Fill in the blank(s) to correctly complete each sentence.

Question

The graph of y = cos (x - π/6) is obtained by shifting the graph of y = cos x ______ unit(s) to the ________ (right/left).

Accepted Answer

Welcome back. Everyone. In this problem, we want to determine which of the following statements correctly describes the procedure for obtaining the graph of Y equals the cosine of X minus 1/12 of pi from the graph of Y equals the cosine of X. For our answer choices. A says that we're going to translate the graph of Y equals cosine of X by 1/12 of pi units to the right B says to translate the graph of Y equals X by 1/12 of pi units to the left C says to stretch the graph of Y equals X vertically by a factor of 12. And D says to reflect, reflect the graph of Y equals the cosine of X across the X axis and translate it by 1/12 of pi units upward. Now, if we want to figure out which statement is correct, we have to understand what this, what's the difference between our graph of Y equals the cosine minus the cosine of X minus a twelveth of pi and Y equals the cosine of X. So first, let's do a quick sketch of that graph Y equals the cosine of X. It would look something like this. Now, when we compare them, what do we notice what's the difference between both of them? Well, here, the difference is that inside the parentheses for the cosine function in our new graph is subtracting 1/12 of pi units. What do we know here? Well, recall, recall that for any graph, any trigonometric function in the form Y equals F of X. OK. Then let me put this in red Y equals F of X minus C. In other words, any other function that's changing something inside the parentheses is a horizontal translation off our graph is a horizontal translation of ye of our original graph Y equals F of X. OK. Where if the value of C is positive when C is positive, then we know that it's going to be a translation, a translation or it moves, it moves see units to the right. OK. Move C units to the right. I'm just trying to make some space here. And if C is negative is less than zero, then our graph is going to move C units, the magnitude of C units to the left. So if we compare or graph to the general form, then we should be able to figure out the value of C and based on its nature tell if it's going to move to the right or left. So let's go ahead and do that. So that means by comparing I comparing Y equals the sine or rather the cosine, that's what we're talking about here. The cosine of X minus 1/12 of pi, OK, two Y equals F of X minus C. Then here, since our new graph is already in the general form, we can see that C OK. C is equal to a third of units. C equals 1/12 of five units. No, because it's positive, that means it's going to be you, it's going to be moving to the right. OK. So it's, we know it's positive. Therefore, the graph of Y equals the cosine of X minus 1/12 of pi can be obtained by translating the graph of Y equals the cosine of X by 1/12 of pi units to the right. When we look back on our answer choices, that would have been answer choice. A thanks a lot for watching everyone. I hope this video helped.

Fill in the blank(s) to correctly complete each sentence.
The graph of y = cos (x - π/6) is obtained by shifting the graph of y = cos x unit(s) to the __ (right/left).

Key Concepts

Horizontal Phase Shift in Trigonometric Functions

Understanding the Argument of the Cosine Function

Graphical Interpretation of Trigonometric Transformations

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