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Graphing Quadratic Equations

The general form of a quadratic equation is y = ax² + bx + c. The graph of such an equation is a parabola, a type of 2-dimensional curve.

The "basic" parabola, y = x², looks like this:

The function of the coefficient a in the general equation is to make the parabola "wider" or "skinnier", or to turn it upside down (if negative):

If the coefficient of x² is positive, the parabola opens up; otherwise it opens down.

The Vertex

The vertex of a parabola is the point at the bottom of the "U" shape (or the top, if the parabola opens downward).

The equation for a parabola can also be written in "vertex form":

y = a(x – h)² + k

In this equation, the vertex of the parabola is the point (h, k).

You can see how this relates to the standard equation by multiplying it out:

y = a(x – h)(x – h) + k

y = ax² – 2ahx + ah² + k

The coefficient of x here is –2ah. This means that in the standard form, y = ax² + bx + c, the expression

gives the x-coordinate of the vertex.

Here, a = 3 and b = 12. So, the x-coordinate of the vertex is:

Substituting in the original equation to get the y-coordinate, we get:

y = 3(–2)² + 12(–2) – 12

= –24

So, the vertex of the parabola is at (–2, –24).

The Axis of Symmetry

The axis of symmetry of a parabola is the vertical line through the vertex. For a parabola in standard form, y = ax² + bx + c, the axis of symmetry has the equation

Note that –b/2a is also the x-coordinate of the vertex of the parabola.

Here, a = 2 and b = 1. So, the axis of symmetry is the vertical line

Intercepts

You can find the y-intercept of a parabola simply by entering 0 for x. If the equation is in standard form, then you can just take c as the y-intercept. For instance, in the above example:

y = 2(0)² + (0) – 1 = –1

So the y-intercept is –1.

The x-intercepts are a bit trickier. You can use factoring, or completing the square, or the quadratic formula to find these (if they exist!).

Domain and Range

As with any function, the domain of a quadratic function f(x) is the set of x-values for which the function is defined, and the range is the set of all the output values (values of f).

Quadratic functions generally have the whole real line as their domain: any x is a legitimate input. The range is restricted to those points greater than or equal to the y-coordinate of the vertex (or less than or equal to, depending on whether the parabola opens up or down).