## Forms of Quadratic Equations

Quadratic equations come in several generic forms (or patterns) but they all have several things in common:

- The highest power of x (the independent variable) is 2 when all expressions are expanded in polynomial form.
- The other integer powers, 1 meaning just x, and 0 meaning a constant term, may or may not be present. But the squared term must be present.
- Other powers (negative integers, and non-integers), cannot be present.

The most common general form of a quadratic equation is \[y=ax^2 +bx+c,\] where the *a*, *b*, and *c* are constants specific in a particular equation. For example, \[y=3x^2 -2x+7.\] Here a = 3, b = -2, and c = 7. There are other forms as well:\[\begin{array}{l}

y=k\left(x+a\right)\left(x+b\right),

y=a{\left(x-h\right)}^2 +k

\end{array},\]where the unspecified constants *a*, and *k* are specific to each form and are not the same numbers when converting between each of these forms. The most basic quadratic equation is \[y=x^2.\]Choosing various values of *x*, then squaring them to get the corresponding *y* values, and plotting these on a Cartesian coordinate grid, creates the following curve:

This shape is called a parabola. All quadratic equations have this shape when plotted but their position, orientation, and scale may be different. Each form of quadratic equations have their advantages. This lesson however, will concentrate on the form \[y=a{\left(x-h\right)}^2 +k.\]

## The ‘a’ Factor

So we will be looking at the quadratic form \[y=a{\left(x-h\right)}^2 +k.\] This form is called the turning point form. Let’s start out simple and look at the effect of *a* alone by setting the *h* and *k* to zero. This leaves us with the equation \[y=ax^2.\] This coefficient in front of the *x*^{2} term scales and orients the parabola. If *a* is negative, all the *y* values are now negative. This flips (reflects) the parabola across the *x*-axis. If *a* is a large number, greater than 1, then the *y* values are larger for a given *x* than the *y* values in the basic *y* = *x*^{2} parabola. This has the effect of making the parabola sharper, that is, it is dilated along the *x*-axis. If *a* is a fraction between -1 and 1, then the *y* values increase more slowly. This has the effect of making the parabola flatter which is also a dilation along the *x*-axis. Below are several graphs of *y* = *ax*^{2} for various values of *a*. The basic parabola is shown (dashed curve) for comparison:

Notice how the negative sign flips the parabola across the *x*-axis.

## The k Effect

Now let’s look at the equation \[y=ax^2 +k.\]I have just added a *k* to the previous equation form. If you add or subtract a constant number to an equation, it just raises or lowers the graph of the equation by k units. This is independent of the effect that *a* has on the curve. Below are examples for two choices of *k* using an *a* that was used above for comparison:

## The h Reaction

Now so far we have scaled and inverted our parabola and moved it up or down. What about moving it right or left? That’s what the *h* does in the form \[y=a{\left(x-h\right)}^2 +k.\] We get to this form by replacing *x* with *x – h*. Whenever this is done in any equation, as well as the quadratic equation, this moves the curve to the right *h* units if *h* is positive or to the left if *h* is negative. But be careful. There is a negative sign in the form *y*=*a*(*x*–*h*)^{2}+*k*, so in *y*=*a*(*x*-3)^{2}+*k*, *h* = 3 so this parabola is moved 3 units to the right. Whereas, *a*(*x*+3)^{2}+*k* can be thought of as *a*(*x*-(-3))^{2}+*k*, so *h* is -3 and this moves the parabola 3 units to the left. The effect of *h* on the graph of a parabola is independent of the effects of *a* or *k*.

Below are two examples of the effect of *h* using the last example above for comparison: