Quadratic Function

Polynomial function of degree two

Not to be confused with Quartic function. If you're confusing them, you might need to recalibrate your understanding of 'degree,' or perhaps just everything.

In the grand, often tedious, theatre of mathematics, a quadratic function—specifically, one of a single variable—presents itself in a form that is both ubiquitous and, frankly, rather obvious once you grasp it. It takes the elegant, or perhaps merely predictable, structure of:

$f(x)=ax^{2}+bx+c,\quad a\neq 0,$

where ⁠ $x$ ⁠ is the lone, often misunderstood, variable dictating the curve's journey, and ⁠ $a$ ⁠, ⁠ $b$ ⁠, and ⁠ $c$ ⁠ are the coefficients that shape its destiny. The critical caveat, $a \neq 0$ , isn't just a minor detail; it’s the entire point. Without it, you're just dealing with a linear function, and frankly, who has time for that kind of simplicity?

The expression itself, ⁠ $ax^{2}+bx+c$ ⁠, when observed as a standalone mathematical object rather than an active function, is precisely what we term a quadratic polynomial. It is, by definition, a polynomial of degree two. In the more… accessible realms of elementary mathematics, the subtle distinctions between a polynomial and its associated polynomial function are often, and somewhat lazily, ignored. Consequently, the terms "quadratic function" and "quadratic polynomial" become nearly synonymous, frequently abbreviated to the rather uninspired "quadratic." One might say it's a simplification born of convenience, not clarity.

A quadratic polynomial with two real roots (crossings of the x axis). A rather dramatic illustration of a function acknowledging its own existence.

The visual manifestation of a real single-variable quadratic function, when plotted on a graph, is an unmistakable parabola. This iconic curve, with its singular turning point, is a testament to the function's second-degree nature. Should one choose to engage in the somewhat existential act of equating a quadratic function with zero, the result is the ever-present quadratic equation. The solutions to this equation are, rather fittingly, known as the zeros (or roots) of the corresponding quadratic function. These roots represent the points where the parabola, in its infinite wisdom, deigns to cross the x-axis. A quadratic function can offer up two distinct roots, a single repeated root (a tangent touch, if you will), or, in a move that often frustrates the less enlightened, zero real roots, implying its journey never quite intersects the x-axis in the real plane. These elusive solutions are elegantly, if somewhat mechanically, described by the venerable quadratic formula.

But why stop at one variable when the universe offers so many? A quadratic polynomial or quadratic function is perfectly capable of embracing more. Consider, for instance, a two-variable quadratic function, involving the variables ⁠ $x$ ⁠ and ⁠ $y$ ⁠. Its form expands, becoming a bit more baroque:

$f(x,y)=ax^{2}+bxy+cy^{2}+dx+ey+f,$

where the coefficients ⁠ $a$ ⁠, ⁠ $b$ ⁠, and ⁠ $c$ ⁠ must not all be simultaneously zero, otherwise, again, you're merely dealing with something less interesting. The zeros of such a function, rather than simple points on a line, describe a conic section—a more sophisticated geometric entity like a circle, an ellipse, another parabola, or a hyperbola within the ⁠ $x$ ⁠–⁠ $y$ ⁠ plane. And if two variables aren't enough to satisfy your intellectual appetites, a quadratic function can indeed accommodate an arbitrarily large number. The set of its zeros then forms a quadric, which manifests as a surface in the case of three variables, and blossoms into a full-blown hypersurface in the general, n-dimensional scenario. It seems even basic algebraic structures enjoy a good expansion.

Etymology

The adjective "quadratic" isn't some abstract, arbitrarily assigned label. It derives directly from the Latin word quadrātum, which, perhaps unsurprisingly, means "square". This linguistic lineage becomes glaringly obvious when you consider a term raised to the second power, such as ⁠ $x^{2}$ ⁠. In the archaic, yet enduring, language of algebra, this is quite literally referred to as a "square" because, conceptually, it represents the area of a square whose side length is ⁠ $x$ ⁠. A rather straightforward, if somewhat uninspired, naming convention.

Terminology

Navigating the precise language of mathematics often feels like tiptoeing through a minefield of potential misunderstandings. Here's what you absolutely must know.

