Equivalent Expressions For Y = -3sin(x) + 2

Hey guys! Today, we're diving into the world of trigonometric functions, specifically looking at the function y = -3sin(x) + 2. Our mission is to figure out which other expressions are equivalent to this one. This is a classic problem in trigonometry that tests your understanding of transformations and trigonometric identities. We'll go through each option step-by-step, so you can clearly see how we arrive at the answers. Understanding these equivalencies is super important for more advanced math and physics, so let's get started!

Understanding the Original Function: y = -3sin(x) + 2

Before we jump into the options, let’s break down the original function, y = -3sin(x) + 2. This will help us understand what transformations are at play and how they affect the graph of the sine function. The basic sine function, y = sin(x), has a range of -1 to 1. The transformations applied here are:

• Amplitude Change: The -3 in front of the sine function changes the amplitude. The absolute value, |–3| = 3, means the graph will stretch vertically, reaching a maximum of 3 and a minimum of -3. The negative sign indicates a reflection over the x-axis. So, instead of starting by going up, our transformed sine function will start by going down.
• Vertical Shift: The +2 at the end shifts the entire graph upwards by 2 units. This means the midline of the sine wave, which is normally at y = 0, will now be at y = 2. This shift is crucial because it affects the overall position of the graph on the coordinate plane.

So, to recap, our function y = -3sin(x) + 2 is a sine wave that's been flipped upside down, stretched vertically by a factor of 3, and then shifted upwards by 2 units. Keeping these transformations in mind will help us evaluate the given options more effectively. It’s like having a mental picture of what the equivalent functions should look like!

Evaluating Option A: y = -3sin(x) - 2

The first option we need to consider is y = -3sin(x) - 2. At first glance, it looks quite similar to our original function, y = -3sin(x) + 2. The key difference here is the constant term: -2 instead of +2. Remember that the +2 in the original function represents a vertical shift upwards by 2 units. So, what does -2 do?

Well, -2 represents a vertical shift downwards by 2 units. This is a significant change! Our original function’s midline is at y = 2, but this new function's midline would be at y = -2. The amplitude and reflection over the x-axis remain the same, but the entire graph is now positioned differently on the coordinate plane. Essentially, this function is a reflection of our original function across the x-axis, followed by a vertical shift downwards by 4 units (since the original midline is at 2, and the new one is at -2).

To really nail this down, think about a specific point on the sine wave. For example, the original function y = -3sin(x) + 2 starts at the point (0, 2) because sin(0) = 0, and the function simplifies to y = 2. In contrast, the function y = -3sin(x) - 2 would start at the point (0, -2). This difference in the starting point clearly shows that the two functions are not equivalent.

In conclusion, Option A, y = -3sin(x) - 2, is not equivalent to our original function y = -3sin(x) + 2 due to the different vertical shift. It’s crucial to pay attention to these constant terms as they have a significant impact on the function's graph!

Evaluating Option B: y = 3sin(-x) + 2

Now, let's tackle the second option: y = 3sin(-x) + 2. This one involves a bit more trigonometric trickery! We've got a negative sign inside the sine function, which means we need to consider how the sine function behaves with negative inputs. Recall that the sine function is an odd function. What does that mean?

An odd function has the property that sin(-x) = -sin(x). This is a fundamental trigonometric identity. So, we can rewrite sin(-x) as -sin(x). Let’s substitute this into our option B:

y = 3sin(-x) + 2 = 3(-sin(x)) + 2 = -3sin(x) + 2

Wow, look at that! By using the property of odd functions, we've transformed y = 3sin(-x) + 2 into y = -3sin(x) + 2, which is exactly our original function. This is a big win! It shows how important it is to remember those trigonometric identities. They can help simplify complex expressions and reveal equivalencies that aren't immediately obvious.

This equivalence makes sense graphically as well. The negative sign inside the sine function represents a reflection over the y-axis. However, because the sine function is odd, reflecting sin(x) over the y-axis is the same as reflecting it over the x-axis and then multiplying by -1. The multiplication by 3 then stretches the graph vertically, and the addition of 2 shifts it up, resulting in the same graph as our original function.

