What is the axis of symmetry? (with photos)

The axis of symmetry is usually taught after students have had a chance to work with quadratic equations.

The axis of symmetry is an idea used to graphically represent certain algebraic expressions that create parabolas, or almost u-shaped shapes. They are called quadratic functions and their form usually looks like this equation: y = ax 2 + bx + c. The variable a cannot be equal to zero. In fact, the simplest of these functions is y = x 2 , where the exact middle vertex or line going down from the parabola, also called the axis of symmetry, would be the y-axis of the graph, or x = 0. It divides directly the parabola in half, and everything on either side of it proceeds symmetrically.

The axis of symmetry is usually taught in the first year of algebra.

People are often asked to graph more complex quadratic functions and the symmetry axis will not be divided as conveniently by the y axis. Instead, it will be on your left or right, depending on the equation, and it might take some manipulation of the function to figure it out. It is important to find the vertex or starting point of the parabola, as its x-coordinate is equal to the axis of symmetry. This makes graphing the rest of the parabola much easier.

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, he can apply a simple formula to derive the vertex and axis of symmetry; remember that the axis passes through the vertex. This takes two parts.

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The first is to set x equal to negative ab divided by 2a: x = -4/2 or -2. This number is the x-coordinate of the vertex and is substituted back into the equation to get the y-coordinate. 4 + 16 + 12 = 32 or y = 32, which derives the vertex as (-2, 32). The axis of symmetry would be plotted along the -2 line, and people would know where to plot it because they knew where the parabola started.

Sometimes the quadratic function is presented in a factored or intercepted form and may look like this: y = a (xm) (xn). Again, the goal is to find x, thus deriving the line of symmetry, and then find y and the vertex by substituting x back into the equation. To get x, it is defined as equal to am + n divided by 2.

Although conceptually this way of graphing and finding the axis of symmetry can take a while, this is a valuable concept in mathematics and algebra. It tends to be taught after students have had some time working with quadratic equations and learning to perform some basic operations such as factoring them. Most students encounter this concept at the end of their first year of algebra, and it can be visited in more complex forms in later mathematical studies.

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