Question

using visual observation, determine whether the graph is symmetric with respect to the (a) x - axis, (b) y - axis, or (c) the origin.
a. yes, because if (a,b) is on the graph, then so is (-a,b).
b. no, because for (\\(\frac{\pi}{2}, - 5\\)), the point (\\(\frac{\pi}{2},5\\)) is not on the graph.
c. yes, because if (a,b) is on the graph, then so is (a, - b).
d. no, because for (\\(\frac{\pi}{2}, - 5\\)), the point (\\(\frac{\pi}{2}, - 5\\)) is not on the graph.

Explanation:

Step1: Recall symmetry about y - axis

A graph is symmetric about the y - axis if for every point \((a,b)\) on the graph, the point \((-a,b)\) is also on the graph.

Step2: Recall symmetry about x - axis

A graph is symmetric about the x - axis if for every point \((a,b)\) on the graph, the point \((a, - b)\) is also on the graph.

Step3: Analyze the given points

We are checking the condition for x - axis symmetry. If a graph is symmetric about the x - axis and \((a,b)\) is on the graph, then \((a,-b)\) is on the graph.

Answer:

C. Yes, because if \((a,b)\) is on the graph, then so is \((a, - b)\)