Young students learn about reflections through pictures and discussions of symmetry in which one half of an object is reflected over an imaginary line to create the other half of the object. If you have ever cut out the shape of a heart by folding a piece of paper in half, you have used basic reflections. The left side of the heart is a reflection of the right side.
In pre-algebra and algebra, reflections apply to geometric figures placed on the coordinate plane. Each vertex of a figure, such as a triangle, can be reflected across either the X-axis (horizontal), the Y-axis (vertical) or other linear marker to create a new figure. In each case, the line of the axis acts like a mirror between the two shapes. To start practicing these reflections, students visually analyze the original figure and the axis in order to draw the new one in the correct position.
Once students have mastered visually reflecting figures, they learn how to mathematically perform reflections of figures and functions over the X- or Y-axis. In this process, the coordinate numbers themselves are manipulated and plotted on the plane to give the new image. If the figure is being reflected over the X-axis, the signs of the Y-coordinates are flipped, and if the figure is being reflected over the Y-axis, the signs of the X-coordinates are flipped. For example, a triangle with vertices (-1, 2), (3, 4) and (5, -6) becomes (1, 2), (-3, 4) and (-5, -6) when reflected over the Y-axis. A figure or function can also be reflected over the line Y = X by switching the X and Y coordinates of each pair.
After students learn to graph more complicated equations, they visit relections again through the topic of inverse functions. Functions that are inverses of each other are reflections of each other over the line Y = X. This visual information helps students when algebraically manipulating a function to find its inverse. For instance, when finding the inverse of a quadratic function, which is a radical function, you would know to indicate that both the positive and negative radical are included because this is what makes the new function look like a reflection of the old function. If only the positive radical function were used, the new graph would look like only half of a reflection of the original.