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Polynomial transformations are changes made to the basic form of a polynomial function to produce a new graph. For the function given in our exercise, the base form is \( y = x^{4} \). The transformations change this basic polynomial graph in specific ways: horizontal shifts, vertical shifts, scaling, and reflections. Transformations can make the graphs look very different from their original shape.
In the given example, the function is transformed with a horizontal shift and vertical scaling. The base graph \( y = x^{4} \) is modified according to the function \( f(x) = \frac{1}{3}(x+3)^{4} \). Understanding these transformations helps you graph even complex polynomial functions easily and accurately by adjusting a simpler, well-known graph.

Horizontal shifts move the graph of a function left or right. This is done by adding or subtracting a value inside the function's argument. For our function \( f(x) = \frac{1}{3}(x+3)^{4} \), we add 3 inside the function's argument. Normally, adding a positive number results in a shift to the left. Adding 3 shifts the base graph of \( y = x^{4} \) three units to the left.
Here's the transformation rule for the horizontal shift:

• Original point on \( y = x^{4} \): \( (x, y) \)
• Shifted point: \( (x-3, y) \)

Applying this rule to key points, like (0, 0), (-1, 1), and (-2, 16), helps to make accurate graphs.

Vertical scaling changes how tall or short the graph appears. It scales every y-coordinate by a specific factor. For our function \( f(x) = \frac{1}{3}(x+3)^{4} \), the factor is \( \frac{1}{3} \). Vertical scaling compresses the graph for factors between 0 and 1 and stretches for factors larger than 1.
Here’s the transformation rule:

• Original point on \( y = x^{4} \): \( (x, y) \)
• Scaled point: \( (x, \frac{y}{3}) \)

Applying this to key points after the horizontal shift gives us new coordinates. For example, the point (1, 16) becomes (1, \( \frac{16}{3}\)). It is crucial to apply vertical scaling after all horizontal shifts to maintain accuracy.