Answer:
[tex]\sf f(x) = -(x + 3.6)^3 + 5.4 [/tex]
Step-by-step Explanation:
To write the function based on the given parent function [tex]\sf y = x^3[/tex] and the specified transformations applied in the given order, let's break down each transformation step by step:
Shift 3.6 units to the left:
To shift the function [tex]\sf 3.6[/tex] units to the left, we replace [tex]\sf x[/tex] with [tex]\sf x + 3.6[/tex]:
[tex]\sf f_1(x) = (x + 3.6)^3 [/tex]
Note: Left means to add and right means to subtract from x.
Reflect across the y-axis:
To reflect the function across the y-axis, we introduce a negative sign outside the function:
[tex]\sf f_2(x) = -(x + 3.6)^3 [/tex]
Shift upwards by 5.4 units:
To shift the reflected function [tex]\sf 5.4[/tex] units upwards, we add [tex]\sf 5.4[/tex] to the function:
[tex]\sf f(x) = -(x + 3.6)^3 + 5.4 [/tex]
Therefore, the function based on the given parent function [tex]\sf y = x^3[/tex] and the specified transformations in the given order is:
[tex]\sf \boxed{f(x) = -(x + 3.6)^3 + 5.4} [/tex]
This function [tex]\sf f(x)[/tex] represents the result of shifting the parent function [tex]\sf y = x^3[/tex] [tex]\sf 3.6[/tex] units to the left, reflecting it across the y-axis, and then shifting it upwards by [tex]\sf 5.4[/tex] units.
