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.
$x^{\frac{2}{3}}+y^{\frac{2}{3}}=1$

Verified step by step guidance

1. Step 1: Understand the types of symmetry. A graph is symmetric about the x-axis if replacing y with -y yields an equivalent equation. It is symmetric about the y-axis if replacing x with -x yields an equivalent equation. It is symmetric about the origin if replacing both x with -x and y with -y yields an equivalent equation.
2. Step 2: Check for symmetry about the x-axis. Replace y with -y in the equation: \(x^{\frac{2}{3}} + (-y)^{\frac{2}{3}} = 1\). Since \((-y)^{\frac{2}{3}} = y^{\frac{2}{3}}\), the equation remains the same, indicating symmetry about the x-axis.
3. Step 3: Check for symmetry about the y-axis. Replace x with -x in the equation: \((-x)^{\frac{2}{3}} + y^{\frac{2}{3}} = 1\). Since \((-x)^{\frac{2}{3}} = x^{\frac{2}{3}}\), the equation remains the same, indicating symmetry about the y-axis.
4. Step 4: Check for symmetry about the origin. Replace both x with -x and y with -y in the equation: \((-x)^{\frac{2}{3}} + (-y)^{\frac{2}{3}} = 1\). Since both terms remain unchanged, the equation is symmetric about the origin.
5. Step 5: Verify by graphing. Graph the equation \(x^{\frac{2}{3}} + y^{\frac{2}{3}} = 1\) to visually confirm the symmetries about the x-axis, y-axis, and the origin.

Key Concepts

Symmetry in Graphs

Symmetry in graphs refers to the property where a graph remains unchanged under certain transformations. A graph is symmetric about the x-axis if replacing y with -y yields the same equation, symmetric about the y-axis if replacing x with -x does, and symmetric about the origin if replacing both x and y with their negatives results in the same equation. Understanding these transformations is crucial for analyzing the symmetry of functions.

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the relationship between variables. This process helps in identifying key features of the function, such as intercepts, slopes, and symmetry. By graphing the given equation, one can visually confirm the symmetry properties and better understand the behavior of the function across different quadrants.

Implicit Functions

Implicit functions are defined by equations that relate variables without explicitly solving for one variable in terms of another. The equation x^(2/3) + y^(2/3) = 1 is an example of an implicit function, where both x and y are intertwined. Analyzing such equations often requires techniques like implicit differentiation or algebraic manipulation to explore their properties, including symmetry.