Therefore, Option B, y = 3sin(-x) + 2, is indeed equivalent to y = -3sin(x) + 2. This highlights the power of understanding and applying trigonometric identities.

Evaluating Option C: y = -3cos(x - π/2) + 2

Alright, let's move on to the third option: y = -3cos(x - π/2) + 2. This option introduces the cosine function, which means we need to think about the relationship between sine and cosine. This relationship often involves phase shifts, which can be a bit tricky, but we'll break it down step by step.

Remember the fundamental identity that connects sine and cosine: cos(x - π/2) = sin(x). This identity tells us that a cosine function shifted to the right by π/2 (90 degrees) is equivalent to the sine function. This is a crucial piece of the puzzle! Now, let's substitute this identity into our option C:

y = -3cos(x - π/2) + 2 = -3sin(x) + 2

Guess what? We’ve done it again! Substituting the identity directly transforms Option C into our original function, y = -3sin(x) + 2. This is fantastic! It demonstrates that understanding phase shifts and trigonometric identities can simplify expressions and reveal hidden equivalencies.

Graphically, this equivalence is also clear. The cosine function is essentially a sine function shifted by π/2. So, when we shift the cosine function in option C to the right by π/2, it aligns perfectly with the sine function in our original expression. The -3 scales and reflects the function, and the +2 shifts it vertically, ensuring the graphs match.

So, Option C, y = -3cos(x - π/2) + 2, is equivalent to our original function y = -3sin(x) + 2. This option is a great example of how different trigonometric forms can represent the same function.

Evaluating Option D: y = 3cos(x + π/2) + 2

Last but not least, let’s analyze Option D: y = 3cos(x + π/2) + 2. This option, like option C, involves the cosine function and a phase shift. However, this time the shift is a positive π/2, which means a shift to the left. Let's see how this plays out.

To tackle this, we’ll use another trigonometric identity: cos(x + π/2) = -sin(x). This identity is very similar to the one we used in option C, but the sign is crucial. It tells us that a cosine function shifted to the left by π/2 is equivalent to the negative of the sine function.

Now, let’s substitute this identity into Option D:

y = 3cos(x + π/2) + 2 = 3(-sin(x)) + 2 = -3sin(x) + 2

Hold on a second… that looks familiar! Just like in options B and C, this substitution has transformed Option D into our original function: y = -3sin(x) + 2. This is pretty cool – it means that Option D is also an equivalent expression.

Graphically, shifting the cosine function to the left by π/2 and then scaling it by 3 and adding 2 results in the same graph as y = -3sin(x) + 2. This reinforces the idea that there can be multiple ways to represent the same trigonometric function through transformations and identities.

Thus, Option D, y = 3cos(x + π/2) + 2, is also equivalent to y = -3sin(x) + 2.

Final Answer: Equivalent Expressions

Alright, guys, we've gone through all the options, and it's time for the grand reveal! We started with the function y = -3sin(x) + 2, and we evaluated four different options. Here’s a quick recap of what we found:

• Option A: y = -3sin(x) - 2 - Not equivalent (different vertical shift).
• Option B: y = 3sin(-x) + 2 - Equivalent (using the identity sin(-x) = -sin(x)).
• Option C: y = -3cos(x - π/2) + 2 - Equivalent (using the identity cos(x - π/2) = sin(x)).
• Option D: y = 3cos(x + π/2) + 2 - Equivalent (using the identity cos(x + π/2) = -sin(x)).

So, the expressions that are equivalent to y = -3sin(x) + 2 are Options B, C, and D. These options demonstrate how different transformations and trigonometric identities can result in the same function. Understanding these relationships is key to mastering trigonometry and calculus!

I hope this breakdown has helped you see how to approach problems involving trigonometric equivalencies. Remember, it's all about breaking down the functions, applying the correct identities, and visualizing the transformations. Keep practicing, and you’ll become a trig whiz in no time!