Parabola Shift: Finding The New Equation & Vertex

Let's dive into understanding how a parabola shifts and how to find its new equation when the vertex changes! We'll specifically tackle a parabola defined by the equation $y = x^2 - 2x + 1$, which intersects the y-axis at the point (0, 1). Our goal is to figure out what happens when this parabola is shifted so its vertex lands at the point (1, 2). This involves understanding transformations of parabolas, finding the original vertex, calculating the shift vector, and determining the equation of the shifted parabola.

Understanding the Initial Parabola

First, let's break down the starting parabola: $y = x^2 - 2x + 1$. To really understand this parabola, we need to identify its key features, especially its vertex. Remember, the vertex is the turning point of the parabola – the minimum or maximum point. Finding the vertex is crucial because it tells us the lowest or highest point on the curve and helps us visualize the parabola's position in the coordinate plane. To find the vertex, we can use a couple of methods. One common way is to complete the square. This method involves rewriting the quadratic equation in vertex form, which makes the vertex coordinates pop right out. The vertex form of a parabola equation is $y = a(x - h)^2 + k$, where (h, k) represents the coordinates of the vertex. Alternatively, we can use the formula $x = -b / 2a$ to find the x-coordinate of the vertex, where a and b are coefficients from the standard form of the quadratic equation ($y = ax^2 + bx + c$). Once we have the x-coordinate, we can plug it back into the original equation to find the corresponding y-coordinate. Understanding the initial parabola’s vertex is the first step in determining how the shift affects the equation. The vertex provides a reference point, allowing us to calculate the horizontal and vertical translations required to move the parabola to its new position. Without knowing the starting point, it's impossible to accurately determine the shift vector and the resulting equation. Additionally, knowing the vertex helps in visualizing the graph of the parabola, making it easier to understand the impact of the transformation. This initial analysis sets the stage for the subsequent steps in solving the problem.

Finding the Original Vertex

To find the original vertex of the parabola $y = x^2 - 2x + 1$, let's use both methods I mentioned earlier: completing the square and using the vertex formula. This will ensure we fully understand the process and can double-check our results. First, let's complete the square. We start with $y = x^2 - 2x + 1$. Notice that the expression $x^2 - 2x + 1$ is already a perfect square! It can be factored as $(x - 1)^2$. So, the equation can be rewritten as $y = (x - 1)^2 + 0$. Now, comparing this to the vertex form $y = a(x - h)^2 + k$, we can see that $h = 1$ and $k = 0$. Therefore, the vertex is at the point (1, 0). Now, let's use the vertex formula as a cross-check. For a quadratic equation in the form $y = ax^2 + bx + c$, the x-coordinate of the vertex is given by $x = -b / 2a$. In our equation, $y = x^2 - 2x + 1$, we have $a = 1$ and $b = -2$. Plugging these values into the formula, we get $x = -(-2) / (2 * 1) = 2 / 2 = 1$. To find the y-coordinate, we substitute $x = 1$ back into the original equation: $y = (1)^2 - 2(1) + 1 = 1 - 2 + 1 = 0$. So, the vertex is indeed at the point (1, 0), confirming our result from completing the square. Finding the original vertex is a crucial step because it serves as the reference point for determining the shift. We know the parabola initially has its vertex at (1, 0), and we want to move it to (1, 2). The difference between these two points will give us the exact translation needed. Without this initial vertex, we wouldn't be able to calculate the shift accurately. It’s like trying to give directions without knowing the starting location – you need a point of reference to guide the way. Identifying the original vertex is the foundation for understanding the transformation the parabola undergoes.

Determining the Shift

Now that we know the original vertex is at (1, 0) and the new vertex is at (1, 2), we can figure out the shift. The shift is simply the vector that describes how we moved the parabola's vertex from its original position to its new position. To find this shift, we look at the difference in the x-coordinates and the difference in the y-coordinates. The x-coordinate of the vertex remained the same (1 in both cases), so there's no horizontal shift. However, the y-coordinate changed from 0 to 2, indicating a vertical shift. To calculate the vertical shift, we subtract the original y-coordinate from the new y-coordinate: 2 - 0 = 2. This means the parabola has been shifted 2 units upward. We can represent this shift as a vector (0, 2), where 0 represents the horizontal shift and 2 represents the vertical shift. Understanding this shift is key to finding the new equation of the parabola. The shift directly affects how we modify the original equation to reflect the new position of the parabola in the coordinate plane. The shift vector (0, 2) tells us exactly how the parabola has been translated, and this information is used to adjust the equation accordingly. It's like having a blueprint for the transformation – the shift vector guides us in redrawing the parabola in its new location. Without correctly determining the shift, we would be unable to accurately write the equation of the transformed parabola. The shift is the bridge between the original parabola and its shifted counterpart.