Coefficients

The coefficients of a quadratic function—those constants ⁠ $a$ ⁠, ⁠ $b$ ⁠, and ⁠ $c$ ⁠ we discussed earlier—are typically assumed to be either real or complex numbers. This is the standard, the path of least resistance. However, for those with a more adventurous spirit, or perhaps a penchant for making things unnecessarily complicated, these coefficients can, in fact, be drawn from any arbitrary ring. When this happens, naturally, the domain and the codomain of the function must also align with this chosen ring. If you're going to play with abstract algebra, at least be consistent (see polynomial evaluation for the gritty details).

Degree

When the term "quadratic polynomial" is invoked, authors occasionally operate with a rather fluid definition of "degree." Sometimes, they mean "having degree exactly 2," which is the most common and, frankly, most sensible interpretation. Other times, they might mean "having degree at most 2," which allows for the inclusion of linear or even constant functions as "degenerate cases" of quadratics. It's a bit like calling a tricycle a degenerate car. Usually, the surrounding context will, if you're lucky, clarify which of these interpretations is intended. If not, well, prepare for ambiguity.

It's also worth noting that the word "order" sometimes gets tossed around with the meaning of "degree," as in "a second-order polynomial." While this might seem harmless, it's generally best to maintain a distinction. The "degree of a polynomial" rigorously refers to the largest degree of any non-zero term within that polynomial. "Order," on the other hand, more typically refers to the lowest degree of a non-zero term, particularly in the context of a power series. Mixing these up is a rookie mistake.

Variables

A quadratic polynomial, in its various manifestations, can involve a singular, solitary variable—let's call it $x$ (the univariate case, which is often the easiest to grasp, much like a child's toy). Or, it can embrace the complexity of multiple variables, perhaps $x$ , $y$ , and $z$ (the multivariate case, which is where things start to get interesting, or at least, less predictable).

The one-variable case

Let's return to the comfortable familiarity of the univariate quadratic polynomial. Any such creature can be expressed in its canonical form:

$ax^{2}+bx+c,$

where $x$ is the variable, and $a$ , $b$ , and $c$ are the coefficients that define its specific characteristics. These polynomials frequently make their grand entrance in the form of a quadratic equation:

$ax^{2}+bx+c=0.$

The solutions to this equation are famously known as the roots of the polynomial, and they can be elegantly, if somewhat tediously, derived in terms of the coefficients via the renowned quadratic formula. Every quadratic polynomial, without exception, has an associated quadratic function, and the graph of this function, as we've established, is always a parabola. It’s a fundamental truth, much like the inevitability of taxes.

Bivariate and multivariate cases

Venturing beyond the simplicity of a single variable, any quadratic polynomial involving two variables can be systematically written as:

$ax^{2}+by^{2}+cxy+dx+ey+f,$

Here, $x$ and $y$ are the variables, and $a, b, c, d, e, f$ are the coefficients. Crucially, at least one of $a$ , $b$ , or $c$ must be non-zero; otherwise, you're merely dabbling in linear functions, which, while useful, lack the dramatic flair of a quadratic. Such polynomials are absolutely fundamental to the study of conic sections. This is because the implicit equation of a conic section is precisely what you get when you set a quadratic polynomial in two variables equal to zero. Consequently, the zeros of a bivariate quadratic function form a conic section—which might be a perfectly formed ellipse, a dashing hyperbola, or even a somewhat less exciting degenerate case.

Extending this concept further, quadratic polynomials with three or more variables elegantly correspond to higher-dimensional geometric objects: quadric surfaces, or, in the most general sense, hypersurfaces.

A special, and rather efficient, subset of quadratic polynomials are those that contain only terms of degree two. These are specifically termed quadratic forms. They strip away the linear and constant terms, focusing purely on the second-degree interactions, a minimalist approach to quadratic expression.

Forms of a univariate quadratic function

A univariate quadratic function, in its various guises, can be presented in three primary formats. Each offers a different perspective, much like looking at a problem from three equally unappealing angles.

