Quadratic Functions

Prerequisites- Linear Functions
Previously, we discussed linear functions. These functions took on the general form:
where
In our math journey, the next kind of functions to learn about are quadratic functions. Quadratic functions include a squared input variable,
where
Don't confuse the quad in the name quadratic with the prefix corresponding to four. The adjective quadratic actually comes from the Latin word quadrātum, which means square.
When graphed, quadratic functions appear as curves.

Quadratic Function Forms
The most common quadratic function form is the standard form:
where
Another form for quadratic functions is called the factored form:
where
The last form for quadratics functions we will discuss is called the vertex form:
where
The 3 forms can be changed into each other per the methods shown in the diagram below.

Distribute and Simplify
The factored and vectored forms can be converted to the standard form simply by evaluating and simplifying.
Consider a factored form quadratic equation:
We can evaluate the multiplication to obtain the standard form of this equation.
Mathematical Algebra Aid (Distributive Multiplication)
Suppose we have the equation:
Because
Additionally, because
Suppose we now wanted to actually obtain the value of this equation. One method would be to undo the steps we just did and evaluate the multiplication:
Another method would be to use the distributive law of multiplication:
Now we can evaluate equation
Naturally, we are inclined to laugh and mock. Sure, this is a mathematically valid method to calculating the value of the equation, but why would anyone ever want to do this!?
Consider we have another equation and we are asked to simplify it:
Clearly, the first approach will not work.
From this example, the value of distributive multiplication should be more clear. A previously unapproachable problem becomes very manageable.
This technique is often called FOIL, which stands for First Outer Inner Last.
When performing distributive multiplication, multiply the first two terms, the outer two terms, the inner two terms, and the last two terms:
This of course resembles the standard quadratic form. The formula for converting from factored to standard form is:
A vertex form quadratic equation can be converted to standard form by a similar process.
This of course resembles the standard quadratic form. The formula for converting from vertex to standard form is:
Completing the Square
The method of completing the square converts a standard form quadratic to vertex form. There is a simple formula to accomplish this:
Derivation of Completing the Square Formula
For a standard form quadratic function
If we expand the vertex form equation, we obtain:
Setting the standard and vertex forms equal to each other, we get:
To solve for the conversion formula, we set each
To solve for
Divide equation
Substitute equation
To solve for
Substitute equation
Consider the quadratic equation:
Using the formula for completing the square, we obtain that:
The vertex form of
Factoring
The factoring method converts a standard form quadratic equation into a factored form quadratic equation.
Factoring requires knowing the roots of the quadratic equation. A root is just a value of
We will discuss solving for roots in the next section, but for a quadratic equation with roots
Derivation of Factoring Formula
The goal of factoring is to convert a standard form quadratic into a factored form quadratic
We assume that we know the roots
The standard quadratic form has three parameters,
Three possible factored forms of this equation are:
One means to get a consistent factored form is to always set
With the assumption
If we expand the generic factored form in
For these to be equivalanet,
Because we know the roots
For equations
We can now solve for
Consider the quadratic equation:
The roots of this equation are
Using the factored conversion formula:
The factored form quadratic is then:
Finding Roots of Quadratic Functions
The roots of a quadratic function are the input values that cause the output to be zero.
For example, the roots of
If a quadratic function is in factored or vertex form, the roots are very easy to determine through direct solving. If a quadratic equation is in standard form, we can use the quadratic formula.
Roots of Factored Form Quadratic Functions
Given a factored quadratic function
Derivation of Roots of Factored Form Quadratic Functions
A factored quadratic equation has the form:
Solving for the roots is as easy as setting each term equal to
For example, consider the factored form quadratic function:
The roots are
Roots of Vertex Form Quadratic Functions
Given a vertex form quadratic function
Derivation of Roots of Vertex Form Quadratic Functions
Set the generic vertex form quadratic equation to
For example, consider the vertex form quadratic function:
The roots are:
Quadratic Formula Method, Roots of Standard Form Quadratic Functions
The easiest method for finding the roots of a quadratic equation is by using the quadratic formula. For a quadratic equation
Derivation of Quadratic Formula, Finding Roots of Quadratic Functions
For a standard form quadratic function
We start by dividing the entire equation by
To simplify things for a moment, let
To solve for
Now we can set the right hand side of equation
Finally, we can substitute
Equation
For example, suppose we are given the quadratic equation:
Evaluating the quadratic equation:
The inputs
Other Methods of Finding Roots
The root finding methods presented in this article aren't the only methods of solving for roots. Quadratic roots can be observed visually by plotting the function and determining the values where
For the example quadratic equation:
The roots

Though crude, the first quadratic root finding method typically taught is "guess and check". Math courses teaching quadratic functions often use as examples quadratic functions with small integer roots that are easy to guess and verify. After correctly guessing the roots of hundreds of simple quadratic functions, one develops an intuition for it.
There are also many general purpose root finding algorithms such as the Bisection method, Newton's method, Secant method, and more. These are algorithms that iteratively solve for the roots rather than through a direct formula. They deserve more time and attention than can be provided here so will be introduced in a future article.
Introduction to Complex Numbers
Consider the following standard form quadratic equation:
If we solve for the roots using the quadratic formula, we obtain the following:
Wait a minute...
Complex numbers are typically represented with the symbol
Fundamentally, there is nothing wrong with the roots of a quadratic being complex. Consider the factored form of equation
We can expand it as expected to get back the standard form:
Also consider if we substitute one of the roots into the standard form:
Even though
Though they are typically introduced at the same time as quadratic functions, there is often a large time gap between first introduction and practical application. This is unfortunate, but at mechatronics.studio, we will embrace the power of complex numbers early.
Practical Applications of Quadratic Functions
Quadratic functions have a number of practical applications. Below are a couple of examples.
Trajectory of Thrown Objects
When an object is thrown, the relationship between the object's horizontal
Note that we have not discussed the physics of how this equation is derived, but for the moment, take a leap of faith that this is the equation that describes the
A graph of the thrown ball is shown below.

Suppose we want to know the distance the ball travels when it hits the ground, when
Clearly the negative root isn't what we are interested in so we choose the positive root and conclude that the distance the ball has traveled when it hits the ground is
Mirrors
When light is reflected on a flat mirror, the angle of reflection is equal to the angle of incidence. Any light traveling in the same direction at the mirror will be reflected back identically.

Light reflects the same way on curved mirrors, but because the surface of the mirror is curved, light traveling in the same direction is reflected at different angles depending on where the light hits the surface. Quadratic mirrors, also called parabolic mirrors, are special because all light coming from one direction is reflected to one point, the focus.

The equation for this mirror is
One very common application of parabolic mirrors is the satellite dish, which reflects radio waves (very low frequency light waves) to a receiver.
