Given the base functionof x^4. Describe (in point form) how each parameter of the function y= - 1/2f(1/3x+2)-6 transforms the graph of f(x)
To describe how each parameter of the function y = -1/2f(1/3x + 2) - 6 transforms the graph of f(x) = x^4, we will break it down into different transformations:
1. Vertical Stretch Factor (-1/2): This parameter vertically compresses the graph of f(x) by a factor of -1/2. It means that any point (x, y) on the original graph will be moved to (x, -1/2y) on the transformed graph. In other words, the y-values are halved, causing the graph to become flatter vertically.
2. Horizontal Compression Factor (1/3): This parameter horizontally expands the graph of f(x) by a factor of 3. It means that any point (x, y) on the original graph will be moved to (3x, y) on the transformed graph. In other words, the x-values are multiplied by 3, causing the graph to become narrower horizontally.
3. Horizontal Translation (2): This parameter shifts the graph of f(x) 2 units to the left. It means that any point (x, y) on the original graph will be moved to (x - 2, y) on the transformed graph. In other words, all x-values are decreased by 2, resulting in a leftward shift of the graph.
4. Vertical Translation (-6): This parameter shifts the graph of f(x) 6 units downward. It means that any point (x, y) on the original graph will be moved to (x, y - 6) on the transformed graph. In other words, all y-values are decreased by 6, causing the graph to shift downwards.
Combining all these transformations together, the overall effect on the graph of f(x) = x^4 can be summarized as follows:
- The graph is vertically compressed by a factor of -1/2.
- The graph is horizontally expanded by a factor of 3.
- The graph is shifted 2 units to the left.
- The graph is shifted 6 units downward.