1. **Problem statement:** Identify the symmetries of each given graph. 2. **Symmetry rules:** - Symmetry about the x-axis means if $(x,y)$ is on the graph, then $(x,-y)$ is also on the graph. - Symmetry about the y-axis means if $(x,y)$ is on the graph, then $(-x,y)$ is also on the graph. - Symmetry about the origin means if $(x,y)$ is on the graph, then $(-x,-y)$ is also on the graph. --- ### (a) Downward curve in first quadrant and negative y region - The graph is only in the first quadrant and negative y-values, not mirrored on the left side. - It is not symmetric about the y-axis (no left side mirror). - It is not symmetric about the x-axis (no reflection above x-axis). - It is not symmetric about the origin (no matching points in opposite quadrants). **Answer:** none of the above. --- ### (b) Circle centered at origin with radius 4 - Circle is symmetric about the x-axis because for every $(x,y)$, $(x,-y)$ is on the circle. - Circle is symmetric about the y-axis because for every $(x,y)$, $(-x,y)$ is on the circle. - Circle is symmetric about the origin because for every $(x,y)$, $(-x,-y)$ is on the circle. **Answer:** x-axis, y-axis, origin. --- ### (c) Sine wave oscillating about x-axis - Sine wave is symmetric about the origin because $\sin(-x) = -\sin(x)$. - It is not symmetric about the x-axis (reflection would change sign of y but sine wave does not have that symmetry). - It is not symmetric about the y-axis (sine is an odd function, not even). **Answer:** origin. --- **Final answers:** - (a) none of the above - (b) x-axis, y-axis, origin - (c) origin