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ACT Math : How to find the range of the cosine

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Example Questions

Example Question #151 : Trigonometry

Which of the following functions has a range of $\displaystyle [-12, -8]$?

Possible Answers:

None of these functions has the specified range.

$\displaystyle f(x) = cos(2x) -5$

$\displaystyle f(x) = 2cos(3x-3) -10$

$\displaystyle f(x) = 2cos(-x -5) +2$

$\displaystyle f(x) = 2cos(x+1) +10$

Correct answer:

$\displaystyle f(x) = 2cos(3x-3) -10$

Explanation:

The range of the function represents the spread of possible answers you can get for f(x) $\displaystyle f(x)$, given all values of $\displaystyle x$. In this case, the ordinary range for a cosine function is [-1, 1] $\displaystyle [-1, 1]$, since the largest value that cosine can solve to is $\displaystyle 1$ (for a cosine of $0/2\pi/360^{\circ}$ $\displaystyle 0/2\pi/360^{\circ}$ or a multiple of one of those values), and the smallest value cosine can solve to is $\displaystyle -1$ (for a cosine of $\pi/180^{\circ}$ $\displaystyle \pi/180^{\circ}$ 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 $f (x) = \alpha cos(bx-c) +d$ $\displaystyle f (x) = \alpha cos(bx-c) +d$, where $\displaystyle d$ represents the vertical shift, $d = \frac{range_{max}+range_{min}}{2}$ $\displaystyle d = \frac{range_{max}+range_{min}}{2}$.

In this case, since our range is $\displaystyle [-12, -8]$, we expect our $\displaystyle d$ to be $\frac{-8-12}{2} = -10$ $\displaystyle \frac{-8-12}{2} = -10$.

Of the answer choices, only $\displaystyle f(x) = 2cos(3x-3) -10$ has d=-10 $\displaystyle d=-10$, so we know this is our correct choice.

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

Which of the following represents a cosine function with a range of $\displaystyle 39$ to $\displaystyle 83$?

Possible Answers:

$\displaystyle f(x)=12cos(5x)+83$

$\displaystyle f(x)=22cos(5x)+61$

$\displaystyle f(x)=44cos(2x)+39$

$\displaystyle f(x)=22cos(61x)$

$\displaystyle f(x)=44cos(39x)$

Correct answer:

$\displaystyle f(x)=22cos(5x)+61$

Explanation:

The range of a cosine wave is altered by the coefficient placed in front of the base equation. So, if you have $\displaystyle f(x)=12cos(x)$, this means that the highest point on the wave will be at $\displaystyle 12$ and the lowest at $\displaystyle -12$; however, if you then begin to shift the equation vertically by adding values, as in, $\displaystyle f(x)=12cos(x)+1$, then you need to account for said shift. This would make the minimum value to be $\displaystyle -11$ and the maximum value to be $\displaystyle 13$.

For our question, the range of values covers $\displaystyle 83-(39)=44$. This range is accomplished by having either $\displaystyle 22$ or $\displaystyle -22$ as your coefficient. ($\displaystyle -22$ merely flips the equation over the $\displaystyle x$-axis. The range "spread" remains the same.) We need to make the upper value to be $\displaystyle 83$ instead of $\displaystyle 22$. To do this, you will need to add 83-22 $\displaystyle 83-22$, or $\displaystyle 61$, to $\displaystyle 22$. This requires an upward shift of $\displaystyle 61$. An example of performing a shift like this is:

$\displaystyle f(x)=cos(x)+61$

Among the possible answers, the one that works is:

$\displaystyle f(x)=22cos(5x)+61$

The $\displaystyle 5x$ parameter does not matter, as it only alters the frequency of the function.

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ACT Math : How to find the range of the cosine

Study concepts, example questions & explanations for ACT Math

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Example Question #151 : Trigonometry

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

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