The axis of symmetry is used to graph quadratic functions, with the vertex being the midline of the parabola. To find the axis of symmetry, manipulate the function or use a formula to determine the vertex’s x-coordinate. This concept is taught in algebra and is useful for graphing more complex functions.
The axis of symmetry is an idea used to graph certain algebraic expressions that create parabolas or nearly U-shaped shapes. These are called quadratic functions, and their shape typically looks like this equation: y = ax2 + bx + c. The variable a cannot be equal to zero. Really the simplest of these functions is y = x2, where the vertex or exact midline running along the parabola, also called the axis of symmetry, would be the y-axis of the graph ox = 0. It directly divides the parabola in half , and everything on its sides proceeds symmetrically.
Quite often people are asked to graph more complex quadratic functions and the axis of symmetry will not conveniently be divided by the y-axis. Instead it will be to the left or right of it, depending on the equation, and you may need to manipulate the function to figure this out. It is important to find out the vertex or starting point of the parabola, since its x coordinate is equal to the axis of symmetry. It makes the rest of the dish much easier graphically.
To make this determination, there are a few ways to approach the problem. When a person is faced with a function like y = x2 + 4x + 12, he can apply a simple formula to find the vertex and axis of symmetry; remember that the axis passes through the vertex. This requires two parts.
The first is to set x equals negative ab divided by 2a: x = -4/2 or -2. This number is the x coordinate of the vertex and is substituted 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 drawn through the -2 line, and people would know where to draw it because they would know where the parabola begins.
Sometimes the quadratic function is presented in factored or intercepted form and might look like this: y = a(xm)(xn). Again, the goal is to compute x, thereby deriving the line of symmetry, and then compute y and the vertex by substituting x into the equation.
To get x, set equal m + n divided by 2.
While conceptually this form of graphing and finding the axis of symmetry may take some time, this is a valuable concept in mathematics and algebra. It tends to be taught after students have spent some time working with quadratic equations and learning how to do some basic operations such as factoring on them. Most students encounter this concept at the end of the first year of algebra and it can be examined in more complex forms in later math studies.
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