ACT Math : How to find the range of the cosine

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #11 : How To Find The Range Of The Cosine

Which of the following functions has a range of ?

Possible Answers:

None of these functions has the specified range.

Correct answer:

Explanation:

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 ?

Possible Answers:

Correct answer:

Explanation:

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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