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Example Question #11 : How To Find The Range Of The Cosine
Which of the following functions has a range of ?
None of these functions has the specified range.
The range of the function represents the spread of possible answers you can get for , given all values of . In this case, the ordinary range for a cosine function is , since the largest value that cosine can solve to is (for a cosine of or a multiple of one of those values), and the smallest value cosine can solve to is (for a cosine of or a multiple of one of those values).
One fast way to match a range to a function is to look for the function which has a vertical shift equal to the mean of the range values. In other words, for the standard trigonometric function , where represents the vertical shift, .
In this case, since our range is , we expect our to be .
Of the answer choices, only has , so we know this is our correct choice.
Example Question #11 : How To Find The Range Of The Cosine
Which of the following represents a cosine function with a range of to ?
The range of a cosine wave is altered by the coefficient placed in front of the base equation. So, if you have , this means that the highest point on the wave will be at and the lowest at ; however, if you then begin to shift the equation vertically by adding values, as in, , then you need to account for said shift. This would make the minimum value to be and the maximum value to be .
For our question, the range of values covers . This range is accomplished by having either or as your coefficient. ( merely flips the equation over the -axis. The range "spread" remains the same.) We need to make the upper value to be instead of . To do this, you will need to add , or , to . This requires an upward shift of . An example of performing a shift like this is:
Among the possible answers, the one that works is:
The parameter does not matter, as it only alters the frequency of the function.
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