Assume that the curve is a semicircle and find the center. If it is in fact the center of a semicircle, the distance from the center to any of the points on the curve will be the same. You will only need to look at three distances. If the curve is part of a hyperbola, only two of the distances could possibly be the same --- if they were equidistant from the axis of the hyperbola. If the curves are so big that the graph does not show the endpoints, the centers are harder to find and you will need a different test.
Connect the endpoints of the curve. The midpoint of the connecting line is the center of the semicircle. If the segment of curve is a part of a hyperbola, this "center" will not be the center of anything --- hyperbolas do not have centers. This test can only be used if both endpoints of the curve are visible on the graph. Draw three lines from the center to the curve and measure them. If the three line segments are equal, the curve is a circle; otherwise, the curve is a hyperbola.
Use chords to find the curvature at three different places on the curve. The curvature of a circle is the same everywhere; the curvature of a hyperbola changes constantly, but is symmetric about the central axis --- that is why three tests are needed. To test for curvature, draw a "chord," a line segment that connects two points on the curve. Make all three chords the same length. Construct a shorter line segment, perpendicular from the midpoint of the chord to the curve. These shorter line segments are "indicators of curvature." If all three indicators are the same, the curve is a circle. If any of the indicators are different lengths, the curve is a hyperbola.