How is the graph of $y=-2(3)^x+4$ translated from the graph of $y=2(3)^x$?
A. reflected across the $x$-axis and 4 units up
B. reflected across the $y$-axis and 4 units left
C. reflected across the $y$-axis and 4 units down
D. reflected across the $x$-axis and 4 units right
Answer (2)
Reflect y = 2 ( 3 ) x across the x-axis to get y = − 2 ( 3 ) x .
Translate y = − 2 ( 3 ) x vertically by 4 units up to get y = − 2 ( 3 ) x + 4 .
The graph of y = − 2 ( 3 ) x + 4 is obtained by reflecting y = 2 ( 3 ) x across the x-axis and translating it 4 units up.
Therefore, the answer is reflected across the x -axis and 4 units up. reflected across the x -axis and 4 units up
Explanation
Understanding the Problem We are given two functions: y = − 2 ( 3 ) x + 4 and y = 2 ( 3 ) x . We want to describe the transformation from the graph of y = 2 ( 3 ) x to the graph of y = − 2 ( 3 ) x + 4 .
Reflection First, consider the transformation from y = 2 ( 3 ) x to y = − 2 ( 3 ) x . This is a reflection across the x-axis.
Vertical Translation Next, consider the transformation from y = − 2 ( 3 ) x to y = − 2 ( 3 ) x + 4 . This is a vertical translation by 4 units.
Conclusion Therefore, the graph of y = − 2 ( 3 ) x + 4 is obtained from the graph of y = 2 ( 3 ) x by a reflection across the x-axis, followed by a vertical translation of 4 units up.
Examples
Imagine you are designing a roller coaster. The function y = 2 ( 3 ) x represents the initial climb, and y = − 2 ( 3 ) x + 4 represents the subsequent drop and adjustment. The reflection across the x-axis simulates the change in direction from climbing to dropping, and the vertical translation of 4 units up ensures the coaster ends at a safe height. Understanding transformations of functions helps engineers design safe and exciting rides.
The graph of y = − 2 ( 3 ) x + 4 is obtained from the graph of y = 2 ( 3 ) x by reflecting it across the x-axis and then translating it 4 units up. The answer is option A: reflected across the x-axis and 4 units up.
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