Question

Determine whether \(x^{2}+4 y^{2}=16\) is symmetric respect to the \(x\) -axis, the \(y\) -axis, and/or the origin.

Short answer

The equation is symmetric with respect to the x-axis, the y-axis, and the origin.

Step-by-step solution

- Test for Symmetry with respect to the x-axis

To test for symmetry with respect to the x-axis, replace y with -y and check if the equation remains unchanged. Starting with the original equation:\[ x^2 + 4y^2 = 16 \]Replace y with -y:\[ x^2 + 4(-y)^2 = 16 \]Since \((-y)^2\) is the same as \(y^2\), the equation becomes:\[ x^2 + 4y^2 = 16 \]The equation remains unchanged, indicating symmetry with respect to the x-axis.

- Test for Symmetry with respect to the y-axis

To test for symmetry with respect to the y-axis, replace x with -x and check if the equation remains unchanged. Starting with the original equation:\[ x^2 + 4y^2 = 16 \]Replace x with -x:\[ (-x)^2 + 4y^2 = 16 \]Since \((-x)^2\) is the same as \(x^2\), the equation becomes:\[ x^2 + 4y^2 = 16 \]The equation remains unchanged, indicating symmetry with respect to the y-axis.

- Test for Symmetry with respect to the origin

To test for symmetry with respect to the origin, replace x with -x and y with -y, and check if the equation remains unchanged. Starting with the original equation:\[ x^2 + 4y^2 = 16 \]Replace x with -x and y with -y:\[ (-x)^2 + 4(-y)^2 = 16 \]Since \((-x)^2\) is the same as \(x^2\) and \((-y)^2\) is the same as \(y^2\), the equation becomes:\[ x^2 + 4y^2 = 16 \]The equation remains unchanged, indicating symmetry with respect to the origin.

Key concepts

Symmetry with Respect to x-axis

Symmetry concerning the x-axis means that if you reflect the graph of an equation over the x-axis, it will look the same. To determine if an equation has this symmetry, replace every instance of y with -y and see if the equation remains unchanged.
In our example, we start with the equation:
\[ x^2 + 4y^2 = 16 \]
Replace y with -y:
\[ x^2 + 4(-y)^2 = 16 \]
Since \((-y)^2\) is the same as \( y^2 \), the equation becomes:
\[ x^2 + 4y^2 = 16 \]
The equation \( x^2 + 4y^2 = 16 \) remains identical, confirming that it is symmetric concerning the x-axis.

If this condition holds true for other equations, they too would exhibit symmetry along the x-axis.

Symmetry with Respect to y-axis

Symmetry related to the y-axis means that the graph will appear the same when reflected over the y-axis. To test for y-axis symmetry, you replace every x with -x and check if the equation stays the same.
Using our equation again:
\[ x^2 + 4y^2 = 16 \]
Replace x with -x:
\[ (-x)^2 + 4y^2 = 16 \]
Because \((-x)^2\) equals \( x^2 \), the equation becomes:
\[ x^2 + 4y^2 = 16 \]
The equation \( x^2 + 4y^2 = 16 \) did not change, showing that it has symmetry with respect to the y-axis.

This gives us a reliable check for identifying symmetry along the y-axis in algebraic equations.

Symmetry with Respect to Origin

Symmetry regarding the origin means if you rotate the graph 180 degrees, it looks unchanged. To check this type of symmetry, replace each x with -x and each y with -y, then see if the equation remains the same.
Beginning with our usual equation:
\[ x^2 + 4y^2 = 16 \]
Replace x with -x and y with -y:
\[ (-x)^2 + 4(-y)^2 = 16 \]
Because \( (-x)^2 \) equals \( x^2 \) and \( (-y)^2 \) equals \( y^2 \), it changes into:
\[ x^2 + 4y^2 = 16 \]
The fact that \( x^2 + 4y^2 = 16 \) remains unchanged proves that the equation has symmetry with respect to the origin.

Checking for origin symmetry helps identify functions or shapes that are centrally symmetric.

Testing Symmetry

Testing symmetry is a systematic process that allows us to analyze and understand the properties of graphs of equations. Here are the steps:

• x-axis Symmetry: Replace y with -y
• y-axis Symmetry: Replace x with -x
• Origin Symmetry: Replace x with -x and y with -y

By going through these steps, we understand better how an equation behaves under reflection or rotation.
This testing helps in simplifying complex equations and understanding their geometric representations.

Algebraic Equations

Algebraic equations form the basis of algebra. They can describe lines, curves, and shapes. Understanding their symmetry helps in analyzing these shapes.
In our example, the equation \( x^2 + 4y^2 = 16 \) depicts an ellipse.
Here are some key types of symmetry in algebraic equations:

• x-axis Symmetry: Simplifies graphing and solving
• y-axis Symmetry: Important for parabolas and circles
• Origin Symmetry: Common in circular and rotational shapes

Understanding these principles allows us to predict and sketch graphs more easily, providing a solid foundation in algebra.