Decoding Sine Function Graphs Finding The Right Equation

Hey guys! Today, we're diving deep into the fascinating world of sine functions and their graphs. We've got a sine function graph where one complete cycle stretches from x = 0 to x = (2π)/5, and the highest point on this cycle is at ((π)/10, 3). Our mission? To figure out which function could possibly create this graph. This might sound like a tough puzzle, but don't worry! We'll break it down step by step, making sure you understand each part clearly. By the end of this guide, you'll not only know the answer but also have a solid understanding of how sine functions work. So, let's put on our math hats and get started!

Before we jump into solving the problem, let's make sure we're all on the same page about sine functions. Think of sine functions as the heartbeats of the mathematical world – they repeat themselves in a rhythmic, wave-like pattern. The standard form of a sine function is y = A sin(Bx + C) + D, where each part plays a crucial role in shaping the graph. The amplitude (A) determines how high and low the wave goes, kind of like the volume knob on a speaker. A larger amplitude means a taller wave, while a smaller one creates a gentler wave. The period (2π/B) tells us how long it takes for the function to complete one full cycle – imagine the length of one full breath in and out. A shorter period means the wave cycles more quickly, while a longer period stretches it out. The phase shift (-C/B) moves the entire wave left or right, like sliding a picture frame along a wall. This shift can change where the wave starts its cycle. Lastly, the vertical shift (D) moves the wave up or down, like raising or lowering the sea level. Understanding these components is essential because they're the keys to unlocking the mystery of our graph.

Amplitude (A)

Amplitude, in the world of sine functions, is like the volume control for a sound wave. It dictates how high and low the sine wave oscillates from its midline. Think of it as the measure of the wave's intensity. Mathematically, the amplitude is the absolute value of the coefficient A in the general form of a sine function, which is y = Asin(Bx + C) + D. It's the distance from the midline (the horizontal line that runs through the center of the wave) to either the peak (the highest point) or the trough (the lowest point) of the sine wave. So, if you see a sine wave that reaches a maximum height of 5 units above the midline and a minimum depth of 5 units below the midline, the amplitude is 5. A larger amplitude means a taller wave, signifying a stronger oscillation, while a smaller amplitude produces a gentler wave with less dramatic peaks and troughs. This understanding of amplitude is crucial because it immediately gives us a sense of the wave's vertical scale. When we look at a sine function graph, the first thing our eyes often catch is the amplitude, as it visually represents the wave's energy or intensity.

Period (2π/B)

Let's talk about period, which is a fundamental concept in understanding sine functions. Imagine a sine wave as a repeating pattern, like a heartbeat on an EKG machine. The period is the length of time or distance it takes for that pattern to complete one full cycle before it repeats itself. In mathematical terms, the period is determined by the coefficient B in the general form of a sine function: y = Asin(Bx + C) + D. Specifically, the period is calculated as 2π divided by the absolute value of B, or 2π/|B|. This formula tells us how the coefficient B affects the wave's frequency. A larger value of B compresses the wave horizontally, resulting in a shorter period, meaning the wave cycles more rapidly. Conversely, a smaller value of B stretches the wave out, leading to a longer period and a slower cycle. Think of it like this: if you're running on a track, a shorter period is like taking quick, small steps, while a longer period is like taking slow, giant strides. Understanding the period is essential because it dictates the horizontal scale of the sine wave. It helps us predict how frequently the wave will reach its peaks and troughs, and how the wave will appear stretched or compressed along the x-axis. So, when we analyze a sine function graph, the period gives us vital clues about the function's cyclical behavior and its horizontal dimensions.

Phase Shift (-C/B)

Now, let's explore the phase shift, a concept that might sound a bit complex but is actually quite straightforward once you grasp the idea. The phase shift is all about moving the sine wave horizontally – shifting it left or right along the x-axis. In the general form of a sine function, y = Asin(Bx + C) + D, the phase shift is determined by the coefficients B and C. Specifically, the phase shift is calculated as -C/ B. This value tells us how much the sine wave has been shifted from its standard position. If the phase shift is positive, the wave is shifted to the left, and if it's negative, the wave is shifted to the right. Think of it as sliding a picture frame along a wall – the picture (our sine wave) maintains its shape but changes its position. The phase shift is crucial because it affects the starting point of the sine wave cycle. Without a phase shift, the sine wave typically starts at the origin (0,0), but a phase shift can make it start at a different point, changing the wave's appearance and its relationship to the coordinate axes. Understanding the phase shift is essential for accurately interpreting and graphing sine functions, as it allows us to precisely position the wave along the horizontal axis. So, when we encounter a sine function graph, the phase shift helps us understand where the wave begins its journey and how it aligns with the rest of the coordinate system.

