Symmetry and Graphs

Graph A: This graph is symmetric about its axis; that is, it is symmetric about the line x = 3. There is no other symmetry. This graph shows a function.

Graph B: This graph is symmetric about the axes; that is, it is symmetric about the lines x = 0 (the y-axis) and y = 0 (the x-axis). It is also symmetric about the origin. Since there exist vertical lines (such as the line x = 2) which will cross this graph twice, what it shows is not a function.

Graph C: This graph is symmetric about the lines x = 1 and y = −2, and symmetric about the point (1, −2). Since a vertical line can be drawn to cross the ellipse twice, this is not a function.

Graph D: This graph is symmetric about slanty lines: y = x and y = −x. It is also symmetric about the origin. Because this hyperbola is angled correctly (so that no vertical line can cross the graph more than once), the graph shows a function.

Graph E: This graph (of a square-root function) shows no symmetry whatsoever, but it is a function.

Graph F: This graph (of a cubic function) is symmetric about the point (−4, −1), but not around any lines. This graph does show a function.

Graph G: This parabola is lying on its side. It is symmetric about the line y = 2. It is not a function.

Graph H: This parabola is vertical, and is symmetric about the y-axis. It is a function; in fact, it is an even function.

Graph A: This linear graph goes through the origin. If I rotate the graph 180° around the origin, I'll get the same picture. So this graph is odd. (The function would not be odd if this line didn't go through the origin.)

Graph B: This parabola's vertex is on the y-axis, so the axis of symmetry is the y-axis. That means that the function is even.

Graph C: This cubic is centered on the origin. If I rotate the graph 180° around the origin, I'll get the same picture. So this graph is odd.

Graph D: This cubic is centered at the point (0, −3). This graph is symmetric, but not about the origin or the y-axis. So this function is neither even nor odd.

Graph E: This cube root is centered on the origin, so this function is odd.

Graph F: This square root has no symmetry. The function is neither even nor odd.

Graph G: This graph looks like a bell-shaped curve. Since it is mirrored around the y-axis, the function is even.

Graph H: This hyperbola is symmetric about the lines y = x and y = −x, but this tells me nothing about evenness or oddness. However, the graph is also symmetric about the origin, so this function is odd.