Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.
$ƒ (x) = x ∣ x ∣$

검증된 단계별 안내

Step 1: Understand the function given, $ f(x) = x |x| $. This function involves the absolute value of $ x $, which affects its symmetry properties.

Step 2: Test for symmetry about the y-axis. Replace $ x $ with $ -x $ in the function to get $ f(-x) = (-x) |-x| = -x |x| $. Since $ f(-x) \neq f(x) $, the function is not symmetric about the y-axis.

Step 3: Test for symmetry about the x-axis. For this, check if $ -f(x) = f(x) $. Since $ -f(x) = -x |x| $ and $ f(x) = x |x| $, $ -f(x) \neq f(x) $, so the function is not symmetric about the x-axis.

Step 4: Test for symmetry about the origin. Check if $ f(-x) = -f(x) $. We have $ f(-x) = -x |x| $ and $ -f(x) = -x |x| $. Since $ f(-x) = -f(x) $, the function is symmetric about the origin.

Step 5: Verify your findings by graphing the function $ f(x) = x |x| $. The graph should confirm that the function is symmetric about the origin, as it will look the same when rotated 180 degrees around the origin.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Symmetry in Graphs

Symmetry in graphs refers to the property where a graph remains unchanged under certain transformations. A graph is symmetric about the y-axis if replacing x with -x yields the same function value, indicating even symmetry. It is symmetric about the x-axis if replacing y with -y gives the same x value, and it is symmetric about the origin if replacing both x and y with their negatives results in the same function.

Absolute Value Function

The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. This function is crucial in determining symmetry because it affects the behavior of the graph, particularly in how it reflects across the axes. For example, the function f(x) = x|x| combines linear and absolute value characteristics, influencing its symmetry properties.

Graphing Techniques

Graphing techniques involve plotting points on a coordinate plane to visualize the behavior of a function. This process helps in identifying symmetry and other characteristics of the graph. By graphing the function f(x) = x|x|, one can observe its shape and confirm its symmetry properties, providing a practical method to validate theoretical findings.

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.ƒ(x)=x∣x∣ƒ(x) = x |x|