Quadratics as Second Degree Polynomials
A quadratic is a second degree polynomial, which means that in the expression of a quadratic, there will be no more than 2 x-terms being multiplied together. In expanded form, the highest exponent you will see is 2, and in factored form, there will only be two factors of x. For example: (x+3)(x–2) is a quadratic, but (x+3)(x–2)(x+1) is not, because there are 3 factors of x.
Forms of Quadratic Equations
There are several different ways to write a quadratic equation. We are going to cover standard form, vertex form, and factored form:
This is the expanded form of quadratic expressions. There is an x-squared term, an x-term, and a constant term. We use the coefficients a, b, and c, which are the same coefficients used to solve quadratic equations using the quadratic formula.
Vertex form is ideal for graphing quadratic equations, because it provides information about the parabola's vertex readily in its equation. The variables h and k represent the x– and y–coordinates to the vertex. The vertex is the point
Equations in factored form allow us to easily identify the x-intercepts of the parabola, which we will later discuss as solutions to the quadratic equation. and represent x-values at which y is equal to zero.
When quadratic equations are graphed, we call the curve a parabola. It has a distinct U shape to it (or an upside down U shape if the parabola opens downward). There is also either a minimum or a maximum point (also dependent upon which direction the parabola opens). This maximum or minimum point is known as the vertex of the parabola, and it lies on a line of symmetry to the graph. This means that one half of the parabola can be reflected about that line of symmetry to match up perfectly with the other half of the parabola.
Here are a few graphs of parabolas. See if you can spot the vertex, and notice its symmetry:
The leading coefficient to the equation (the value of "a" in each of the three forms above) determine whether the parabola opens up or down. If a is positive, the parabola opens up (and has a U shape). If a is negative, the parabola opens down (and has an upside down U shape).
Solutions to Quadratic Equations
A solution to a quadratic equation is also referred to as a zero, or a root. This is because solutions are x-values that make y equal to zero. Graphically, these are x-intercepts to the parabola. Quadratic equations can have zero, one, or two real solutions. There will never be three real solutions to the equation. This is because parabolas can intersect the x-axis at most 2 times.
There are several different methods to solving a quadratic equation. The most common ways are by using the Zero Factor Property, and using the Quadratic Formula. These methods are covered in greater detail elsewhere, but in general:
the shape of a quadratic equation on a graph; it is symmetric at the vertex
a second degree polynomial, with an x-squared term as its highest degree term
the maximum or minimum point of a parabola located on the axis of symmetry