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

Question

The graph of y = -5 + 2 cos x is obtained by shifting the graph of y = 2 cos x ________ unit(s) __________ (up/down).

Accepted Answer

Welcome back. Everyone. In this problem, we want to figure out which of the following statements correctly describes the procedure for obtaining the graph of Y equals negative 13 plus eight, multiplied by the cosine of X from the graph of Y equals eight multiplied by the cosine of X. For our answer choices. A says we're going to translate the graph of Y equals eight cosine X by 13 units downward. B says we are going to translate it by 13 units to the left C says we are going to reflect it across the X axis and stretch it vertically by a factor of 13. And D says we are going to reflect it across the X axis and compress it vertically by a factor of 13. Now, what are we really trying to figure out here? Well, what we're saying is that let's imagine that we have the graph of Y. Let me try it here. Y equals it multiplied by the cosine of X. OK. Let's say it looks, oh no, it's a cosine graph. So let's say it looks something like this. What we're really saying is that we are trying to figure out what will happen if we subtract 13 from our graph. Because by comparison, we can tell that the difference between our original function Y equals eight multiplied by the cosine of X. And our function after our translation Y equals negative 13 plus eight cosine X which we can rewrite as eight cosine X minus 13. We can rearrange the terms. Notice that the difference between the both of them is minus 13. So what does that negative 13 mean? Well, recall recall that for any graph that's written in the form Y equals FX plus C, this graph is a vertical translation. So this is a right here is a vertical translation of Y equals F of X. OK. So essentially what it's saying is that if we have one function without a constant compared to the other with a constant added to the function that is a vertical translation, the translation is upward if C is positive and downward if C is negative. So when we compare our two graphs here or, or, or two functions here, rather, we can tell that the difference or constant is negative 13. So since it's negative 13, that means our graph is going to move downward by 13 units. In other words, if we were to put our graph on our grid, it would probably look something like this. So this would be the graph of Y equals negative 13 plus the cosine or plus eight multiplied by the cosine of X OK. So it's a shift of 13 units downward. And that tells us then that the correct answer is answer choice. A we translate the graph of Y equals eight cosine X by 13 units downward to get the graph of Y equals negative 13 plus eight cosine X. 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 = -5 + 2 cos x is obtained by shifting the graph of y = 2 cos x unit(s) __ (up/down).

Key Concepts

Vertical Shifts in Trigonometric Graphs

Amplitude and Vertical Translation

Function Transformation Notation

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