Question

7 which of the following functions have rotational symmetry? select two correct answers.
□ absolute value functions
□ quadratic functions
□ cube root functions
□ square root functions
□ cubic functions
clear all

Explanation:

Step1: Recall Rotational Symmetry

A function has rotational symmetry (about the origin) if \( f(-x)= -f(x) \) (odd function) or rotating its graph \( 180^\circ \) about the origin maps it onto itself.

Step2: Analyze Absolute Value Functions

Absolute value function \( y = |x| \) is even (\( f(-x)=f(x) \)), symmetric about y - axis, not rotational about origin.

Step3: Analyze Quadratic Functions

Quadratic function \( y = ax^2+bx + c \), vertex - form \( y=a(x - h)^2 + k \), symmetric about vertical line \( x = h \), not rotational about origin.

Step4: Analyze Cube Root Functions

Cube root function \( y=\sqrt[3]{x} \). \( f(-x)=\sqrt[3]{-x}=-\sqrt[3]{x}=-f(x) \), so it is odd. Rotating its graph \( 180^\circ \) about origin maps it onto itself.

Step5: Analyze Square Root Functions

Square root function \( y = \sqrt{x} \) (domain \( x\geq0 \)), not defined for \( x<0 \) in real numbers, no rotational symmetry about origin.

Step6: Analyze Cubic Functions

Cubic function (e.g., \( y = x^3 \)): \( f(-x)=(-x)^3=-x^3=-f(x) \), so it is odd. Rotating its graph \( 180^\circ \) about origin maps it onto itself.

Answer:

Cube Root Functions, Cubic Functions