Question

Describe the transformations between y = Vx and y = - (Vx+1+5).

Answer 1

To describe the transformations between the two functions \( y = \sqrt{x} \) and \( y = -(\sqrt{x} + 1 + 5) \), we first need to analyze each function.

1. **Starting Function**:
\[
y = \sqrt{x}
\]
This is the basic square root function, which starts at the origin \((0,0)\) and increases monotonically. Its graph is characterized by a curve that rises gently to the right.

2. **Transformed Function**:
\[
y = -(\sqrt{x} + 1 + 5)
\]
This function can be simplified to:
\[
y = -(\sqrt{x} + 6)
\]
This means we can analyze the transformations step by step.

### Steps of Transformation:
1. **Vertical Shift**:
- The term \(+6\) indicates a vertical shift. Specifically, \( \sqrt{x} \) shifts upward by 6 units. So:
\[
y = \sqrt{x} + 6
\]
- This moves the whole graph of \( y = \sqrt{x} \) upwards.

2. **Reflection**:
- The negative sign in front of the entire expression indicates a reflection over the x-axis. Therefore:
\[
y = -(\sqrt{x} + 6)
\]
- Reflecting the graph of \( y = \sqrt{x} + 6 \) over the x-axis produces a graph that now opens downwards.

3. **Final Transformation**:
- Combining these transformations, we can summarize that we:
- Move the graph of \( y = \sqrt{x} \) up 6 units to get \( y = \sqrt{x} + 6 \).
- Then, reflect the resulting graph across the x-axis to get \( y = -(\sqrt{x} + 6) \).

### Summary of Transformations:
- Start with \( y = \sqrt{x} \).
- Shift upward by 6 units: \( y = \sqrt{x} + 6 \).
- Reflect across the x-axis: \( y = -(\sqrt{x} + 6) \).

So, the transformation from \( y = \sqrt{x} \) to \( y = -(\sqrt{x} + 6) \) involves a vertical shift upwards followed by a reflection across the x-axis.