$f(x)=ax^{2}+bx+c$ This is known as the standard form. It’s the default, the one you’re given most often, like a baseline level of existential dread.
$f(x)=a(x-r_{1})(x-r_{2})$ This is the factored form. Here, $r_1$ and $r_2$ are the roots of the quadratic function, which are also the solutions to the corresponding quadratic equation. This form is particularly useful if you care about where the function crosses the x-axis, which, admittedly, some people do.
$f(x)=a(x-h)^{2}+k$ This is the vertex form. In this configuration, $h$ and $k$ are the x and y coordinates of the vertex, respectively. It’s the most direct way to identify the function's turning point, which, for a parabola, is its most defining characteristic.

It's crucial to note that the coefficient $a$ remains steadfastly the same value across all three forms; it's the underlying scalar that dictates the parabola's fundamental shape and orientation. To transition from the standard form to the factored form, one merely needs to invoke the quadratic formula to unearth the two roots, $r_1$ and $r_2$ . To convert from standard form to vertex form, you must engage in the somewhat ritualistic process known as completing the square—a method often described as "elegant" by those who enjoy mathematical suffering. Conversely, converting from factored form (or vertex form) back to standard form is a more straightforward, if slightly tedious, exercise in multiplication, expansion, and distribution of terms. It's essentially just algebra, but with more steps.

Graph of the univariate function

$f(x)=ax^{2}|_{a\in \{0.1,0.3,1,3\}}$ $f(x)=x^{2}+bx|_{b\in \{1,2,3,4\}}$ $f(x)=x^{2}+bx|_{b\in \{-1,-2,-3,-4\}}$

Regardless of which specific format you choose to present it in, the graph of any self-respecting univariate quadratic function, defined as ⁠ $f(x)=ax^{2}+bx+c$ ⁠, will invariably be a parabola. This is not a suggestion; it's a mathematical law. Equivalently, this is the graph of the bivariate quadratic equation ⁠ $y=ax^{2}+bx+c$ ⁠, which just means we're mapping the function's output to the vertical axis.

If $a > 0$ , the parabola, in a gesture of optimism, opens upwards. It has a minimum point.
If $a < 0$ , the parabola, perhaps more realistically, opens downwards. It has a maximum point.

The coefficient $a$ is the master sculptor of the graph's curvature. A larger absolute magnitude of $a$ results in a parabola that is more "closed" or sharply curved, while a smaller magnitude renders it wider, more spread out, as if it's had too much coffee.

The coefficients $b$ and $a$ , working in tandem, dictate the precise location of the parabola's axis of symmetry. This invisible vertical line also conveniently marks the x-coordinate of the vertex and corresponds to the $h$ parameter in the vertex form. Its position is given by the formula:

$x=-{\frac {b}{2a}}.$

Finally, the coefficient $c$ has a simpler, more direct role: it controls the vertical displacement of the parabola. More specifically, it represents the exact height at which the parabola, in its upward or downward journey, gracefully intercepts the y-axis. It’s the function’s initial condition, if you will, before $x$ starts to exert its influence.

Vertex

The vertex of a parabola is not just a point; it is the point. It's where the function, having made its ascent or descent, decides to turn around. Hence, it is also rather aptly referred to as the turning point. If the quadratic function is already in its vertex form, ⁠ $f(x)=a(x-h)^{2}+k$ ⁠, then identifying the vertex is trivially simple: it’s $(h, k)$ .

For those less fortunate, starting with the standard form ⁠ $f(x)=ax^{2}+bx+c$ ⁠, one must employ the method of completing the square to transform it into vertex form. The process unfolds thus:

${\begin{aligned}f(x)&=ax^{2}+bx+c\\&=a(x-h)^{2}+k\\&=a\left(x-{\frac {-b}{2a}}\right)^{2}+\left(c-{\frac {b^{2}}{4a}}\right),\\\end{aligned}}$

From this, the coordinates of the vertex, $(h, k)$ , for a parabola presented in standard form, are revealed to be:

$\left(-{\frac {b}{2a}},c-{\frac {b^{2}}{4a}}\right).$

Alternatively, if the quadratic function is conveniently provided in its factored form, ⁠ $f(x)=a(x-r_{1})(x-r_{2})$ ⁠, the x-coordinate of the vertex can be found by simply taking the average of the two roots:

${\frac {r_{1}+r_{2}}{2}}$

Consequently, the vertex $(h, k)$ in this case is:

$\left({\frac {r_{1}+r_{2}}{2}},f\left({\frac {r_{1}+r_{2}}{2}}\right)\right).$

The vertex also holds the distinction of being either the absolute maximum point of the function if $a < 0$ (a parabola opening downwards, reaching its peak before its inevitable decline), or the absolute minimum point if $a > 0$ (a parabola opening upwards, finding its lowest ebb before rising again).