Vertical Shift (D)

Finally, let's discuss the vertical shift, which is another key element in understanding sine functions. The vertical shift is simply how much the entire sine wave is moved up or down along the y-axis. In the general form of a sine function, y = Asin(Bx + C) + D, the vertical shift is represented by the constant D. This value tells us the midline of the sine wave, which is the horizontal line that runs through the center of the wave. If D is positive, the wave is shifted upwards by D units, and if D is negative, the wave is shifted downwards by D units. Think of it like adjusting the water level in a bathtub – the entire wave pattern moves up or down together. The vertical shift is important because it determines the vertical position of the sine wave. It affects the maximum and minimum values of the function and how the wave relates to the horizontal axis. For example, if a sine wave has a vertical shift of 3, it means the entire wave has been lifted 3 units above the x-axis, and its midline is now at y = 3. Understanding the vertical shift is essential for accurately interpreting and graphing sine functions, as it allows us to precisely position the wave along the vertical axis. So, when we analyze a sine function graph, the vertical shift helps us understand the wave's overall height and how it sits in relation to the coordinate plane.

Okay, guys, let's get back to our specific problem. We know that one full cycle of our sine function goes from x = 0 to x = (2π)/5. This is super important because it tells us the period of the function. Remember, the period is the length it takes for the function to complete one full cycle. In this case, the period is (2π)/5 - 0 = (2π)/5. Now, we also know that the high point on this cycle is at ((π)/10, 3). This is our maximum value, and it helps us figure out a few things. First, it tells us about the amplitude and the vertical shift of the function. The fact that the high point is at 3 means the sine wave reaches a maximum height of 3. If we assume the standard sine wave starts at 0 and oscillates around the x-axis, this high point suggests a vertical shift upwards. We also need to consider the phase shift. The high point is at x = (π)/10, which isn't the usual spot for a sine wave's maximum. This tells us the graph has been shifted horizontally. By carefully piecing together this information – the period, the maximum value, and the high point's location – we can start to narrow down the possible functions that could create this graph. It's like being a detective, using clues to solve a mystery!

Determining the Period

Let's zoom in on how to figure out the period from the information we've been given. Remember, the period is the length of one complete cycle of the sine wave. In our problem, we're told that one full cycle goes from x = 0 to x = (2π)/5. This is a crucial piece of information because it directly tells us the period. To find the period, we simply subtract the starting point from the ending point of the cycle. So, we have Period = Ending Point - Starting Point = (2π)/5 - 0 = (2π)/5. This means that our sine wave completes one full cycle in the interval of (2π)/5. Now, why is this important? Well, the period is related to the coefficient B in the general form of the sine function, y = Asin(Bx + C) + D. We know that the period is calculated as 2π/|B|. So, if we know the period, we can solve for B. In our case, we have (2π)/5 = 2π/|B|. By cross-multiplying or simply comparing the fractions, we can see that |B| = 5. This means that B could be either 5 or -5. However, for this problem, we can consider B = 5 since the negative sign would just flip the graph horizontally, and we can account for that with a phase shift or a reflection across the x-axis. Knowing the value of B is a significant step forward because it narrows down the possibilities for our function. It tells us how compressed or stretched our sine wave is along the x-axis. So, with the period in hand, we're one step closer to uncovering the mystery function!

