Describe the transformations from 𝑓(𝑥) to 𝑔(𝑥)
where:
𝑓(𝑥) = 𝑥^4 and 𝑔(𝑥) =1/3(𝑥 − 2)^4 + 7

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.

Describe the transformations from 𝑓(𝑥) to 𝑔(𝑥) where 𝑓(𝑥) = 𝑥^4 and 𝑔(𝑥) = 1/3(𝑥 − 2)^4 + 7.

Explore the step-by-step transformations from f(x) = x^4 to g(x) = 1/3(x - 2)^4 + 7. Learn about horizontal shifts, vertical compressions, and vertical translations. Perfect for students in Grades 10-12 studying function transformations and polynomial graphs.

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