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

Question

The graph of y = 3 + 5 cos (x + π/5) is obtained by shifting the graph of y = cos x horizontally ________ unit(s) to the __________, (right/left) stretching it vertically by a factor of ________, and then shifting it vertically ________ unit(s) __________. (up/down)

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 eight plus two, cosine two multiplied by the cosine of X plus 1/7 of pi from the graph of Y equals the cosine of X A says we're going to translate the graph of Y equals the cosine of X by 1/7 of pi units to the right stretch it vertically by a factor of two and translate it by eight units downward. B says we'll translate the graph by 1/7 of pi units to the left and stretch it vertically by a factor of eight C says we'll translate the graph 1/7 of pi units to the right compress it vertically by a factor of two and translate it by eight units upward. And this is we will translate the graph by 1/7 of pi units to the left stretched vertically by a factor of two and translated by eight units upward. Now, what are we, what are we really trying to figure out here? Well, what we're really saying is that let's imagine we have the graph of Y equals the cosine of X. OK. So our graph would look something like this. Now, what we're really saying is that our new graph of Y equals eight plus two multiplied by the cosine of X plus 1/7 of pi can be found by looking at parts of the graph. In other words, we don't need to plot the new points on the graph. Instead, we can look at aspects of that new equation and tell what we're going to do to our original graph of Y equals the cosine of X. So that's what we're really trying to figure out. So what do we know that can help us? Well, let's write down our graph here. So the graph that we have OK is, is Y equals eight plus two multiplied by the cosine of X plus 1/7 of pi. Now, if you notice here, we have our cosine function that's being multiplied by a coefficient. Uh The angle is changing from X and it's the entire function is being added to a constant. So generally, then this graph, if we were to generalize, generally, this graph would look like Y equals a multiplied by F of X minus C plus D. OK. Because again, as you notice in both graphs, something is multiplying the function inside the trigonometric function is changing and it's being added to a constant. So this graph here recall that it represents a horizontal and vertical translation and a vertical stretch or compression of the graph of Y equals F of X. We know that because A helps us to figure out whether our graph is being stretched or compressed. OK. C helps us to figure out whether our graph is shifted horizontally. So it's a horizontal shift and the D helps us to figure out if our graph is being shifted vertically. So if we can figure out the values of A C and D from our, from our equation, then we should be able to tell what the horizontal translation is what the vertical translation is. And if the graph is being stretched or compressed, now, if we're comparing both of these equations, right, let's put this in red here. Notice that I could rewrite my equation to look more like the general form, which means that here I could write my cosine term first. So instead I could write this as two multiplied by the cosine of X plus 1/7 of pie plus eight. OK. And here notice in our general term, we're subtracting inside the bracket. So we could rewrite a positive seventh of pi, we could rewrite that as minus negative seventh of pi because minus negative seventh of pi is the same thing as plus 1/7 of pi. No, it's in the general form. What do you notice here? What do you, what do you see? Well, here notice that A equals two. OK. Yeah. A equals two. We can see it here A equals two C equals negative 1/7 of pi and eight equals D. So let's write that down. OK. But these are all our value. Now, how is that going to help us find the stretch, the horizontal shift and the vertical shift? Well, here A equals two. And if A is positive, that means it's a vertical stretch. And, well, if A is positive, yes, if it is positive, that means it's a vertical stretch. And if the magnitude of A is greater than one, that also means that it's, we're sure it's going to be a stretch. So if a equals two, this tells us that it's a vertical stretch by a factor of two. OK. So let's write that it's, our graph is going to be vertically stretched by a factor of two. OK. Next, right. Next, when we think about our horizontal shift C helps us and since C equals negative 1/7 of pi because it's negative, that means it's going to be translated horizontally to the left. So our graph is going to move 1/7 of pi units to the left. OK. And finally, since D equals eight, because D is positive, that means we're going to translate it upward. OK. So that means we're going to move our graph eight units upward. So when we compare both of our graphs, we are going to translate the graph of Y equals the cosine of X by vertically stretching it by a factor of two, moving it 1/7 of pi units to the left and moving it eight units upward to get the graph of Y equals eight plus two multiplied by the cosine of X plus 1/7 of pi. When we look back on our answer choices that would have been answer choice. D Thanks a lot for watching everyone. I hope this video helps.

Fill in the blank(s) to correctly complete each sentence.
The graph of y = 3 + 5 cos (x + π/5) is obtained by shifting the graph of y = cos x horizontally unit(s) to the , (right/left) stretching it vertically by a factor of , and then shifting it vertically unit(s) ____. (up/down)

Key Concepts

Horizontal Phase Shift in Trigonometric Functions

Vertical Stretching and Amplitude

Vertical Shift of Trigonometric Graphs

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