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

Question

The graph of y = sin (x + π/4) is obtained by shifting the graph of y = sin x ______ unit(s) to the ________ (right/left).

Accepted Answer

Welcome back everyone. In this problem, we want to figure out which of the following statements correctly describes the procedure of obtaining the graph of Y equals the sine of X plus 2/5 of pi from the graph of Y equals sine, X. For our answer choices, A says to translate the graph of Y equals sine X by 2/5 of pi units downward. B says to stretch the graph of Y equals sine X vertically by a factor of five halves. C says to translate the graph of Y equals sine X by 2/5 pi units to the left. And the D says to translate the graph of Y equals sine X by 2/5 pi units to the right. No, if we're going to figure out what our new graph will look like. First, let's try to get an idea of what our original graph looks like. So here's a sketch of Y equals sine X look something like that. Now, essentially what we are saying is that we want to figure out how we can move from Y equals sine X two. Our new function Y equals the sine of X plus 2/5 of pi if we compare both of those formula or both of those functions notice that the difference is plus 2/5 of pi inside the bracket. So the question is, how is this or what difference does this make for both functions? Well, what do we know? Recall? OK. Recall that if Y equals the, or Y equals FX for a trigonometric function, then if we have another function Y equals F of X minus C. In other words, we're changing something to the value of our angle X. Then this function is a horizontal translation. It's a horizontal translation of our original graph. And for that horizontal translation, if C is positive, that means it moves to the right, the C is positive, it was, it moves to the right. OK. And if C is negative, that is C is less than zero. OK, then that means it moves to the left, our function moves to the left. OK? By the magnitude of C units. OK. Likewise here, if it was positive, it will move to the right by C units. OK. Now, when we compare them, we'll notice that this one, it kind of does look like our equation in red here. So if we can compare them, get them in the same exact form, then we'll know how much our graph is moved horizontally or how much it's translated horizontally. So let's try to compare them. OK. Now, when we compare, by comparing Y equals the sine of X plus 2/5 of pi two Y equals F of X minus C. What do you notice? Well, like we said earlier, they're almost looking the same but the difference is in our function, it's positive, the change is positive the 2/5 of pi whereas in our general form, it's negative. So we need to rewrite our general, our equation or, or function in the general form. And we can do that by subtracting the negative value of 2/5 of pi. In other words, we can write it as the sine of X minus negative 2/5 of pi and, and that makes sense, right? Because if we're adding 2/5 of pi, it's the same thing as subtracting negative 2/5 of pi. No. When we compare them, what do you notice then now we notice that the value of CC is going to be equal to negative 2/5 of pi. Now remember C tells us how much it's translated by because C is negative, that means it's going to be moving to the left. OK. And it is going to be moving to the left by 2/5 of pi units. So that means the graph of Y equals the sine of X plus 2/5 of pi can be obtained by translating the graph of Y equals sine X by 2/5 of pi units to the left. When we look back on our answer choices, that would have been answer choice. C let me put that in yellow here. 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 = sin (x + π/4) is obtained by shifting the graph of y = sin x unit(s) to the __ (right/left).

Key Concepts

Phase Shift

Sine Function Properties

Graph Transformations

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