - 1 - 1 2 0 1 2 1 3 2 2 5 2 3 = y x 2 | 9 25 4 4 9 4 1 1 4 0 1 4 1 9 4 4 25 4 9 = y + x 2 1 | 10 29 4 5 13 4 2 5 4 1 5 4 2 13 4 5 29 4 10 = y + x 2 2 | 11 33 4 6 17 4 3 9 4 2 9 4 3 17 4 6 33 4 11 = y - x 2 2 | 7 17 4 2 1 4 - 1 - 7 4 - 2 - 7 4 - 1 1 4 2 17 4 7 = y - x 2 4 | 5 9 4 0 - 7 4 - 3 - 15 4 - 4 - 15 4 - 3 - 7 4 0 9 4 5

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Note that the x intercepts of the curve (graph of) = y - x 2 4 are = x + 2. This example suggests that, for a 0, the graph of = y + x 2 a can be obtained by shifting (or translating) the graph of = y x 2 through a distance of a units vertically through a distance of a units upwards if a is positive and downwards if a is negative. _____________________________ More generally, for a 0, the graph of = y + ( ) f x a is obtained by shifting or translating the graph of = y ( ) f x : vertically upwards through a distance of a units if a > 0. vertically downwards through a ditance of a units if a < 0.
Example 2 : This example shows some vertical translations of the graph of the equation = y x . x | - 3 - 2 - 1 0 1 2 3 = y x | 3 2 1 0 1 2 3 = y + x 1 | 4 3 2 1 2 3 4 = y + x 2 | 5 4 3 2 3 4 5 = y - x 2 | 1 0 - 1 - 2 - 1 0 1 Example 3 : The graph of = y + x 1 ( blue curve) is obtained by translating the graph of = y x upwards through a distance of 1 unit. The graph of = y - x 2 ( orange curve) is obtained by translating the graph of = y x downwards through a distance of 2 units.

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translations - Translations of graphs Shifting a graph...

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