The axis of symmetry is an idea used in graphing certain algebraic expressions that create parabolas, or nearly u-shaped forms. These are called quadratic functions and their form typically looks like this equation: y = ax^{2} + bx + c. The variable a cannot equal zero. Truly the simplest of these functions is y = x^{2}, in which the vertex or the exact middle line running down the parabola, also called the axis of symmetry, would be the graph’s y-axis or x = 0. It directly divides the parabola in half, and everything on either side of it proceeds in a symmetrical manner.
Very often people are asked to graph more complex quadratic functions and the axis of symmetry won’t be as conveniently divided by the y-axis. Instead it will be to the left or right of it, depending on the equation, and may need to some manipulation of the function to figure out. It is important to find out the parabola’s vertex or starting point, as it’s x-coordinate is equal to the axis of symmetry. It makes graphing the rest of the parabola much easier.
In order to make this determination, there are a few ways to approach the problem. When a person is faced with a function like y= x^{2} + 4x + 12, they can apply a simple formula to derive the vertex and the axis of symmetry; remember the axis runs through the vertex. This takes two parts.
The first is to set x equal to negative b divided by 2a: x = -4/2 or -2. This number is the x coordinate of the vertex and it is substituted back into the equation to obtain the y coordinate. 4 + 16 + 12 = 32, or y =32, which derives the vertex as (-2, 32). The axis of symmetry would be drawn through the line -2, and people would know where to draw it because they’d know where the parabola began.
Sometimes the quadratic function is presented in factored or intercept form, and might look like this: y = a(x-m)(x-n). Again, the goal is to figure out x, thus deriving the line of symmetry, and then figure out y and the vertex by substituting x back into the equation. To obtain x, it is set as equal to m + n divided by 2.
Though conceptually this form of graphing and finding the axis of symmetry can take a little time, this is a valuable concept in mathematics and in algebra. It tends to be taught after students have had some time working with quadratic equations and learning how to perform some basic operations like factoring on them. Most students encounter this concept in the late first year of algebra, and it may be visited in more complex forms in later math studies.