
describe how the graph of each of these functions is related to the graph of y=f(x) find the stretch, reflection, and translation (if there are any of these present) y= f(2x-2)

To describe how the graph of \( y = f(2x - 2) \) is related to the graph of \( y = f(x) \), we need to break down the transformation by considering the effects of the stretch, reflection, and translation on the graph of \( y = f(x) \).

### Step 1: Factor the inside of the function
The function \( y = f(2x - 2) \) can be rewritten to make transformations more obvious by factoring the argument of the function as follows:

\[
y = f(2(x - 1))
\]

Now, we can see the transformations clearly.

### Step 2: Identify the transformations
1. **Horizontal Compression (Stretch)**:
   - The factor of \( 2 \) inside the function affects the **horizontal scaling** of the graph. 
   - Specifically, the graph is **compressed horizontally by a factor of 2**. This means that every \( x \)-coordinate of the points on the graph of \( y = f(x) \) is divided by 2 (or the graph appears to be stretched towards the \( y \)-axis).

2. **Horizontal Translation (Shift)**:
   - The term \( (x - 1) \) indicates a **horizontal shift** to the right by 1 unit. 
   - This is because \( x - 1 \) means we are moving every point on the graph of \( f(x) \) 1 unit to the right.

### Summary of Transformations:
- **Horizontal Compression by a factor of 2** (the graph gets narrower).
- **Shift right by 1 unit**.

### There is no reflection or vertical translation present in this transformation.

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Would you like a detailed explanation or have any questions about these transformations?

Here are 5 related questions to expand on the concept:
1. How would the graph of \( y = f(x - 3) \) be related to the graph of \( y = f(x) \)?
2. What happens if the function is \( y = f(-x) \)? How does this affect the graph?
3. How would the graph change if the transformation was \( y = f(\frac{1}{2}x) \)?
4. What effect does a vertical stretch or compression have on a function?
5. What happens if the function is \( y = f(2x) + 3 \)?

**Tip**: Always consider horizontal transformations (inside the function) before vertical transformations (outside the function).

describe how the graph of each of these functions is related to the graph of y=f(x) find the stretch, reflection, and translation (if there are any of these present) y= f(2x-2)

Explore how the graph of y=f(2x-2) is transformed from the graph of y=f(x). This problem covers horizontal compression, translation, and other key function transformations. Ideal for students in Grades 9-11 studying Algebra.

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