Questions: Determine whether the following statement is true or false. We graph (y=sin x+5 cos x) for (0 leq x leq 2 pi) by adding x -coordinates. Is the statement true or false? A. False. We graph (y=sin x+5 cos x) for (0 leq x leq 2 pi) by multiplying the (y)-coordinates by 5. B. False. We graph (y=sin x+5 cos x) for (0 leq x leq 2 pi) by multiplying the (x)-coordinates by 5. C. True. D. False. We graph (y=sin x+5 cos x) for (0 leq x leq 2 pi) by adding (y)-coordinates.

Determine whether the following statement is true or false. We graph (y=sin x+5 cos x) for (0 leq x leq 2 pi) by adding x -coordinates. Is the statement true or false? A. False. We graph (y=sin x+5 cos x) for (0 leq x leq 2 pi) by multiplying the (y)-coordinates by 5. B. False. We graph (y=sin x+5 cos x) for (0 leq x leq 2 pi) by multiplying the (x)-coordinates by 5. C. True. D. False. We graph (y=sin x+5 cos x) for (0 leq x leq 2 pi) by adding (y)-coordinates.
Transcript text: Determine whether the following statement is true or false. We graph $y=\sin x+5 \boldsymbol{\operatorname { c o s }} \mathrm{x}$ for $0 \leq x \leq 2 \pi$ by adding x -coordinates. Is the statement true or false? A. False. We graph $y=\boldsymbol{\operatorname { s i n }} x+5 \boldsymbol{\operatorname { c o s }} x$ for $0 \leq x \leq 2 \pi$ by multiplying the $y$-coordinates by 5 . B. False. We graph $y=\sin x+5 \boldsymbol{\operatorname { c o s }} x$ for $0 \leq x \leq 2 \pi$ by multiplying the $x$-coordinates by 5 . C. True. D. False. We graph $y=\sin x+5 \cos x$ for $0 \leq x \leq 2 \pi$ by adding $y$-coordinates.
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Solution

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Solution Steps

To determine whether the given statement is true or false, we need to understand how the function \( y = \sin x + 5 \cos x \) is graphed over the interval \( 0 \leq x \leq 2\pi \). Specifically, we need to check if the graph is created by adding x-coordinates or by some other method.

  1. The function \( y = \sin x + 5 \cos x \) is a combination of the sine and cosine functions.
  2. To graph this function, we calculate the y-values for each x-value in the interval \( 0 \leq x \leq 2\pi \).
  3. The statement suggests that the graph is created by adding x-coordinates, which is not correct. The correct approach involves calculating the y-values for each x-value.

Therefore, the correct answer is D: False. We graph \( y = \sin x + 5 \cos x \) for \( 0 \leq x \leq 2\pi \) by adding y-coordinates.

Step 1: Define the Function

We start by defining the function \( y = \sin x + 5 \cos x \). This function is a combination of the sine and cosine functions.

Step 2: Generate \( x \) Values

We generate \( x \) values in the interval \( 0 \leq x \leq 2\pi \). This interval is chosen because it represents one complete cycle of the sine and cosine functions.

Step 3: Calculate \( y \) Values

For each \( x \) value, we calculate the corresponding \( y \) value using the function \( y = \sin x + 5 \cos x \).

Step 4: Plot the Function

We plot the function \( y = \sin x + 5 \cos x \) over the interval \( 0 \leq x \leq 2\pi \). The plot helps us visualize the behavior of the function.

Step 5: Analyze the Statement

The statement claims that the graph of \( y = \sin x + 5 \cos x \) for \( 0 \leq x \leq 2\pi \) is created by adding \( x \)-coordinates. This is incorrect because the graph is created by calculating the \( y \)-values for each \( x \)-value, not by adding \( x \)-coordinates.

Final Answer

\(\boxed{\text{D}}\)

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