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Unformatted text preview: t to the reader to verify that the domain of j is the same as f ,
but the range of j is [−1, 3]. What we have discussed is generalized in the following theorem.
Theorem 1.2. Vertical Shifts. Suppose f is a function and k is a positive number.
• To graph y = f (x) + k , shift the graph of y = f (x) up k units by adding k to the
y -coordinates of the points on the graph of f .
• To graph y = f (x) − k , shift the graph of y = f (x) down k units by subtracting k from
the y -coordinates of the points on the graph of f .
The key to understanding Theorem 1.2 and, indeed, all of the theorems in this section comes from
an understanding of the Fundamental Graphing Principle for Functions. If (a, b) is on the graph
of f , then f (a) = b. Substituting x = a into the equation y = f (x) + k gives y = f (a) + k = b + k .
Hence, (a, b + k ) is on the graph of y = f (x) + k , and we have the result. In the language of ‘inputs’
and ‘outputs’, Theorem 1.2 can be paraphrased as “Adding to, or subtracting...
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