To describe the transformations from $f(x) = x^4$ to $g(x) = \frac{1}{3}(x - 2)^4 + 7$, we can analyze the changes step by step. Here’s how $f(x)$ is transformed into $g(x)$:

1. Horizontal Shift:

• The function is shifted to the right by 2 units.
• This is represented by the $(x - 2)^4$ term. In general, replacing $x$ with $x - h$ shifts the graph of the function to the right by $h$ units (here, $h = 2$).

2. Vertical Stretch:

• The function is vertically compressed by a factor of $\frac{1}{3}$.
• The $\frac{1}{3}$ coefficient in front of $(x - 2)^4$ causes the graph to stretch vertically. This reduces the steepness of the curve.

3. Vertical Shift:

• The function is shifted upwards by 7 units.
• The $+7$ outside the parentheses moves the entire graph up by 7 units.

Summary of Transformations:

• Shift right by 2 units (horizontal translation).
• Vertical compression by a factor of $\frac{1}{3}$.
• Shift upward by 7 units (vertical translation).

Combined Effect:

The graph of $f(x) = x^4$ is first shifted 2 units to the right, then compressed vertically by a factor of $\frac{1}{3}$, and finally shifted upward by 7 units to become $g(x) = \frac{1}{3}(x - 2)^4 + 7$.

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Would you like more details or have any questions?

Here are 5 related questions you might find useful:

1. How does a vertical reflection affect a function like $f(x) = x^4$?
2. What happens if the horizontal shift in $g(x)$ is negative?
3. How would a horizontal compression affect $g(x)$?
4. Can you describe the effect of combining horizontal and vertical transformations on $f(x) = x^4$?
5. How does the degree of the polynomial function affect its shape?

Tip: Horizontal shifts inside parentheses change the $x$-values, while vertical shifts outside affect the $y$-values directly.