Question

Test the equation \(y=5 x^{2}-1\) for symmetry with respect to the \(x\) -axis, the \(y\) -axis, and the origin.

Short answer

The equation is symmetric with respect to the y-axis only.

Step-by-step solution

- Symmetry with respect to the x-axis

To test for symmetry with respect to the x-axis, replace y by -y in the equation and see if the resulting equation is equivalent to the original equation.Starting equation: \(y = 5x^{2} - 1\)Replace y by -y: \(-y = 5x^{2} - 1\)Rearrange to solve for y: \(y = -5x^{2} + 1\)Since the resulting equation \(y = -5x^{2} + 1\) is not equivalent to the original equation, the given equation is not symmetric with respect to the x-axis.

- Symmetry with respect to the y-axis

To test for symmetry with respect to the y-axis, replace x by -x in the equation and see if the resulting equation is equivalent to the original equation.Starting equation: \(y = 5x^{2} - 1\)Replace x by -x: \(y = 5(-x)^{2} - 1\)Simplify: \(y = 5x^{2} - 1\)Since the resulting equation is the same as the original equation, the given equation is symmetric with respect to the y-axis.

- Symmetry with respect to the origin

To test for symmetry with respect to the origin, replace x by -x and y by -y in the equation and see if the resulting equation is equivalent to the original equation.Starting equation: \(y = 5x^{2} - 1\)Replace x by -x and y by -y: \(-y = 5(-x)^{2} - 1\)Simplify: \(-y = 5x^{2} - 1\)Rearrange to solve for y: \(y = -5x^{2} + 1\)Since the resulting equation \(y = -5x^{2} + 1\) is not equivalent to the original equation, the given equation is not symmetric with respect to the origin.

Key concepts

Testing Symmetry

Symmetry in algebra helps us understand how a function or equation behaves under certain transformations. Testing for symmetry involves checking if changing variables like x or y results in the same equation. This process reveals whether a graph is a mirror image over a particular axis or point. Symmetry can simplify graphing and analyzing functions.

Here are the main types of symmetry we test:

• Symmetry with respect to the x-axis
• Symmetry with respect to the y-axis
• Symmetry with respect to the origin

Each type of symmetry has specific steps to determine if a given equation displays that symmetry. Let's explore these in detail.

Symmetry with Respect to the X-Axis

Symmetry with respect to the x-axis means that for every point (x, y) on the graph, the point (x, -y) is also on the graph. In simpler terms, the graph is a mirror image over the x-axis.

To determine if an equation is symmetric with respect to the x-axis:

• Replace y with -y in the equation.
• Check if the new equation is equivalent to the original.

Let's apply this to our example equation:

Starting with the equation: \(y = 5x^{2} - 1\)
Replace y with -y:\(-y = 5x^{2} - 1\)
Rearrange to solve for y: \(y = -5x^{2} + 1\)

Since \(y = -5x^{2} + 1\) is not equivalent to \(y = 5x^{2} - 1\), the equation is not symmetric with respect to the x-axis.

Symmetry with Respect to the Y-Axis

Symmetry with respect to the y-axis means the graph remains unchanged if flipped over the y-axis. In other words, for every point (x, y), there is a corresponding point (-x, y).

To test for y-axis symmetry:

• Replace x with -x in the equation.
• Check if the new equation is equivalent to the original.

Let's use the same example equation:

Starting with: \(y = 5x^{2} - 1\)
Replace x with -x: \(y = 5(-x)^{2} - 1\)
Simplify: \(y = 5x^{2} - 1\)

Since \(y = 5x^{2} - 1\) is the same as the original equation, the equation is symmetric with respect to the y-axis.

Symmetry with Respect to the Origin

Symmetry with respect to the origin means that rotating the graph 180 degrees around the origin leaves the graph unchanged. For every point (x, y), there exists a point (-x, -y).

To test for origin symmetry:

• Replace x with -x and y with -y in the equation.
• Verify if the resulting equation is equivalent to the original equation.

Using our example, we proceed as follows:

Starting equation: \(y = 5x^{2} - 1\)
Replace x with -x and y with -y: \(-y = 5(-x)^{2} - 1\)
Simplify: \(-y = 5x^{2} - 1\)
Rearrange: \(y = -5x^{2} + 1\)

Since \(y = -5x^{2} + 1\) differs from \(y = 5x^{2} - 1\), the given equation is not symmetric with respect to the origin.