Introduction:
In the realm of mathematics, understanding how different transformations affect functions and their graphs is essential. Translating a function involves shifting it horizontally, vertically, or both, resulting in a new graph with distinct characteristics. In this article, we delve into the translation from the graph y = 6x^2 to y = 6(x + 1)^2, aiming to identify the phrase that best describes this transformation and its implications on the graph.
Understanding the Translation:
To analyze the translation from y = 6x^2 to y = 6(x + 1)^2, we need to focus on the effect of the term “(x + 1)” within the equation. This term represents a horizontal translation, specifically a shift to the left by one unit. It indicates that every x-value in the original function is decreased by one in the translated function.
Describing the Translation Phrase:
- Left Shift/Horizontal Translation: The most accurate phrase to describe the transformation in this case is a “left shift” or “horizontal translation to the left by one unit.” This phrase aptly conveys the specific nature of the transformation, indicating that the graph has been horizontally moved towards the left by a distance of one unit.
Implications on the Graph:
Understanding the implications of this translation on the graph is crucial to fully grasp the transformation. Shifting the graph one unit to the left means that every point on the original graph is now situated to the left by one unit in the translated graph.
For example, the vertex of the original graph at (0, 0) will now be located at (-1, 0) in the translated graph. Similarly, any point on the original graph that had an x-coordinate of x will have an x-coordinate of (x – 1) in the translated graph.
Overall, the translation to the left by one unit results in a horizontal shift of the graph, altering the location of each point on the x-axis.
Conclusion:
The translation from y = 6x^2 to y = 6(x + 1)^2 involves a horizontal shift to the left by one unit. This transformation can be accurately described as a “left shift” or “horizontal translation to the left by one unit.” Understanding the implications of this translation helps visualize how each point on the graph has shifted horizontally. By recognizing and comprehending such transformations, we gain a deeper understanding of how mathematical functions behave and how different equations can result in distinct graphical representations.
