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Quadratics and the Quadratic Formula: Theory and Graphing

**Three
types of quadratic equations**** · Solving the equation · Graphing the equation · Changing the value of** **x _{0}**

**The three types of
quadratic equations**

Every quadratic equation describes a parabola, which looks like this graph.

The graph looks a little like a cup, and the bottom of the cup is called the **vertex**.
The mouth of the cup keeps getting larger to infinity. For the most part, we will be
interested in the region around the vertex.

The cup is upright (vertex down) when a > 0, upside down (vertex up) when a < 0. The case a = 0 renders the equation linear, not quadratic, so we won’t consider that case here.

There are three important cases of quadratics depending on where the graph crosses the
x-axis (these points are called **roots** or **zeros** of the equation). In case I,
two distinct, real roots, the vertex lies on the opposite side of the x-axis from the rest
of the graph and so the curve must cross the x-axis exactly twice. One can see exactly
where the roots are from the graph, and they are clearly real numbers. Analytically, this
case corresponds to the portion of the quadratic formula under the radical (the **discriminant**)
being strictly positive:

Case I, two real, distinct roots

The graph above is an example of Case I.

If the vertex is exactly on the x-axis, there is only one real root (case II). Analytically, this corresponds to the discriminant equal to zero:

**Case II, exactly one real root**

If the vertex lies entirely on the same side of the x-axis as the rest of the graph, then the curve never touched the x-axis, never crossed either. There are no real roots, but there are complex roots and they always come in matched pairs called complex conjugates. This is case III. Analytically, this corresponds to negative values of the discriminant:

Case III, two complex conjugate roots

There are many ways to play with the equations to emphasize different qualities we might want to consider in detail. In general, there are two big things we want to do with a quadratic equation: solve it for all possible values of x, or graph the equation.

There are three main methods of solving quadratics: **Guessing the solutions** (also
known as the double parentheses method), **completing the
square**, and using the **quadratic formula**. Each has advantages and
disadvantages in any given situation. We have dealt with methods of solving quadratics in
each of these ways elsewhere (follow the links for a detailed discussion).

There is a special form of a quadratic that is best for graphing the equation. We'll analyze it thoroughly here.

Here **y _{0}** is the

Notice that the vertex is at (x_{0}, y_{0}) = (0, 0) and that y(-1) =
y(1) = 1. Thus, the vertex is at the origin and the parabola has unit amplitude.

When the value of x_{0} is changed, the parabola moves to the left (x_{0}
negative) or to the right (x_{0} positive) a distance | x_{0} |. This is
easy to get mixed up since the value of x_{0} is subtracted in the equation, which
makes it look like the sign has been reversed. Some examples will help clear this up.

In the graph below, x_{0} = -1 is negative, so the vertex is moved one unit to
the left.

The equation, however, looks as though we are adding 1 to x:

In other words,

In short, positive values added to x in the equation move the graph to the left, toward
more negative values. The value of x_{0} is negative, but the appearance of the
number in the equation is positive. This takes some practice to remember.

Conversely, when x_{0} is positive, the number appears to be subtracted from x
and the graph looks like this:

And so a value subtracted from x in the equation moves the graph to the right, toward
more positive values. The value of x_{0} is positive, but the appearance of the
number in the equation is negative. This is a great source of confusion to students, and
is well worth going over again and again until very well understood.

The situation is similar with the y_{0} value - things appear backwards due to
the way the equation is customarily written. Again, there is nothing wrong with the
equation, since it is subtracted, the sign is apparently reversed. When y_{0} is
positive, the number appears in the equation to be negative, and the graph is higher by y_{0}
number of units. When y_{0} is negative, the number appears to be positive, and
the graph is lower.

Here are two examples.

In this case y_{0} = -1, and the graph has been lowered by one unit. In the
other case,

y_{0} = 1, and the graph has been raised one unit.

Making **"a"** smaller produces a fatter parabola, while making **"a"**
larger produces a skinnier parabola. When **"a"** is positive, the cup faces
up, while negative causes the cup to face down. In each case, it is best to compare with a
normal parabola.

Notice first that negative values of **"a"** produce upside-down
parabolas, but the vertices are all in the same place. Compared with |a| = 1, the two
parabolas with |a| = 0.5 are fatter, while the two parabolas with |a| = 2 are skinnier.
The numbers with smaller absolute values make the parabola apparently larger (although
really the values at each x are smaller by comparison) while the numbers with larger
absolute values make the parabola appear smaller (although the actual values at each x are
larger). Again, this can be confusing, but careful consideration and lots of practice will
make it clear.

Analytical connection between the standard forms

Before trying to put all the information above together, we should go through the algebraic connection between the standard form for graphing and the standard form for solving via the quadratic equation. Please note that we are ignoring the standard form for completing the square, since it is pretty easy to get this once the quadratic form is known.

Starting with the standard graphing form, we can manipulate the algebra

Putting this form equal to the standard quadratic formula gives us

This allows us to translate from the graphical form of the equation to the quadratic form. To go the other way, we invert the equations to get

Admittedly, these relations are a little awkward, but they do allow straightforward conversion from one form of the quadratic to the other. Now we show how to use these methods to graph some quadratics.

Example - Graphing a quadratic

Graph the equation

Solution

First, we need to identify the three constants: a = 2, b = -4 and c = 3. Then, these
are plugged into the conversions for x_{0} and y_{0}:

Putting these results into the standard graphical form yields

This is an upright parabola (a > 0) with vertex at (x_{0}, y_{0}) =
(1, 1) which is skinny compared with a "normal" parabola (a = 2 instead of a =
1). If you like, you can plot three points: x = 1 => y = 1, x = 0 => y = 3 and x = 2
=> y = 3. The graph looks like this

Example - Finding a quadratic from a graph

Looking at this graph, find the standard quadratic form for the equation.

Solution

This type of problem is best done "naively", that is, let's assume that we see what we think we see and avoid getting too technical about the points on the graph. It appears that we can identify three points on the graph:

(x, y) = (-2, 0), (-1, -1) and (0, 0)

Since the vertex is clearly at (-1, -1), it seems clear that x_{0} = -1 and y_{0}
= -1. We still need the amplitude, so let's take one of the points, say (0, 0), and plug
it into the graphical form. In that case,

Now we have all the graphical form constants, we can just plug into the conversion equations to get

So our final answer, the quadratic form, comes out to be

This is one area in particular where it really pays to do a very large number of problems. Graphing skills are extremely useful and are constantly used in more advanced math, chemistry, physics, business and engineering. The ability to go from an equation to a graph and vice-versa is literally an indispensable skill. Getting very good at it now will save a great deal of trouble later, and will probably serve you very well in your career later.

College Algebra (Schaum's Outlines)

The classic algebra problem book - very light on theory, plenty of problems with full solutions, more problems with answers

Schaum's Easy Outline: College Algebra

A simplified and updated version of the classic Schaum's Outline. Not as complete as the previous book, but enough for most students