3.1: Quadratic Functions

What Are We Doing?

In Chapter 2 we studied functions in general.

Chapter 3 will dive into a specific type of function called a polynomial function.

polynomial function
A polynomial function of degree $n$ is a function of the form \[P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \qquad a_n \neq 0\] where $a_n, a_{n-1}, \dots, a_1, a_0 \in \mathbb{R}$. $n$ is an integer.

The degree tells you the highest power of $x$ that is present.

$P(x) = 3$ is a degree 0 polynomial.
$P(x) = 2x + 3$ is a degree 1 polynomial.

This section studies quadratic, or degree 2 polynomials.

Quadratic Functions

quadratic function
A quadratic function is a polynomial function of degree 2, with form \[f(x) = ax^2 + bx + c \qquad a\neq 0\]

Q: How can we convert the form $ax^2 + bx + c$ into transformations from $x^2$?

A: To convert $ax^2 + bx + c$ into transformations, we need a technique called completing the square.

Completing the Square

To complete the square for $x^2 + bx$, add and subtract $\left(\dfrac{b}{2}\right)^2$.

Complete the square for $x^2 + 4x$.

In the fact, the coefficient of $x^2$ is $1$. If needed, you need to use GCF to factor out the coefficient of $x^2$ before completing the square.

Complete the square for $2x^2 + 12x$.

Standard Form

The standard form of $ax^2 + bx + c$ is just completing the square, putting it into transformation form.

standard form, vertex
A quadratic function $f(x) = ax^2 + bx + c$ can be expressed in the standard form \[f(x) = a(x - h)^2 + k\] by completing the square.
The vertex of the parabola is $(h, k)$.

Suppose $f(x) = 2x^2 - 12x + 13$. Express $f$ in standard form and sketch a rough graph.