The vertical line described by the equation ⁠ $x=h=-{\frac {b}{2a}}$ ⁠, which passes directly through the vertex, is not just a casual demarcation; it is the fundamental axis of symmetry for the entire parabola. Everything on one side is a mirror image of the other, a perfect, if somewhat predictable, balance.

Maximum and minimum points

For those who prefer a more sophisticated, or perhaps just more laborious, approach, calculus offers another path to pinpointing the vertex—the function’s maximum or minimum point. This is achieved by finding the roots of the function's derivative. Given our standard quadratic function:

$f(x)=ax^{2}+bx+c\quad \Rightarrow \quad f'(x)=2ax+b$

The critical point (the vertex's x-coordinate) occurs where the slope of the tangent line is zero, meaning $f'(x) = 0$ . Solving for $x$ yields:

$x=-{\frac {b}{2a}}$

Substituting this $x$ value back into the original function gives us the corresponding function value (the y-coordinate of the vertex):

$f(x)=a\left(-{\frac {b}{2a}}\right)^{2}+b\left(-{\frac {b}{2a}}\right)+c=c-{\frac {b^{2}}{4a}},$

Thus, using the elegant machinery of calculus, we arrive once again at the vertex point coordinates, $(h, k)$ , expressed as:

$\left(-{\frac {b}{2a}},c-{\frac {b^{2}}{4a}}\right).$

It’s satisfying, if not entirely surprising, that different paths lead to the same predictable destination.

Roots of the univariate function

Graph of y = ax 2 + bx + c , where a and the discriminant b 2 − 4 ac are positive, with

Roots and y -intercept in red
Vertex and axis of symmetry in blue
Focus and directrix in pink

Visualisation of the complex roots of y = ax 2 + bx + c : the parabola is rotated 180° about its vertex (orange). Its x -intercepts are rotated 90° around their mid-point, and the Cartesian plane is interpreted as the complex plane (green). A clever way to make the imaginary, well, visible.

Further information: Quadratic equation

Exact roots

The roots (or zeros), denoted as $r_1$ and $r_2$ , of the univariate quadratic function are the specific values of $x$ for which $f(x) = 0$ . In its expanded and factored forms, this looks like:

${\begin{aligned}f(x)&=ax^{2}+bx+c\\&=a(x-r_{1})(x-r_{2}),\\\end{aligned}}$

When the coefficients $a$ , $b$ , and $c$ are real or complex, the roots are precisely given by the infamous quadratic formula:

$r_{1}={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}},$

and

$r_{2}={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}.$

The term $b^2 - 4ac$ , known as the discriminant, holds the key to the nature of these roots. If it’s positive, you get two distinct real roots. If it’s zero, one repeated real root. If it’s negative, you’re dealing with a pair of complex conjugate roots, which, while not visible on the real number line, are no less valid in the grand scheme of things.

Upper bound on the magnitude of the roots

For those concerned with the outer limits of a quadratic's roots, there's a rather elegant, if somewhat unexpected, upper bound on their modulus. The magnitude of the roots of a quadratic ⁠ $ax^{2}+bx+c$ ⁠ can never exceed:

${\frac {\max(|a|,|b|,|c|)}{|a|}}\times \phi ,$

where ⁠ $\phi$ ⁠ is the illustrious golden ratio, a number that seems to pop up everywhere, from art to the structure of galaxies, and apparently, even in the constraints of quadratic roots:

${\frac {1+{\sqrt {5}}}{2}}.$

It's a small comfort, perhaps, knowing that even chaos has its boundaries.