Calculating the Shift Vector

As we discussed, the shift vector tells us exactly how the parabola has moved. To calculate the shift vector, we compare the coordinates of the original vertex (1, 0) and the new vertex (1, 2). The shift vector is found by subtracting the coordinates of the original vertex from the coordinates of the new vertex. Let's break this down. For the x-coordinate, we subtract the original x-coordinate (1) from the new x-coordinate (1): 1 - 1 = 0. This means there is no horizontal shift. For the y-coordinate, we subtract the original y-coordinate (0) from the new y-coordinate (2): 2 - 0 = 2. This tells us the parabola has been shifted 2 units upward. Combining these, we get the shift vector (0, 2). This vector represents the translation applied to the parabola. A shift vector of (0, 2) indicates a pure vertical translation, meaning the parabola has moved directly upwards without any horizontal movement. This simplifies the process of finding the new equation, as we only need to adjust the y-component of the equation. The shift vector acts as a precise instruction for how to move the parabola. It encapsulates both the direction and the magnitude of the translation, making it an essential piece of information for transforming the parabola. Without the shift vector, we would only have partial information about the transformation. It’s like trying to solve a puzzle with missing pieces – the shift vector provides the crucial information needed to complete the picture. It is the key to unlocking the transformation and accurately determining the new equation.

Finding the New Equation

Now that we know the shift vector is (0, 2), we can find the equation of the shifted parabola. Remember, the original equation was $y = x^2 - 2x + 1$, which we simplified to $y = (x - 1)^2$. Since the shift is a vertical translation of 2 units upwards, we need to adjust the y-component of the equation. A vertical shift of k units is represented by adding k to the entire equation. In our case, k = 2. So, we add 2 to the original equation: $y = (x - 1)^2 + 2$. This is the equation of the shifted parabola. We can also expand this equation to the standard form to verify our answer. Expanding $(x - 1)^2$, we get $x^2 - 2x + 1$. Adding 2 to this gives us $y = x^2 - 2x + 1 + 2$, which simplifies to $y = x^2 - 2x + 3$. Both forms, $y = (x - 1)^2 + 2$ and $y = x^2 - 2x + 3$, represent the same shifted parabola. The first form, the vertex form, immediately shows us the vertex at (1, 2), which is what we expected. The new equation reflects the transformation of the original parabola. It incorporates the shift we calculated, accurately positioning the parabola in its new location. The fact that we can express the new equation in both vertex form and standard form provides flexibility and allows us to check our work. Finding the new equation is the culmination of the entire process. It demonstrates our understanding of how transformations affect the equation of a parabola. This ability to transform equations is a powerful tool in mathematics, enabling us to analyze and manipulate graphs with precision.

Verifying the New Equation

To verify that our new equation $y = (x - 1)^2 + 2$ (or $y = x^2 - 2x + 3$) is correct, we can check that the vertex is indeed at (1, 2) and that the shape of the parabola remains the same. Let's start with the vertex form, $y = (x - 1)^2 + 2$. As we mentioned before, the vertex form $y = a(x - h)^2 + k$ directly tells us the vertex coordinates (h, k). In this case, $h = 1$ and $k = 2$, so the vertex is at (1, 2), which matches the given information. Now, let's consider the standard form, $y = x^2 - 2x + 3$. We can use the vertex formula $x = -b / 2a$ to find the x-coordinate of the vertex. Here, $a = 1$ and $b = -2$, so $x = -(-2) / (2 * 1) = 1$. To find the y-coordinate, we substitute $x = 1$ into the equation: $y = (1)^2 - 2(1) + 3 = 1 - 2 + 3 = 2$. Again, this confirms that the vertex is at (1, 2). Another way to verify is to compare the coefficient of the $x^2$ term in the original and new equations. In both equations, the coefficient is 1. This means that the parabola's shape (its width and direction of opening) has not changed, only its position. Verifying the new equation is a crucial step to ensure that we have correctly applied the transformation. It’s like proofreading a document – we want to catch any errors before finalizing our work. By checking the vertex and the shape of the parabola, we can confidently say that our new equation accurately represents the shifted parabola. This step reinforces our understanding of the concepts involved and builds our confidence in our solution. It’s the final piece of the puzzle, solidifying our grasp on parabola transformations.

Conclusion

In summary, we've successfully determined the shift and found the new equation of the parabola. We started with the equation $y = x^2 - 2x + 1$, found its original vertex at (1, 0), and then calculated the shift vector (0, 2) needed to move the vertex to (1, 2). Finally, we applied this shift to the original equation to obtain the new equation, $y = (x - 1)^2 + 2$ or $y = x^2 - 2x + 3$. This process demonstrates a fundamental concept in mathematics: transforming graphs by shifting them. Understanding how to shift parabolas and other functions is a valuable skill in algebra and calculus. It allows us to analyze and manipulate equations to fit different scenarios. The key steps involve identifying the original vertex, determining the shift vector, and applying the shift to the equation. By following these steps, you can confidently handle any parabola shifting problem. Remember, math is like building with blocks – each concept builds upon the previous one. Mastering transformations like this one opens the door to understanding more complex mathematical ideas in the future. So, keep practicing and keep exploring the fascinating world of mathematics! This exercise has provided a comprehensive understanding of how to shift a parabola and find its new equation. The ability to perform transformations on graphs is essential in many areas of mathematics and its applications.