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.