Question

Describe the transformations between y equals square root of x and y equals negative ( square root of x + 1 + 5)

Answer 1

To describe the transformations between the function \( y = \sqrt{x} \) and \( y = -(\sqrt{x} + 1) + 5 \), we can break down the transformations step by step.

1. **Starting function**:
\[
y = \sqrt{x}
\]

2. **First transformation**:
Adding 1 to \( \sqrt{x} \) inside the parentheses:
\[
y = \sqrt{x} + 1
\]
This shifts the graph of \( y = \sqrt{x} \) **upward by 1 unit**.

3. **Second transformation**:
Negating the entire expression:
\[
y = -(\sqrt{x} + 1)
\]
This reflects the graph across the x-axis, turning it upside down.

4. **Third transformation**:
Adding 5 after negating:
\[
y = -(\sqrt{x} + 1) + 5
\]
This can be simplified further to:
\[
y = -\sqrt{x} - 1 + 5 = -\sqrt{x} + 4
\]
The addition of 5 shifts the graph **upward by 5 units**.

5. **Final transformation**:
Together, the transformations from \( y = \sqrt{x} \) to \( y = -(\sqrt{x} + 1) + 5 \) result in:
- **Shift up by 1 unit** (to get \( y = \sqrt{x} + 1 \))
- **Reflection across the x-axis** (to get \( y = -(\sqrt{x} + 1) \))
- **Shift up by 5 units** (to apply the final form).

Therefore, the complete transformation sequence is:
- Shift the graph of \( y = \sqrt{x} \) **up by 1 unit**.
- **Reflect across the x-axis**.
- Finally, **shift up by 5 units** to produce the final function \( y = -\sqrt{x} + 4 \).

### Summary:
The transformation from \( y = \sqrt{x} \) to \( y = -(\sqrt{x} + 1) + 5 \) involves a shift up by 1, reflection across the x-axis, and a subsequent shift up by 5.