Rewrite \( y=\sqrt{4 x+16}+5 \) to make it easy to graph using a translation. Describe the graph.

To rewrite the equation \( y = \sqrt{4x + 16} + 5 \) in a form that highlights the translation, we can start by simplifying the expression under the square root. First, factor out the 4 from the expression inside the square root: \[ y = \sqrt{4(x + 4)} + 5 \] Next, we can simplify this further: \[ y = \sqrt{4} \cdot \sqrt{x + 4} + 5 \] Since \( \sqrt{4} = 2 \), we can rewrite the equation as: \[ y = 2\sqrt{x + 4} + 5 \] Now, we can identify the transformations applied to the basic function \( y = \sqrt{x} \): 1. **Horizontal Translation**: The term \( x + 4 \) indicates a horizontal shift to the left by 4 units. 2. **Vertical Stretch**: The coefficient 2 in front of the square root indicates a vertical stretch by a factor of 2. 3. **Vertical Translation**: The \( +5 \) at the end indicates a vertical shift upward by 5 units. ### Description of the Graph The graph of the function \( y = 2\sqrt{x + 4} + 5 \) is derived from the basic square root function \( y = \sqrt{x} \): - The basic square root function \( y = \sqrt{x} \) starts at the origin (0,0) and increases gradually. - After applying the transformations: - The graph shifts left by 4 units, so the new starting point is at (-4, 5). - The graph is stretched vertically by a factor of 2, making it rise faster than the basic square root function. - Finally, the entire graph is shifted up by 5 units. Thus, the graph will start at the point (-4, 5) and will rise to the right, becoming steeper than the standard square root function due to the vertical stretch. The overall shape will still resemble the right half of a parabola, opening to the right.