Question

Think about a graph symmetric with respect to the y-axis. how do the coordinates of the points on the right half of the graph compare with the corresponding coordinates on the left half? analyze a graph symmetric about the origin in a similar manner. what do you notice?

Answer 1

See discussion below.

Explanation:

By definition, for a graph symmetric with respect to the y axis, we have that:

(x, y) is a point on the graph if and only if (-x, y) is a point on the graph. Thus, if (-x, y) is a point on the left half of the graph (note that x > 0), then (x, y) is a point on the right half of the graph, and if (x, y) is a point on the right half of the graph, then (-x, y) is a point on the left half of the graph.

For a graph symmetric with respect to the origin, by definition, (x, y) is a point on the graph if and only if (-x, -y) is a point on the graph. Thus, if (x, y) is a point on the graph in the 1st quadrant (x>0, y>0), then (-x, -y) is a point on the graph in the 3rd quadrant. If (x, y) is a point on the graph in the 4th quadrant (x>0, y<0), then (-x, -y) is a point on the graph in the 2nd quadrant.