Sure, let's break down the transformations needed to graph the function [tex]\( y = -3\sqrt{x-6} \)[/tex] from the parent function [tex]\( y = \sqrt{x} \)[/tex].
1. Horizontal Translation:
- The term [tex]\( \sqrt{x-6} \)[/tex] indicates a horizontal shift.
- Specifically, [tex]\( x-6 \)[/tex] means that the graph is translated 6 units to the right.
2. Reflection Across the x-axis:
- The negative sign in front of the entire function ([tex]\( -3 \sqrt{x-6} \)[/tex]) indicates a reflection across the x-axis.
- This means every point on the graph of [tex]\( y = 3\sqrt{x-6} \)[/tex] is reflected over the x-axis, flipping it upside down.
3. Vertical Stretch:
- The coefficient 3 in [tex]\( -3 \sqrt{x-6} \)[/tex] indicates a vertical stretch.
- This means that the graph is stretched vertically by a factor of 3. Every y-value of the parent function [tex]\( y = \sqrt{x} \)[/tex] is multiplied by 3, making the peaks and troughs steeper by that factor.
So, the parent function [tex]\( y = \sqrt{x} \)[/tex] goes through the following transformations to become [tex]\( y = -3\sqrt{x-6} \)[/tex]:
- Translated 6 units to the right.
- Reflected across the x-axis.
- Vertically stretched by a factor of 3.
Thus, the detailed transformations are translation 6 units to the right, reflection across the x-axis, and vertical stretch by a factor of 3.