The square root of a univariate quadratic function

Taking the square root of a univariate quadratic function doesn't just produce another function; it quite dramatically gives rise to one of the four classic conic sections. This is almost always either an ellipse or a hyperbola, with circles being a special case of ellipses.

If the leading coefficient $a > 0$ , then the equation ⁠ $y=\pm {\sqrt {ax^{2}+bx+c}}$ ⁠ describes a hyperbola. This becomes clear once you square both sides, revealing the characteristic form. The orientation of the hyperbola's axes—whether its major axis (the one passing through its vertices) is horizontal or vertical—is determined by the ordinate (y-coordinate) of the minimum point of the corresponding parabola, ⁠ $y_{p}=ax^{2}+bx+c$ ⁠. If this ordinate is negative, the hyperbola's major axis is horizontal. If it's positive, the major axis is vertical. A subtle detail, but one that completely changes the picture.

Conversely, if $a < 0$ , the equation ⁠ $y=\pm {\sqrt {ax^{2}+bx+c}}$ ⁠ describes either a circle or some other form of ellipse, or, in a rather anticlimactic turn, nothing at all. The deciding factor here is the ordinate of the maximum point of the corresponding parabola, ⁠ $y_{p}=ax^{2}+bx+c$ ⁠. If this ordinate is positive, then its square root indeed sketches out an ellipse (or a circle, if the coefficients align perfectly). However, if the ordinate is negative, then the expression under the square root will always be negative, leading to an empty locus of points in the real plane. It simply doesn't exist, which is a rather definitive statement.

Iteration

To iterate a function ⁠ $f(x)=ax^{2}+bx+c$ ⁠ is to apply it repeatedly, taking the output from one application and feeding it back as the input for the next. It's a process that can quickly spiral into complexity, much like human relationships.

One cannot always, to the dismay of many, deduce the neat, analytic form of ⁠ $f^{(n)}(x)$ ⁠, which represents the $n$ -th iteration of ⁠ $f(x)$ ⁠. (And yes, the superscript can extend to negative numbers, referring to the iteration of the inverse function, assuming one exists. Always assume the inverse exists until proven otherwise, or until it becomes too much trouble.) Yet, amidst this general intractability, there are a few analytically tractable cases, shining like beacons of order in the mathematical wilderness.

For instance, consider the iterative equation of the form:

$f(x)=a(x-c)^{2}+c$

This form, despite its apparent simplicity, can be quite revealing. It can be expressed as a conjugate, a transformation from one system to another:

$f(x)=a(x-c)^{2}+c=h^{(-1)}(g(h(x))),$

where the component functions are:

$g(x)=ax^{2}$ and $h(x)=x-c.$

Through the magic of induction, we can then derive the $n$ -th iteration:

$f^{(n)}(x)=h^{(-1)}(g^{(n)}(h(x)))$

And the $n$ -th iteration of $g(x)$ is, mercifully, easy to compute:

$g^{(n)}(x)=a^{2^{n}-1}x^{2^{n}}.$

Finally, by combining these, we arrive at the closed-form solution for $f^{(n)}(x)$ :

$f^{(n)}(x)=a^{2^{n}-1}(x-c)^{2^{n}}+c$

For a deeper dive into the fascinating, if often mind-bending, relationship between functions like $f$ and $g$ , you might want to look into Topological conjugacy. And for a glimpse into the truly chaotic behavior that can emerge from the general iteration of quadratic polynomials, the Complex quadratic polynomial offers a rather unsettling journey.

A particularly famous example of iteration is the logistic map:

$x_{n+1}=rx_{n}(1-x_{n}),\quad 0\leq x_{0}<1$

This deceptively simple equation, with its parameter $r$ typically ranging between 2 and 4, can exhibit both predictable and utterly chaotic behavior. It's a microcosm of the universe, if you squint hard enough. In certain specific cases, this map can be solved analytically. When $r = 4$ , the system descends into a beautifully unpredictable state of chaos. The solution in this chaotic regime is:

$x_{n}=\sin ^{2}(2^{n}\theta \pi )$

Here, the initial condition parameter ⁠ $\theta$ ⁠ is derived from the initial value $x_0$ :

$\theta ={\tfrac {1}{\pi }}\sin ^{-1}(x_{0}^{1/2})$

If ⁠ $\theta$ ⁠ happens to be a rational number, then after a finite number of iterations, ⁠ $x_n$ ⁠ will fall into a perfectly periodic sequence. However, the vast majority of ⁠ $\theta$ ⁠ values are irrational, and for these, ⁠ $x_n$ ⁠ never truly repeats itself. It is non-periodic and, more dramatically, exhibits sensitive dependence on initial conditions. This means even the tiniest variation in $x_0$ will lead to wildly divergent outcomes over time, hence its classification as chaotic. It's a stark reminder that precision is often an illusion.

