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What are the transformations in the equation [tex] y = (x - 2)^2 + 4 [/tex]?
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.
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