Which of the following describes the graph of $y=\sqrt[3]{8 x-64}-5$ compared to the parent cube root function?
A. stretched by a factor of 2 and translated 64 units right and 5 units down
B. stretched by a factor of 8 and translated 8 units right and 5 units down
C. stretched by a factor of 2 and translated 8 units right and 5 units down
D. stretched by a factor of 8 and translated 64 units right and 5 units down
Answer (1)
Rewrite the function: y = 3 8 x − 64 − 5 = 3 8 ( x − 8 ) − 5 = 2 3 x − 8 − 5 .
Identify the vertical stretch: The factor of 2 stretches the graph vertically by a factor of 2.
Identify the horizontal translation: The ( x − 8 ) term translates the graph 8 units to the right.
Identify the vertical translation: The − 5 term translates the graph 5 units down. Therefore, the graph is stretched by a factor of 2 and translated 8 units right and 5 units down. stretched by a factor of 2 and translated 8 units right and 5 units down .
Explanation
Understanding the Problem We are given the function y = 3 8 x − 64 − 5 and we want to describe how its graph compares to the parent cube root function y = 3 x . The possible transformations are stretches and translations.
Rewriting the Function First, we rewrite the given function to make the transformations more apparent. We can factor out an 8 from the expression inside the cube root: y = 3 8 ( x − 8 ) − 5
Separating the Cube Root Next, we can use the property n ab = n a n b to separate the cube root: y = 3 8 3 x − 8 − 5
Simplifying the Function Since 3 8 = 2 , we can simplify the function to: y = 2 3 x − 8 − 5
Identifying the Transformations Now we can identify the transformations. The factor of 2 in front of the cube root represents a vertical stretch by a factor of 2. The ( x − 8 ) inside the cube root represents a horizontal translation 8 units to the right. The − 5 represents a vertical translation 5 units down.
Conclusion Therefore, the graph of y = 3 8 x − 64 − 5 is stretched by a factor of 2 and translated 8 units right and 5 units down.
Examples
Understanding transformations of functions is crucial in many fields. For example, in physics, understanding how graphs of motion change with different parameters (like initial velocity or acceleration) helps predict the behavior of objects. In engineering, transformations are used to design and analyze systems, such as signal processing where shifting and scaling signals are common operations. Even in economics, understanding how supply and demand curves shift due to external factors can be seen as a transformation of functions.
