How do you determine whether the graph of #y=sqrt(2-x^2)# is symmetric with respect to the x, y axis?

Question

Sadie Ackerman · Accepted Answer

See explanation

Square both sides giving #y^2=2-x^2# #x^2+y^2=2# This is the equation of a circle centred at the origin Thus is symmetrical about both the x and y axis ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Note that the question does NOT use the format #y=+-sqrt(2-x^2)#

Ariana Alvarez · Answer

undefined To determine whether the graph of $ y = \sqrt{2 - x^2} $ is symmetric with respect to the x-axis or y-axis, you can examine its properties.

1. **Symmetry with respect to the x-axis**: A function is symmetric with respect to the x-axis if replacing $ y $ with $ -y $ in the equation leaves the equation unchanged.

For $ y = \sqrt{2 - x^2} $, if replacing $ y $ with $ -y $ results in the same equation, the graph is symmetric with respect to the x-axis.

2. **Symmetry with respect to the y-axis**: A function is symmetric with respect to the y-axis if replacing $ x $ with $ -x $ in the equation leaves the equation unchanged.

For $ y = \sqrt{2 - x^2} $, if replacing $ x $ with $ -x $ results in the same equation, the graph is symmetric with respect to the y-axis.

To determine symmetry:
- Replace $ y $ with $ -y $ in the equation. If the equation remains unchanged, the graph is symmetric with respect to the x-axis.
- Replace $ x $ with $ -x $ in the equation. If the equation remains unchanged, the graph is symmetric with respect to the y-axis.

For $ y = \sqrt{2 - x^2} $:
- Replacing $ y $ with $ -y $ gives $ -y = \sqrt{2 - x^2} $, which is not the same as the original equation. Thus, it's not symmetric with respect to the x-axis.
- Replacing $ x $ with $ -x $ gives $ y = \sqrt{2 - (-x)^2} $, which simplifies to $ y = \sqrt{2 - x^2} $, the original equation. Thus, it is symmetric with respect to the y-axis.

Therefore, the graph of $ y = \sqrt{2 - x^2} $ is symmetric with respect to the y-axis.