How to find the range of the cosine - ACT Math

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.

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 subtract , or , from . This requires an downward 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.

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 . 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 range of a sine or cosine function spans from the negative amplitude to the positive amplitutde. The amplutide is given by in the equation . Thus the range for our function is

For the sine and cosine funcitons, the range is equal to the negative amplitude to the positive amplitude.

The amplitude is found by taking from the general equation:

.

We see in our equation that

(when no coefficient is written, it is a 1).

Thus we get that the amplitude is

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).

However, in this case our final answer is increased by after the cosine is applied to . This results in a final range of to (or, in other words, to , plus ).

So, our final range is .

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).

However, in this case our final answer is multiplied by -3 after the cosine is applied to . This results in a final range of to (or, in other words, to , multiplied by ).

So, our final range is .

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).

However, in this case our final answer is first multiplied by , then decreased by after the cosine is applied to . Multiplying the initial range by results in a new range of . Next, subtracting from this range gives us a new range of .

Note that the does not change our range. This is because, irrespective of other multipliers, a cosine operation can only return values between and . To think of this a different way, will give us the same returns as , only we will move around the unit circle five times as much before finding our answer.

Thus, our final range is .

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.

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.

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.