Determine if the following function [tex]f(x)=2 x(x+1)(x-2)[/tex] has line symmetry, point symmetry or neither?
Answer (1)
Expand the function f ( x ) = 2 x ( x + 1 ) ( x − 2 ) to get f ( x ) = 2 x 3 − 2 x 2 − 4 x .
Find f ( − x ) = − 2 x 3 − 2 x 2 + 4 x .
Find − f ( x ) = − 2 x 3 + 2 x 2 + 4 x .
Since f ( − x ) e q − f ( x ) and f ( − x ) e q f ( x ) , the function has neither point symmetry nor line symmetry. The answer is n e i t h er .
Explanation
Understanding the Problem The problem asks us to determine the symmetry of the function f ( x ) = 2 x ( x + 1 ) ( x − 2 ) . We need to check for point symmetry and line symmetry. A function has point symmetry if f ( − x ) = − f ( x ) , and it has line symmetry about the y-axis if f ( − x ) = f ( x ) .
Expanding the Function First, let's expand the function: f ( x ) = 2 x ( x + 1 ) ( x − 2 ) = 2 x ( x 2 − 2 x + x − 2 ) = 2 x ( x 2 − x − 2 ) = 2 x 3 − 2 x 2 − 4 x
Finding f(-x) Now, let's find f ( − x ) :
f ( − x ) = 2 ( − x ) 3 − 2 ( − x ) 2 − 4 ( − x ) = − 2 x 3 − 2 x 2 + 4 x
Finding -f(x) Next, let's find − f ( x ) :
− f ( x ) = − ( 2 x 3 − 2 x 2 − 4 x ) = − 2 x 3 + 2 x 2 + 4 x
Checking for Point Symmetry Now, we check if f ( − x ) = − f ( x ) . Comparing the expressions for f ( − x ) and − f ( x ) , we see that: f ( − x ) = − 2 x 3 − 2 x 2 + 4 x − f ( x ) = − 2 x 3 + 2 x 2 + 4 x Since f ( − x ) e q − f ( x ) , the function does not have point symmetry.
Checking for Line Symmetry Now, we check if f ( − x ) = f ( x ) . Comparing the expressions for f ( − x ) and f ( x ) , we see that: f ( − x ) = − 2 x 3 − 2 x 2 + 4 x f ( x ) = 2 x 3 − 2 x 2 − 4 x Since f ( − x ) e q f ( x ) , the function does not have line symmetry about the y-axis.
Conclusion Since the function has neither point symmetry nor line symmetry about the y-axis, we can conclude that the function has neither point symmetry nor line symmetry.
Examples
Understanding symmetry is crucial in various fields. In physics, symmetry principles underlie conservation laws, such as the conservation of energy and momentum. In architecture and design, symmetry is used to create aesthetically pleasing and balanced structures. For example, the symmetry of a butterfly's wings or the point symmetry in a snowflake are visually appealing and mathematically significant. Recognizing symmetry helps in simplifying complex problems and appreciating the beauty in the world around us.
