Which of the following transforms $y=x^2$ to the graph of $y=(x+5)^2$?
A. a translation 5 units to the right
B. a translation 5 units to the left
C. a translation 5 units down
D. a translation 5 units up

The transformation that maps the graph of y = x 2 to y = ( x + 5 ) 2 is a translation 5 units to the left. Therefore, the correct answer is option B. This is due to the replacement of x with ( x + 5 ) , indicating a shift to the left.
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Answered by Anonymous | 2025-07-08

The original function is y = x 2 .
The transformed function is y = ( x + 5 ) 2 .
Replacing x with ( x + 5 ) represents a horizontal translation.
The graph translates 5 units to the left: a translation 5 units to the left ​ .

Explanation

Understanding the Problem We are given the original function y = x 2 and the transformed function y = ( x + 5 ) 2 . We need to determine the transformation that maps the original function to the transformed function.

Identifying the Transformation The transformation involves replacing x with ( x + 5 ) in the original function. This means we are looking at a horizontal translation.

Determining the Direction and Magnitude of Translation Recall that replacing x with ( x + a ) in a function results in a horizontal translation of ∣ a ∣ units. If 0"> a > 0 , the translation is to the left, and if a < 0 , the translation is to the right. In our case, x is replaced with ( x + 5 ) , so a = 5 .

Concluding the Transformation Since 0"> a = 5 > 0 , the transformation is a horizontal translation of 5 units to the left. Therefore, the graph of y = x 2 is translated 5 units to the left to obtain the graph of y = ( x + 5 ) 2 .


Examples
Consider a satellite dish. If we model the cross-section of the dish as a parabola y = x 2 , shifting the parabola horizontally, like in this problem, corresponds to physically repositioning the dish. Understanding these transformations is crucial in ensuring the signal is focused correctly. For example, moving the parabola to y = ( x + 5 ) 2 means the vertex of the parabola is shifted, which affects the focal point of the dish.

Answered by GinnyAnswer | 2025-07-08