For each function, give the amplitude, period, vertical transl...

Question

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.

y = -½ cos 3x

Accepted Answer

Welcome back. I am so glad you're here. We are asked to find the amplitude period, vertical translation and phase shift for the following trigonometric function. Our given function is Y equals negative 1/4 cosine of six X. Our answer choices are answer choice. A amplitude equals 1/4 and period equals pi answer choice. B amplitude equals negative 1/4 and period equals pi divided by three answer choice. C amplitude equals 1/4 and period equals pi divided by three and answer choice D amplitude equals four and period equals pi divided by three. All right. So we take a look at our cosine function here and we recognize that the standard form for a cosine function would give us Y equals C plus A cosine open bracket. B multiplied by the quantity of X minus D close parentheses, close bracket. And so let's take a look at our given function and identify our A's and B's and C's and D's. So A here is what's being multiplied directly in front of our cosine function. A here is negative 1/4 our B term that's what's being multiplied directly to our X. And here B is equal to six, our C term is what's being added indirectly to that function. And here nothing is being added or subtracted from our cosine. So our C term is equal to zero and our D term that's what's being added or subtracted directly from our X inside of the cosine function. And here nothing is being added or subtracted directly from that X. So our D value is also zero. Well, that's great. But what does it all mean? All right. So for our A, a has to do with the amplitude, the amplitude of a trigonometric function is found by taking the absolute value of A. And since we know that A equals negative 1/4 we're taking the absolute value of negative 1/4 which is a positive 1/4. So there, our amplitude is a positive 1/4. How about that B term B has to do with the period? We know that the period of a trigonometric function is found by taking two pi and these are four, our cosine functions is taking two pi divided by B. And since we know our B is six, that's two pi divided by six which simplifies to be pi divided by three. So here the period is pi divided by three. Now our C and D terms, I'll quickly go over what those mean, but they are equal to zero. Here, the C terms are related to vertical translation. And since the C term is zero, there is none. And our D term is related to the phase shift because our D value is zero, there is no phase shift. So we look at our answer choices for an amplitude of 1/4 and a period of pi divided by three. And that matches with answer choice C well done. We'll catch you on the next one.

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = -½ cos 3x

Key Concepts

Amplitude of a Trigonometric Function

Period of a Trigonometric Function

Phase Shift and Vertical Translation

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