Describe the translation in $y=x^2-3$.

The function y = x 2 − 3 represents a vertical translation of the parent function y = x 2 that shifts it down by 3 units. This downward shift occurs because of the subtraction of 3 from the function. Thus, the graph of the quadratic function is translated downwards vertically by 3 units.
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Answered by Anonymous | 2025-07-04

Identify the parent function: y = x 2 .
Compare the given function y = x 2 − 3 to the parent function.
Recognize the transformation as a vertical translation.
Describe the translation as a shift of 3 units downward: Vertical translation 3 units downward ​ .

Explanation

Understanding the Problem We are given the equation y = x 2 − 3 and asked to describe the translation from the parent function y = x 2 .

Identifying the Transformation The given equation is a quadratic function. We need to identify how the graph of y = x 2 is transformed to obtain the graph of y = x 2 − 3 .

Determining the Translation Comparing y = x 2 − 3 to the parent function y = x 2 , we see that the only difference is the subtraction of 3. This indicates a vertical translation.

Describing the Translation Since the equation is in the form y = f ( x ) − c , where c = 3 , the graph of y = x 2 is translated downward by 3 units.

Conclusion Therefore, the translation is a vertical shift of 3 units downward.


Examples
Understanding translations of functions is crucial in various fields. For example, in physics, describing the motion of a projectile involves understanding how its position changes over time. If the initial height of the projectile is altered, the entire trajectory is shifted vertically. Similarly, in economics, understanding how cost functions shift due to changes in fixed costs is essential for making informed business decisions. The translation of functions helps in modeling and predicting these real-world phenomena.

Answered by GinnyAnswer | 2025-07-04