When you with respect to the x-axis then this is like saying with reference to the x-axis. You are using the x-axis as a guide.
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No, but they are symmetric with respect to a line parallel to the y-axis - which could be the y-axis itself.
Graphing a line on the coordinate plane is the two-dimensional equivalent to marking a point on a number line. It just means to graph your line with respect to the x and y axes.
On a graph, the slope does tell you the rate of change of y with respect to x. If the slope is steep, that means that there is a high rate of change of y with respect to x. If the slope is shallow, then y is not changing that rapidly with respect to x.
Any point on the graph can be the center of a circle. If the center is on the x-axis, then the circle is symmetric with respect to the x-axis.
Differentiating with respect to one variable means thinking of all the other variables as constants. The answer is thus y=0.