Question #d16d6

Jun 2, 2017

$x$ is not in the equation, $\theta$ is!

Explanation:

Basically what we are trying to do is to put the equation in the standard form of a sin function:

$f \left(x\right) = a \cdot \sin b \left(x - h\right) + k$

Where $a$ is the vertical stretch/compression (if any), $b$ is the horizontal stretch/compression, $h$ is the horizontal shift, and $k$ is the vertical shift.

Now we can put the given equation in that form:

$y - 2 \sin \theta = 3$
$y = 2 \sin \theta + 3$
$f \left(\theta\right) = 2 \sin \theta + 3$

Notice the $f \left(\theta\right)$. This is correct because you were never given $x$ in the original equation, you were given $\theta$. Hence, your answer is:

$f \left(\theta\right) = 2 \sin \theta + 3$ or number $4$ on your list of $4$ possible solutions.

I hope that helps!

Jun 2, 2017

From your answer choice, I can see that you don't have trouble understanding what the function should be rewritten as (I will quickly rewrite it below anyway, just in case).

$y - 2 \sin \theta = 3$

$y - 2 \sin \theta + 2 \sin \theta = 3 + 2 \sin \theta$

$y = 2 \sin \theta + 3$

Now, for the error:

Think about what $f \left(x\right)$ means. Read out literally, it is "function of x". So when you write $f \left(x\right)$, you are telling the reader that you are giving them a function into which they can put a number ($x$), and the output will be whatever is on the other side of the equals sign.

However, in this case, choice C states that:

$f \left(x\right) = 2 \sin \theta + 3$

The problem with this is that $x$ is not used in the output of the function. In order to be written correctly, the answer should be:

$f \left(\theta\right) = 2 \sin \theta + 3$

Since it is $\theta$ that is being manipulated in the equation, not $x$.

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To help understand why using the correct variable in a function is important, think of it this way:

You are given a recipe for a cake. It includes the following ingredients:

Butter
Flour
Eggs
Sugar

The instructions for the recipe are as follows:

Take 3 cups of cake mix and pour it into a large bowl. Next, add 2 cups of water and a tablespoon of salt. Mix together until it is a single consistency. Pour the mix in a cake pan and bake in the oven for 30 minutes at 375 degrees.

I'm sure this would produce an absolutely horrible cake, but that's besides the point (I have no idea how to actually make a cake properly). Where do we get the water, salt, or cake mix from? And what do we need butter, flour, eggs, and sugar for if we're not going to use them in the recipe?

Think of making a cake like this when you write functions. $f \left(x\right)$ gives you your "ingredient list": $x$ and nothing else. If your "instructions" ($2 \sin \theta + 3$) require you to use other ingredients (variables) than what the ingredients list (function) told you to use, the output will not work. Make sure you use your ingredients in your instructions, and don't put anything else in there!