which phrase best describes the translation from the graph y = (x + 2)2 to the graph of y = x2 + 3?

2025

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29-03-2025

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The transformation from the graph of y = (x + 2)² to the graph of y = x² + 3 involves a combination of shifts, specifically a horizontal and a vertical shift. Understanding these shifts is key to accurately describing the transformation. Let's break it down step-by-step.

Understanding the Transformations

To understand the translation, let's analyze each function individually.

y = (x + 2)²

This parabola is a horizontal shift of the parent function y = x². The "+2" inside the parentheses indicates a shift of 2 units to the left.

y = x² + 3

This parabola is a vertical shift of the parent function y = x². The "+3" outside the parentheses indicates a shift of 3 units upwards.

Visualizing the Transformation

Imagine the graph of y = (x+2)². To transform it into y = x² + 3, we need to perform two actions:

1. Horizontal Shift: Move the graph 2 units to the right. This cancels out the initial leftward shift of 2 units from y = (x+2)². The equation would become y = (x)²

2. Vertical Shift: Move the resulting graph 3 units upwards. This adds the vertical shift from y = x² + 3.

Describing the Transformation

Therefore, the most accurate phrase to describe the translation from y = (x + 2)² to y = x² + 3 is a combination of horizontal and vertical shifts:

• A horizontal shift of 2 units to the right, followed by a vertical shift of 3 units upward.

This description fully captures the sequence of transformations needed to map one graph onto the other. Other descriptions, such as "a translation of 2 units to the right and 3 units up" are acceptable. However, explicitly stating the order (horizontal then vertical) is more precise for mathematical clarity.

Alternative Descriptions to Avoid

While the above description is accurate, avoid phrases that might be misleading:

• "A translation of 5 units": This implies a single diagonal shift.

• "A shift of 2 units right and 3 units up (simultaneously)": This, while seemingly correct, doesn't reflect the accurate order of operations in transforming the graph. The order matters!

Conclusion

The transformation from y = (x + 2)² to y = x² + 3 is best described as a horizontal shift of 2 units to the right followed by a vertical shift of 3 units upward. This accurately reflects the process and avoids ambiguity. Understanding the individual components of the transformations is crucial for correctly identifying the overall translation.