Quadratic Functions
Contents
5. Quadratic Functions#
Quadratic functions (also called “quadratics”) are functions like \(f(x) = x^2 - 2x + 1\). In fact, every quadratic function can be written as \(f(x) = ax^2 + bx + c\), where \(a, b,\) and \(c\) are constants. When \(f\) is written this way, we say that it’s in general form.
5.1. Zoom In#
We’re going to study three “forms” - different ways of writing the same quadratic function. Every quadratic can be written in any of these forms. In this set of notes, we’ll see how to transition from one form to another.
Vertex Form |
General Form |
Factored Form |
|---|---|---|
\(f(x) = a(x - h)^2 + k\) |
\(f(x) = ax^2 + bx + c\) |
\(f(x) = a(x - r_1)(x - r_2)\) |
- vertex form#
A quadratic is in vertex form if it is written as \(f(x) = a(x - h)^2 + k\) for some constants \(a, h,\) and \(k\). In this form, the vertex is easily identified: it is \((h, k)\).
- general form#
A quadratic is in general form if it is written as \(f(x) = ax^2 + bx + c\) for some constants \(a, b,\) and \(c\).
It is easy to convert to general form by just expanding any products and collecting like terms. Converting from general form to something else is a little tougher, but we’ll discuss how to do it below.
- factored form#
A quadratic is in factored form if it is written as \(f(x) = a(x - r_1)(x - r_2)\) for some constants \(a, r_1,\) and \(r_2\). In this form, the roots (zeros) are easily identified: they are \(x = r_1, r_2\).
Remark 5
Note that all of these forms use the letter \(a\). This \(a\) is called the leading coefficient, and it remains the same when switching between forms. For example, if \(a = 2\) for your function when it’s written in vertex form, then \(a\) will still be 2 if we convert to general or factored form.