In Exercises 17-22, write a function g whose graph represents the indicated transformation of the graph off. Use a graphing calculator to check your answer

To tackle problems involving transformations of functions, let's first understand what transformations mean in this context. Transformations generally include shifts (translations), reflections, stretches, or compressions of the graph of a function \( f(x) \).

Here’s a step-by-step guide to writing a function \( g(x) \) that represents the indicated transformation of the graph of \( f(x) \):

### Common Transformations:
1. **Vertical Shift**: 
   - **Upward** by \( c \): \( g(x) = f(x) + c \)
   - **Downward** by \( c \): \( g(x) = f(x) - c \)

2. **Horizontal Shift**: 
   - **Right** by \( c \): \( g(x) = f(x - c) \)
   - **Left** by \( c \): \( g(x) = f(x + c) \)

3. **Vertical Stretch/Compression**:
   - **Stretch** by a factor of \( a \): \( g(x) = a \cdot f(x) \) (where \( a > 1 \))
   - **Compression** by a factor of \( a \): \( g(x) = a \cdot f(x) \) (where \( 0 < a < 1 \))

4. **Horizontal Stretch/Compression**:
   - **Stretch** by a factor of \( b \): \( g(x) = f\left(\frac{x}{b}\right) \) (where \( b > 1 \))
   - **Compression** by a factor of \( b \): \( g(x) = f\left(\frac{x}{b}\right) \) (where \( 0 < b < 1 \))

5. **Reflection**:
   - **Across the x-axis**: \( g(x) = -f(x) \)
   - **Across the y-axis**: \( g(x) = f(-x) \)

### Example Problem:
If you’re asked to find \( g(x) \) for a specific transformation, let’s consider a few scenarios:

1. **Shift \( f(x) \) upward by 3 units**: 
   \[
   g(x) = f(x) + 3
   \]

2. **Shift \( f(x) \) to the right by 2 units**:
   \[
   g(x) = f(x - 2)
   \]

3. **Reflect \( f(x) \) across the x-axis and then shift it downward by 4 units**:
   \[
   g(x) = -f(x) - 4
   \]

4. **Stretch \( f(x) \) vertically by a factor of 2**:
   \[
   g(x) = 2f(x)
   \]

5. **Compress \( f(x) \) horizontally by a factor of 1/3**:
   \[
   g(x) = f(3x)
   \]

### Verification:
To verify these transformations using a graphing calculator:
1. Enter the original function \( f(x) \) into the calculator.
2. Enter the transformed function \( g(x) \).
3. Compare the graphs of \( f(x) \) and \( g(x) \) to ensure the transformation was applied correctly.

Would you like to see details on a specific example? You can upload an image or describe a particular function \( f(x) \) and transformation you're working on.

### Follow-Up Questions:
1. What transformations would reflect the graph across the y-axis and then shift it 2 units left?
2. How does a vertical stretch by a factor of 3 affect the graph of \( f(x) \)?
3. What happens to the graph of \( f(x) \) when it’s reflected across both axes?
4. If \( f(x) \) is shifted down by 5 units and then compressed horizontally by a factor of 2, what’s the resulting \( g(x) \)?
5. How does a shift to the right by 4 units change the equation of \( f(x) \)?

### Tip:
When shifting graphs horizontally, remember that positive values inside the function \( f(x - c) \) shift it to the right, while negative values shift it to the left.

Learn how to write a function g(x) that represents the indicated transformation of the graph of function f(x). This detailed guide covers vertical and horizontal shifts, stretches/compressions, and reflections of graphs using function transformations. Suitable for high school students studying functions and graph transformations.

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