Question

Given the function check all the transformations that occurred from the graph of the parent function
f(x)= 1/2(x_+7)^2 + 4
Up 7 units
Up 4 units
Down 7 units
Down 4 units
Leafy 7 units
Left 4 units
Right 7 units
Right 4 units
Vertical stretch
Reflection x-axis
Reflection y-axis

Answer 1

To analyze the transformations that occurred from the graph of the parent function \( f(x) = x^2 \) to the given function \( f(x) = \frac{1}{2}(x + 7)^2 + 4 \), we can break it down into parts by looking at how it differs from the parent function.

1. **Parent function**: \( f(x) = x^2 \)

2. **Horizontal Shift**:
- The term \( (x + 7)^2 \) indicates a shift to the **left** by 7 units. The standard transformation for horizontal shifts is that if \( g(x) = (x - h)^2 \), then the graph shifts right by \( h \) units. Here, because of the plus sign, it shifts left.

3. **Vertical Stretch/Compression**:
- The coefficient \( \frac{1}{2} \) indicates a **vertical compression** (not a stretch). A factor less than 1 compresses the graph vertically.

4. **Vertical Shift**:
- The \( + 4 \) at the end indicates a shift **up** by 4 units.

Putting this together, we can summarize the transformations:

- Left 7 units
- Vertical compression by a factor of \( \frac{1}{2} \)
- Up 4 units

Therefore, the transformations that occurred are:

- **Left 7 units**
- **Up 4 units**
- **Vertical compression** (not listed in your options, but important to note)

The following transformations are **not** applicable:
- Up 7 units
- Down 7 units
- Down 4 units
- Right 7 units
- Right 4 units
- Vertical stretch
- Reflection x-axis
- Reflection y-axis

So, the correct answers are that the graph of \( f(x) \) is shifted left 7 units and up 4 units, and it experiences a vertical compression.