Transformations Of Quadratic Functions Describing The Translation From Y=(x-5)^2+7 To Y=(x+1)^2-2

This article delves into the intricate world of quadratic function transformations, specifically focusing on horizontal and vertical shifts. We will dissect the given problem: describing the translation from the graph of $y=(x-5)^2+7$ to the graph of $y=(x+1)^2-2$. This problem is a classic example of how understanding the vertex form of a quadratic equation can unlock insights into graphical transformations. By carefully examining the changes in the vertex coordinates, we can accurately determine the magnitude and direction of the translation.

Deconstructing Quadratic Functions in Vertex Form

Before we dive into the specific problem, let's first establish a firm understanding of the vertex form of a quadratic equation. The general form is given by $y = a(x-h)^2 + k$, where $(h, k)$ represents the vertex of the parabola. The vertex is a crucial point because it signifies the minimum or maximum value of the quadratic function, depending on the sign of the coefficient 'a'. In our case, both quadratic equations have $a = 1$, indicating that both parabolas open upwards, and their vertices represent the minimum points.

The beauty of the vertex form lies in its direct representation of transformations. The 'h' value dictates the horizontal shift, and the 'k' value dictates the vertical shift. It's essential to note that the horizontal shift is counterintuitive: a positive 'h' value shifts the parabola to the right, while a negative 'h' value shifts it to the left. The vertical shift is more straightforward: a positive 'k' value shifts the parabola upwards, and a negative 'k' value shifts it downwards. This understanding is the cornerstone for solving our translation problem.

To further illustrate this, let's consider a simple example. Suppose we have the basic parabola $y = x^2$. Its vertex is at the origin (0, 0). Now, let's transform it into $y = (x-2)^2 + 3$. The vertex has shifted from (0, 0) to (2, 3). This means the parabola has been translated 2 units to the right and 3 units upwards. Conversely, if we transform $y = x^2$ into $y = (x+1)^2 - 4$, the vertex shifts to (-1, -4), indicating a translation of 1 unit to the left and 4 units downwards. These examples highlight the direct relationship between the vertex form and the graphical transformations, which will be instrumental in solving our main problem.

Identifying the Vertices of the Given Parabolas

Now, let's apply our knowledge to the given problem. We have two quadratic equations: $y = (x-5)^2 + 7$ and $y = (x+1)^2 - 2$. Our first task is to identify the vertices of these parabolas. By directly comparing these equations with the vertex form $y = a(x-h)^2 + k$, we can easily extract the vertex coordinates.

For the first equation, $y = (x-5)^2 + 7$, we can see that $h = 5$ and $k = 7$. Therefore, the vertex of this parabola is (5, 7). This point represents the minimum value of the function and serves as our starting point for the translation.

Similarly, for the second equation, $y = (x+1)^2 - 2$, we need to be careful with the sign of 'h'. Since the equation is in the form $(x+1)$, it can be rewritten as $(x - (-1))$. Thus, $h = -1$ and $k = -2$. The vertex of this parabola is (-1, -2). This is our destination point after the translation.

Having identified the vertices (5, 7) and (-1, -2), we now have a clear picture of the transformation that has occurred. We need to determine how to move from the point (5, 7) to the point (-1, -2). This involves calculating the horizontal and vertical shifts required, which will lead us to the correct answer among the given options. Visualizing these points on a coordinate plane can further enhance our understanding of the translation and help us avoid any sign errors in our calculations.

Calculating the Horizontal and Vertical Shifts

With the vertices identified as (5, 7) and (-1, -2), our next crucial step is to calculate the horizontal and vertical shifts required to move from the first vertex to the second. This calculation will directly reveal the translation that transforms the graph of $y=(x-5)^2+7$ into the graph of $y=(x+1)^2-2$.

To determine the horizontal shift, we compare the x-coordinates of the two vertices. We are moving from an x-coordinate of 5 to an x-coordinate of -1. The difference between these values is -1 - 5 = -6. This negative value indicates a shift to the left. Specifically, the graph shifts 6 units to the left.

Next, we calculate the vertical shift by comparing the y-coordinates of the vertices. We are moving from a y-coordinate of 7 to a y-coordinate of -2. The difference here is -2 - 7 = -9. This negative value signifies a shift downwards. The graph shifts 9 units down.

Therefore, we have determined that the translation consists of a shift of 6 units to the left and 9 units down. This understanding is paramount to selecting the correct answer from the given options and demonstrates a clear grasp of how changes in the vertex form of a quadratic equation manifest as graphical transformations. The ability to accurately calculate these shifts is a fundamental skill in understanding and manipulating functions.

Matching the Shifts to the Correct Option

Having meticulously calculated the horizontal and vertical shifts, we now arrive at the final step: matching our findings to the correct answer option. We determined that the graph translates 6 units to the left and 9 units down. Now, let's revisit the provided options:

A. 6 units left and 9 units down B. 6 units right and 9 units down C. 6 units left and 9 units up D. 6 units right and 9 units up

By directly comparing our calculated shifts with the options, it becomes clear that option A, "6 units left and 9 units down," perfectly aligns with our results. This confirms our understanding of how the vertex form of a quadratic equation dictates the graph's position and transformations in the coordinate plane.

Options B, C, and D can be confidently eliminated as they do not accurately represent the shifts we calculated. Option B incorrectly states a shift to the right, while options C and D misrepresent the vertical shift as upwards instead of downwards. This process of elimination reinforces the importance of precise calculations and a thorough understanding of the concepts involved.

Therefore, the correct answer is A. 6 units left and 9 units down. This conclusion underscores the significance of analyzing the vertex form of quadratic equations to decipher graphical transformations, particularly horizontal and vertical translations.

Conclusion: Mastering Quadratic Function Translations

In conclusion, this exercise has provided a comprehensive exploration of translating quadratic functions. By meticulously analyzing the vertex form of the equations, we successfully determined the horizontal and vertical shifts required to transform one graph into another. The key takeaways from this problem are:

• Understanding the vertex form, $y = a(x-h)^2 + k$, and its direct relationship to the vertex (h, k).
• Recognizing that the 'h' value dictates horizontal shifts (positive for right, negative for left) and the 'k' value dictates vertical shifts (positive for up, negative for down).
• Accurately calculating the horizontal and vertical shifts by comparing the vertices of the original and transformed graphs.
• Applying these calculations to select the correct answer option from a set of choices.

This problem serves as a valuable illustration of how a solid grasp of fundamental concepts can unlock the ability to solve complex problems in mathematics. By mastering the transformations of quadratic functions, you gain a powerful tool for analyzing and manipulating graphs, which is essential for success in algebra and beyond. The ability to confidently navigate these transformations not only enhances problem-solving skills but also fosters a deeper appreciation for the elegance and interconnectedness of mathematical concepts.

Keywords: quadratic functions, vertex form, horizontal shift, vertical shift, graphical transformations, translation, vertex, parabola, equation, mathematics