IDNLearn.com: Where your questions meet expert answers and community support. Ask your questions and receive reliable and comprehensive answers from our dedicated community of professionals.
Sagot :
Sure! Let's break down the transformation of the parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex] to the function [tex]\( g(x) = f(x+2) - 4 \)[/tex].
1. Parent Function: The parent function is given as [tex]\( f(x) = \sqrt[3]{x} \)[/tex]. This represents a cube root function.
2. Horizontal Shift: The function [tex]\( g(x) = f(x+2) - 4 \)[/tex] modifies the input of the parent function.
- [tex]\( f(x + 2) \)[/tex] indicates a horizontal shift. Specifically, [tex]\( x + 2 \)[/tex] means that every [tex]\( x \)[/tex] value is shifted to the left by 2 units. This is because adding inside the function moves the graph in the opposite direction of the sign.
3. Vertical Shift: The function [tex]\( g(x) = f(x+2) - 4 \)[/tex] also modifies the output of the parent function.
- The subtraction by 4 ([tex]\( - 4 \)[/tex]) outside the function indicates a vertical shift downward. This means that the entire graph is shifted downward by 4 units.
In summary:
- The transformation [tex]\( f(x + 2) \)[/tex] shifts the graph of [tex]\( f(x) = \sqrt[3]{x} \)[/tex] 2 units to the left.
- The transformation [tex]\( - 4 \)[/tex] shifts the graph downward by 4 units.
Thus, the graph of [tex]\( g(x) = \left( \sqrt[3]{x + 2} \right) - 4 \)[/tex] is a vertical shift of the cube root function [tex]\( x^{1/3} \)[/tex] down by 4 units and a horizontal shift to the left by 2 units.
Therefore, to select the correct graph [tex]\( g(x) \)[/tex] from the given options, look for the graph that has the cube root shape and is moved 2 units to the left and 4 units down.
1. Parent Function: The parent function is given as [tex]\( f(x) = \sqrt[3]{x} \)[/tex]. This represents a cube root function.
2. Horizontal Shift: The function [tex]\( g(x) = f(x+2) - 4 \)[/tex] modifies the input of the parent function.
- [tex]\( f(x + 2) \)[/tex] indicates a horizontal shift. Specifically, [tex]\( x + 2 \)[/tex] means that every [tex]\( x \)[/tex] value is shifted to the left by 2 units. This is because adding inside the function moves the graph in the opposite direction of the sign.
3. Vertical Shift: The function [tex]\( g(x) = f(x+2) - 4 \)[/tex] also modifies the output of the parent function.
- The subtraction by 4 ([tex]\( - 4 \)[/tex]) outside the function indicates a vertical shift downward. This means that the entire graph is shifted downward by 4 units.
In summary:
- The transformation [tex]\( f(x + 2) \)[/tex] shifts the graph of [tex]\( f(x) = \sqrt[3]{x} \)[/tex] 2 units to the left.
- The transformation [tex]\( - 4 \)[/tex] shifts the graph downward by 4 units.
Thus, the graph of [tex]\( g(x) = \left( \sqrt[3]{x + 2} \right) - 4 \)[/tex] is a vertical shift of the cube root function [tex]\( x^{1/3} \)[/tex] down by 4 units and a horizontal shift to the left by 2 units.
Therefore, to select the correct graph [tex]\( g(x) \)[/tex] from the given options, look for the graph that has the cube root shape and is moved 2 units to the left and 4 units down.
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Your questions deserve precise answers. Thank you for visiting IDNLearn.com, and see you again soon for more helpful information.