Consider the curve given by the equation

What symmetries does this curve display? Check all that are
applicable.
symmetry about the -axis
symmetry about the -axis
no symmetry about the origin

Ask by French Erickson. in the United States

Mar 28,2025

Question

Consider the curve given by the equation

What symmetries does this curve display? Check all that are
applicable.
symmetry about the -axis
symmetry about the -axis
no symmetry about the origin

Ask by French Erickson. in the United States

Mar 28,2025

UpStudy AI · Accepted Answer

To determine the symmetries of the curve given by the equation $$ -4 x^{2}+4 x+4 y^{2}=78 $$, we will check for the following symmetries:

1. **Symmetry about the $$ x $$-axis**: A curve is symmetric about the $$ x $$-axis if replacing $$ y $$ with $$ -y $$ in the equation yields an equivalent equation.

2. **Symmetry about the $$ y $$-axis**: A curve is symmetric about the $$ y $$-axis if replacing $$ x $$ with $$ -x $$ in the equation yields an equivalent equation.

3. **Symmetry about the origin**: A curve is symmetric about the origin if replacing $$ x $$ with $$ -x $$ and $$ y $$ with $$ -y $$ in the equation yields an equivalent equation.

Let's check each symmetry one by one.

### 1. Symmetry about the $$ x $$-axis
Replace $$ y $$ with $$ -y $$:
$$
-4 x^{2} + 4 x + 4 (-y)^{2} = 78
$$
This simplifies to:
$$
-4 x^{2} + 4 x + 4 y^{2} = 78
$$
This is the same as the original equation, so the curve is symmetric about the $$ x $$-axis.

### 2. Symmetry about the $$ y $$-axis
Replace $$ x $$ with $$ -x $$:
$$
-4 (-x)^{2} + 4 (-x) + 4 y^{2} = 78
$$
This simplifies to:
$$
-4 x^{2} - 4 x + 4 y^{2} = 78
$$
This is not the same as the original equation, so the curve is not symmetric about the $$ y $$-axis.

### 3. Symmetry about the origin
Replace $$ x $$ with $$ -x $$ and $$ y $$ with $$ -y $$:
$$
-4 (-x)^{2} + 4 (-x) + 4 (-y)^{2} = 78
$$
This simplifies to:
$$
-4 x^{2} - 4 x + 4 y^{2} = 78
$$
This is not the same as the original equation, so the curve is not symmetric about the origin.

### Conclusion
The symmetries of the curve are:
- Symmetry about the $$ x $$-axis: **Yes**
- Symmetry about the $$ y $$-axis: **No**
- Symmetry about the origin: **No**

Thus, the applicable symmetry is symmetry about the $$ x $$-axis.
The curve is symmetric about the $$ x $$-axis.

Consider the curve given by the equation

What symmetries does this curve display? Check all that are
applicable.
symmetry about the -axis
symmetry about the -axis
no symmetry about the origin

Upstudy AI Solution

Answer

Solution

Mind Expander

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Consider the curve given by the equation What symmetries does this curve display? Check all that are applicable. symmetry about the -axis symmetry about the -axis no symmetry about the origin

Upstudy AI Solution

Answer

Solution

Mind Expander

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Latest Geometry Questions

Consider the curve given by the equation

What symmetries does this curve display? Check all that are
applicable.
symmetry about the -axis
symmetry about the -axis
no symmetry about the origin