Shifting Circles Upwards How The Equation (x+3)^2+(y-2)^2=36 Changes

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Hey there, math enthusiasts! Today, we are diving into the fascinating world of circles and exploring what happens when we shift them around on the coordinate plane. Our main focus is on the circle defined by the equation (x+3)2+(y-2)2=36, and we're going to investigate the results of moving this circle upwards by 3 units. Get ready to uncover some key insights into how geometric transformations affect the algebraic representation of circles.

Understanding the Circle Equation

Before we jump into the shift, let's quickly refresh our understanding of the circle equation. The general form of a circle's equation is (x-h)^2 + (y-k)^2 = r^2, where (h, k) represents the center of the circle and r is the radius. In our case, the equation is (x+3)2+(y-2)2=36. By comparing this to the general form, we can identify that the center of our circle is at (-3, 2) and the radius is the square root of 36, which is 6. So, we have a circle centered at (-3, 2) with a radius of 6. This foundational understanding is crucial as we explore the effects of shifting this circle upwards.

When we talk about shifting a circle, we're essentially performing a geometric transformation known as a translation. A translation moves every point of the shape the same distance in the same direction. In this scenario, we're specifically focusing on a vertical translation—moving the circle up. This means every point on the circle, including its center, will move 3 units upwards. Now, how does this movement affect the equation of the circle? That's the question we're going to dissect.

The Impact of Vertical Shifts on the Center

Let's hone in on the center of the circle, which, as we established, is at (-3, 2). When we shift the circle upwards by 3 units, the x-coordinate of the center remains unchanged because we're only moving vertically. However, the y-coordinate will increase by 3. So, the new center of the circle will be at (-3, 2 + 3), which simplifies to (-3, 5). This is a critical observation. The y-coordinate of the center has indeed changed, and it has increased by the amount of the upward shift. The x-coordinate, on the other hand, remains the same because the shift is purely vertical.

This change in the center's coordinates directly affects the equation of the circle. Remember, the center (h, k) appears in the equation as (x-h)^2 + (y-k)^2 = r^2. Since the new center is (-3, 5), we'll be substituting h = -3 and k = 5 into the equation. This will give us the new equation of the shifted circle, which we'll examine in more detail shortly. For now, it's crucial to grasp that the shift in the center's y-coordinate is a direct consequence of the vertical translation, and this change will be reflected in the circle's equation.

Analyzing the Radius and the Equation

Now that we've pinpointed the new center of the circle, what about the radius? Does shifting the circle upwards change its size? The answer is a resounding no. A translation, whether vertical or horizontal, preserves the shape and size of the figure. Think of it as sliding the circle around without stretching or compressing it. Therefore, the radius of our circle remains the same, which is 6 units. This is a key characteristic of translations—they are rigid transformations that maintain the dimensions of the object being transformed.

With the new center at (-3, 5) and the radius still at 6, we can now write the equation of the shifted circle. Plugging these values into the general form (x-h)^2 + (y-k)^2 = r^2, we get: (x - (-3))^2 + (y - 5)^2 = 6^2. Simplifying this, we have (x + 3)^2 + (y - 5)^2 = 36. Notice how the equation has changed compared to the original. The (x + 3)^2 term remains the same, which reflects the unchanged x-coordinate of the center. However, the (y - 2)^2 term has transformed into (y - 5)^2, indicating the shift in the y-coordinate of the center from 2 to 5. The radius, represented by 6^2 = 36, stays constant, reinforcing that the size of the circle hasn't changed.

Visualizing the Shift

To truly solidify our understanding, let's visualize this shift. Imagine the original circle centered at (-3, 2) on a coordinate plane. Now, picture picking up this circle and sliding it straight up by 3 units. The circle moves upwards, maintaining its shape and size. The center, initially at (-3, 2), lands at (-3, 5). The entire circle has simply been displaced vertically. This mental image is a powerful tool for grasping the concept of translations and how they affect geometric figures. Visualizing the transformation helps connect the algebraic representation (the equation) with the geometric reality (the circle's position).

Moreover, visualizing the shift helps us anticipate the changes in the equation. We know that moving the circle up will affect the y-coordinate of the center, which in turn will alter the term involving y in the equation. Conversely, since the x-coordinate remains constant, the term involving x should stay the same. This predictive ability is a hallmark of a strong understanding of mathematical concepts. By visualizing the transformations, we can anticipate the algebraic consequences and vice versa.

Analyzing the Given Options

Now, let's circle back to the question at hand: Which of the following is a result of shifting the circle with equation (x+3)2+(y-2)2=36 up 3 units? We're presented with a few options, and our task is to identify the correct one based on our understanding of circle shifts.

Option A states: The y-coordinate of the center of the circle increases by 3. This statement aligns perfectly with our analysis. We've clearly established that shifting the circle upwards by 3 units indeed causes the y-coordinate of the center to increase by 3. The original center was at (-3, 2), and the new center is at (-3, 5), a direct increase of 3 in the y-coordinate. This option captures the essence of the vertical shift's impact on the circle's center.

Option B suggests: The y-coordinate of the center of the circle decreases by 3. This is the opposite of what actually happens. Shifting the circle upwards means the center moves in the positive y direction, leading to an increase, not a decrease, in the y-coordinate. Therefore, this option is incorrect.

Option C might present other possibilities related to the x-coordinate or the radius. However, we've already established that the x-coordinate remains unchanged during a vertical shift, and the radius is also unaffected by translations. Thus, any option focusing on changes to the x-coordinate or the radius would be incorrect.

Conclusion

In conclusion, shifting the circle with equation (x+3)2+(y-2)2=36 up 3 units results in the y-coordinate of the center increasing by 3. This is because a vertical translation moves the circle upwards, directly affecting the vertical position of its center. The correct answer is option A. By understanding the circle equation, the concept of translations, and visualizing the shift, we can confidently analyze the effects of geometric transformations on circles. So next time you encounter a circle shifting problem, remember these key principles, and you'll be well-equipped to tackle it!