How is the graph of $y=-\sqrt[3]{x-4}$ transformed to produce the graph of $y=-\sqrt[3]{2 x}-4$?
A. The graph is stretched horizontally by a factor of 2 and then moved right 4 units.
B. The graph is compressed horizontally by a factor of 2 and then moved down 4 units.
C. The graph is compressed horizontally by a factor of 2, moved left 4 units, and moved down 4 units.
D. The graph is stretched horizontally by a factor of 2, moved left 4 units, and moved down 4 units.
Answer (2)
The graph is compressed horizontally by a factor of 2 1 due to the 2 x inside the cube root.
There is a vertical shift down by 4 units due to the − 4 outside the cube root.
Therefore, the graph is compressed horizontally by a factor of 2 and then moved down 4 units.
The final answer is: The graph is compressed horizontally by a factor of 2 and then moved down 4 units. $\boxed{The graph is compressed horizontally by a factor of 2 and then moved down 4 units.}
Explanation
Understanding the Problem We are given the original function y = − 3 x − 4 and the transformed function y = − 3 2 x − 4 . We want to describe the transformation from the original to the transformed function.
Analyzing Horizontal Transformations First, let's analyze the horizontal transformations. The original function has a horizontal shift of 4 units to the right, represented by x − 4 inside the cube root. The transformed function has 2 x inside the cube root. This indicates a horizontal compression by a factor of 2 1 .
Analyzing Vertical Transformations Next, let's analyze the vertical transformations. The original function has no vertical shift. The transformed function has − 4 outside the cube root, which represents a vertical shift down by 4 units.
Conclusion Therefore, the graph of y = − 3 x − 4 is transformed to y = − 3 2 x − 4 by a horizontal compression by a factor of 2 1 and a shift down by 4 units.
Examples
Imagine you are stretching or compressing a spring. Compressing the graph horizontally is like squeezing the spring, making it shorter. Shifting the graph down is like pulling the spring downwards. Understanding these transformations helps in various fields like physics, engineering, and computer graphics, where manipulating functions and graphs is essential for modeling real-world phenomena.
The transformation involves a horizontal compression by a factor of 2 because of the 2 x inside the cube root and a vertical shift down by 4 units due to the − 4 outside the cube root. Therefore, the selected answer is option B.
;
