Select the correct answer. The parent square root function, f, is transformed to create function g. [tex]g(x) = -\sqrt{x+2} + 1[/tex] Which statement describes the transformation? A. The graph of f is translated 1 unit to the right and 2 units down. B. The graph of f is translated 2 units to the right and 1 unit up. C. The graph of f is translated 1 unit to the left and 2 units up. D. The graph of f is translated 2 units to the left and 1 unit up.
Answer (2)
The parent function f ( x ) = x is transformed to g ( x ) = − x + 2 + 1 .
The term ( x + 2 ) indicates a horizontal translation of 2 units to the left.
The negative sign indicates a reflection across the x-axis.
The + 1 indicates a vertical translation of 1 unit up.
Therefore, the graph of f is translated 2 units to the left and 1 unit up. The correct answer is D .
Explanation
Understanding the Problem We are given the parent square root function f ( x ) = x and a transformed function g ( x ) = − x + 2 + 1 . We need to describe the transformation from f to g .
Horizontal Translation The transformation involves several steps. First, let's consider the term inside the square root, ( x + 2 ) . This represents a horizontal translation. Since we have ( x + 2 ) , the graph is translated 2 units to the left.
Reflection Next, we have a negative sign in front of the square root, which means the graph is reflected across the x-axis.
Vertical Translation Finally, we have + 1 outside the square root, which represents a vertical translation. Since it is + 1 , the graph is translated 1 unit up.
Overall Transformation Combining these transformations, we have a translation 2 units to the left, a reflection across the x-axis, and a translation 1 unit up.
Matching the Description with Options Now, let's match this description with the given options. Option D states: 'The graph of f is translated 2 units to the left and 1 unit up.' This matches our description, except for the reflection across the x-axis. However, since the question only asks about translations, we can consider the vertical translation of 1 unit up as the relevant part of the transformation.
Final Answer Therefore, the correct answer is D.
Examples
Understanding transformations of functions is crucial in many fields. For example, in physics, understanding how graphs of motion change with different initial conditions or forces is essential. Similarly, in economics, understanding how supply and demand curves shift with changes in market conditions is vital for making informed decisions. Transformations help us model and predict these changes effectively. For instance, if we know the basic function representing the trajectory of a projectile, we can use transformations to account for factors like wind resistance or changes in launch angle, allowing us to accurately predict where the projectile will land. This is done by applying translations, reflections, and stretches to the original function.
The transformation of the function g(x) involves moving the graph 2 units to the left and 1 unit up from the parent function f(x). The negative sign indicates a reflection, but the focus of the question is on translations. Therefore, the correct answer is option D.
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