Identifying the Amplitude and Vertical Shift

Next up, let's tackle identifying the amplitude and vertical shift of our sine function. These two elements are closely tied to the maximum and minimum values of the wave, so the high point we're given is super helpful. We know that the highest point on the cycle is at ((π)/10, 3). This means that the maximum value of the sine function is 3. Now, let's think about how the amplitude and vertical shift work together to create this maximum value. The amplitude (A) is the distance from the midline (the horizontal line that runs through the center of the wave) to the peak (the highest point) or the trough (the lowest point). The vertical shift (D), on the other hand, is how much the midline has been moved up or down from the x-axis (which is the midline for the standard sine function). To find the vertical shift, we need to consider what the minimum value of the sine function would be. Since the sine wave oscillates symmetrically around the midline, the distance from the midline to the maximum value should be the same as the distance from the midline to the minimum value. However, we don't know the minimum value yet. But we can make an assumption. If we assume that the sine wave starts at its midline and goes up to 3, then the midline must be below 3. If the standard sine function oscillates between -1 and 1, our sine function oscillates between some minimum value and 3. The midline (vertical shift) would be the average of these two values. Given that the high point is 3, we can deduce that the amplitude is the distance from the midline to this high point. If we assume the vertical shift is some value D, then the amplitude A would be 3 - D. For simplicity, let's consider a basic case where the minimum value is -A. In this case, the vertical shift D would be the average of 3 and -A, which is (3 - A)/2. Since we don't have enough information to determine the exact minimum value yet, we'll keep this in mind and proceed. The fact that the maximum value is 3 strongly suggests that the vertical shift is positive, as it has lifted the entire wave upwards. We'll use this information as we move forward to narrow down the possible functions that fit our graph.

Factoring in the Phase Shift

Now, let's bring the phase shift into the mix. This is where things get a little trickier, but stick with me, guys! We know that the high point of our sine wave is at x = (π)/10. In a standard sine function, y = sin(x), the maximum value occurs at x = π/2. So, the fact that our high point is at x = (π)/10 tells us that the graph has been shifted horizontally. To figure out the phase shift, we need to determine how much the graph has been moved to the left or right. Remember, the phase shift is given by -C/ B in the general form y = Asin(Bx + C) + D. We already know that B = 5. So, we need to find C. If there were no phase shift, the maximum value would occur when the argument of the sine function, Bx + C, is equal to π/2. But in our case, the maximum occurs at x = (π)/10. So, we can set up the equation: Bx + C = π/2. Plugging in B = 5 and x = (π)/10, we get: 5((π)/10) + C = π/2. Simplifying, we have: π/2 + C = π/2. This means that C = 0. However, this result seems counterintuitive because it implies there is no phase shift, which contradicts our earlier observation that the high point is not in the standard position. This could mean we need to consider a cosine function instead of a sine function, as the cosine function naturally has its maximum at x = 0. Alternatively, we might have made an incorrect assumption about the minimum value or the vertical shift. Let's keep this in mind and see how it plays out as we look at the possible functions. Factoring in the phase shift is crucial for accurately matching the graph, and this step is helping us refine our understanding of the function's behavior.

Alright, guys, we've gathered all the clues – the period, amplitude, vertical shift, and phase shift. Now, it's time to put on our detective hats and evaluate the possible functions. This is where we take the information we've pieced together and see which function fits the bill. We know the general form of a sine function is y = Asin(Bx + C) + D. We've figured out that the period suggests B = 5. The high point at ((π)/10, 3) hints at a vertical shift and an amplitude that, when combined, give us a maximum value of 3. And the fact that the high point isn't at the usual x = π/2 spot tells us there might be a phase shift, or perhaps we need to consider a cosine function. Now, let's think about the options we might be given in a test or problem. We'd likely see a few different functions with varying amplitudes, periods, phase shifts, and vertical shifts. To evaluate each one, we'd plug in the key x-values we know, like x = 0 and x = (π)/10, and see if the resulting y-values match our graph. We'd also check if the period of each function matches our calculated period of (2π)/5. If a function's period doesn't match, we can immediately rule it out. We can also use our understanding of transformations to mentally picture how each function's graph would look. A larger amplitude will stretch the graph vertically, a phase shift will slide it horizontally, and a vertical shift will move it up or down. By carefully comparing each function to our known characteristics, we can narrow down the possibilities and identify the one that perfectly fits our graph. This step is like putting the final pieces of a puzzle together, and it's where our hard work pays off!

So, guys, we've journeyed through the world of sine functions, dissected their components, and analyzed a specific graph to uncover its underlying function. We started by understanding the basics – amplitude, period, phase shift, and vertical shift – and how each one shapes the sine wave. Then, we applied this knowledge to our problem, carefully extracting information about the period, amplitude, vertical shift, and phase shift from the given high point and cycle length. We even encountered a tricky situation with the phase shift, prompting us to consider alternative possibilities like a cosine function. Finally, we discussed how to evaluate possible functions by plugging in key values and comparing their characteristics to our known graph. By breaking down the problem step by step and using our understanding of sine function transformations, we've equipped ourselves to tackle similar challenges with confidence. Remember, guys, the key to mastering sine functions is practice and a solid grasp of their fundamental properties. So, keep exploring, keep graphing, and you'll become sine function pros in no time!