To identify the transformations applied to the basic function [tex]\( y = x^2 \)[/tex] to obtain the function [tex]\( y = (x-2)^2 + 4 \)[/tex], follow these steps:
1. Start with the basic function [tex]\( y = x^2 \)[/tex]:
- This is the standard parabola opening upwards with its vertex at the origin [tex]\((0,0)\)[/tex].
2. Horizontal Shift:
- The function [tex]\( (x - 2)^2 \)[/tex] modifies the basic function [tex]\( y = x^2 \)[/tex].
- The expression [tex]\( x - 2 \)[/tex] indicates a horizontal shift.
- Specifically, replacing [tex]\( x \)[/tex] with [tex]\( x - 2 \)[/tex] shifts the graph to the right by 2 units.
3. Vertical Shift:
- The function [tex]\( (x-2)^2 + 4 \)[/tex] further modifies the expression [tex]\( (x - 2)^2 \)[/tex].
- Adding 4 to the entire function results in a vertical shift.
- Specifically, adding 4 moves the graph upwards by 4 units.
Taking these transformations into account, the complete set of transformations applied to the basic function [tex]\( y = x^2 \)[/tex] to obtain the function [tex]\( y = (x-2)^2 + 4 \)[/tex] is:
- Shift right by 2 units
- Shift up by 4 units
These are the transformations applied to the given function.
