What are the horizontal shift and vertical shift for the function below? Use complete sentences and explain your reasoning.
y = √x + 2 - 3
Answer (2)
The function is y = x + 2 − 3 .
The horizontal shift is determined by the term inside the square root: x + 2 which means a shift of 2 units to the left.
The vertical shift is determined by the constant term: − 3 which means a shift of 3 units downward.
The horizontal shift is 2 units to the left and the vertical shift is 3 units downward, so the answer is Horizontal shift: 2 units left, Vertical shift: 3 units down .
Explanation
Understanding the Problem The given function is y = x + 2 − 3 . We need to identify the horizontal and vertical shifts of this function compared to the base function y = x .
Understanding Shifts To understand the shifts, we compare the given function to the base function y = x . The general form of a square root function with horizontal and vertical shifts is y = x − h + k , where h represents the horizontal shift and k represents the vertical shift.
Horizontal Shift In our function, y = x + 2 − 3 , we can rewrite x + 2 as x − ( − 2 ) . Thus, h = − 2 . This means the horizontal shift is 2 units to the left.
Vertical Shift The constant term outside the square root is − 3 , so k = − 3 . This means the vertical shift is 3 units downward.
Conclusion Therefore, the horizontal shift is 2 units to the left, and the vertical shift is 3 units downward.
Examples
Understanding horizontal and vertical shifts is crucial in various real-world applications. For instance, in physics, when analyzing projectile motion, shifts help model the trajectory of an object launched from a height or at an angle. In economics, shifts in supply and demand curves can be visualized and understood using these concepts, aiding in predicting market changes. Similarly, in computer graphics, transformations like translation (shifting) are fundamental in positioning objects on the screen. These shifts allow us to adapt and modify base functions to fit specific scenarios, providing a powerful tool for modeling and analysis.
The function y = x + 2 − 3 has a horizontal shift of 2 units to the left and a vertical shift of 3 units downward. These shifts indicate how the graph of the function is translated from the basic square root function. Understanding these shifts is essential for visualizing the graph accurately.
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