To determine the graph of the transformed function [tex]\( g(x) = -2 f(x) \)[/tex] from the parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex], we'll need to understand how the transformations affect the graph.
1. Parent Function Analysis:
The parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex] is a cube root function. It has the following characteristics:
- It passes through the origin [tex]\((0, 0)\)[/tex].
- It is an increasing function.
- It is symmetric with respect to the origin (odd function).
2. Applying the Transformation [tex]\( g(x) = -2 \sqrt[3]{x} \)[/tex]:
The function [tex]\( g(x) = -2 f(x) \)[/tex] applies two transformations to the parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex]:
- Vertical Stretch by a Factor of 2: This increases the distance of all points from the x-axis by a factor of 2. So, if a point [tex]\( (x, y) \)[/tex] is on the graph of [tex]\( \sqrt[3]{x} \)[/tex], it will move to [tex]\( (x, 2y) \)[/tex] on the graph of [tex]\( 2 \sqrt[3]{x} \)[/tex].
- Reflection Across the x-axis: The negative sign in front of [tex]\( 2 f(x) \)[/tex] reflects the graph across the x-axis. Hence, the transformed point from the vertical stretch [tex]\( (x, 2y) \)[/tex] moves to [tex]\( (x, -2y) \)[/tex].
3. Visualizing the Transformations Step-by-Step:
- Start with [tex]\( f(x) = \sqrt[3]{x} \)[/tex].
- Apply the vertical stretch to get [tex]\( 2 \sqrt[3]{x} \)[/tex].
- Apply the reflection to get [tex]\( g(x) = -2 \sqrt[3]{x} \)[/tex].
4. Characteristics of [tex]\( g(x) \)[/tex]:
- It will still pass through the origin [tex]\((0, 0)\)[/tex] because neither stretching nor reflecting affects this point.
- Due to the reflection, the graph that was increasing in [tex]\( f(x) \)[/tex] will become decreasing in [tex]\( g(x) \)[/tex].
Conclusion:
The graph of the function [tex]\( g(x) = -2 \sqrt[3]{x} \)[/tex] will resemble the graph of [tex]\( f(x) = \sqrt[3]{x} \)[/tex], but it will be stretched vertically by a factor of 2 and reflected over the x-axis, causing the previously increasing graph to now be decreasing.
Given the options A and B (without visuals provided), you would select the graph that shows:
- The origin [tex]\((0, 0)\)[/tex] as a point.
- A decrease in [tex]\( y \)[/tex] values as [tex]\( x \)[/tex] moves from left to right (indicating the reflection and stretching).
Since we don't have visuals to refer to here, the hint is to look for these described properties in the options provided.