Contrast this with the solution of the logistic map when $r = 2$ , a much calmer scenario:

$x_{n}={\frac {1}{2}}-{\frac {1}{2}}(1-2x_{0})^{2^{n}}$

This holds true for ⁠ $x_{0}\in [0,1)$ ⁠. In this case, since ⁠ $(1-2x_{0})\in (-1,1)$ &#8288> for any $x_0$ other than the unstable fixed point 0, the term ⁠ $(1-2x_{0})^{2^{n}}$ ⁠ rapidly approaches 0 as $n$ tends towards infinity. Consequently, ⁠ $x_n$ ⁠ converges gracefully to the stable fixed point, ⁠ ${\tfrac {1}{2}}$ &#8288>. A predictable, reassuring outcome, for those who prefer their mathematics to be less dramatic.

Bivariate (two variable) quadratic function

Further information: Quadric and Quadratic form

Moving into higher dimensions, a bivariate quadratic function is simply a polynomial of the second degree that involves two variables. It takes the general form:

$f(x,y)=Ax^{2}+By^{2}+Cx+Dy+Exy+F,$

where $A, B, C, D$ , and $E$ are fixed coefficients that sculpt the surface, and $F$ is the constant term that merely shifts it vertically. Such a function, when visualized, describes a quadratic surface in three-dimensional space. If one were to set ⁠ $f(x,y)$ &#8288> equal to zero, you would be describing the intersection of this surface with the plane ⁠ $z=0$ &#8288>. This intersection forms a locus of points that is precisely equivalent to a conic section. It’s how 2D geometry emerges from 3D forms, for those who appreciate such transformations.

Minimum/maximum

Determining the minimum or maximum of a bivariate quadratic function is a bit more involved than its univariate counterpart, primarily due to the increased complexity of its shape. The behavior is dictated by the value of the discriminant-like term $4AB - E^2$ :

If ⁠ $4AB-E^{2}<0$ ⁠, the function exhibits no true maximum or minimum point. Instead, its graph forms a hyperbolic paraboloid—that distinctive saddle shape that perpetually rises in one direction while falling in another. It's a surface of endless ambiguity.
If ⁠ $4AB-E^{2}>0$ ⁠, then the function does possess a definite maximum or minimum. Specifically, it has a minimum if both $A > 0$ and $B > 0$ (a bowl-like shape opening upwards), and a maximum if both $A < 0$ and $B < 0$ (an inverted bowl-like shape). This graph forms an elliptic paraboloid. In this more agreeable scenario, the minimum or maximum occurs at the point ⁠ $(x_{m},y_{m})$ &#8288>, where the coordinates are given by:

$x_{m}=-{\frac {2BC-DE}{4AB-E^{2}}},$

and $y_{m}=-{\frac {2AD-CE}{4AB-E^{2}}}.$
If ⁠ $4AB-E^{2}=0$ ⁠ and ⁠ $DE-2CB=2AD-CE\neq 0$ &#8288>, then the function once again has no single maximum or minimum point. Its graph, in this instance, forms a parabolic cylinder—a trough or ridge extending infinitely. A rather uninspired shape, if you ask me.
Finally, if ⁠ $4AB-E^{2}=0$ &#8288> and ⁠ $DE-2CB=2AD-CE=0$ &#8288>, the function achieves its maximum or minimum not at a single point, but along an entire line. It will be a minimum if $A > 0$ and a maximum if $A < 0$ . This also forms a parabolic cylinder, but one where the "turning point" is extended into a continuous ridge or valley. It's a reminder that sometimes, the answers aren't as singular as you might hope.