SL 2.6—Quadratic function: IB DP Math AA Notes

What is a Quadratic Function?

Quadratic Function

A polynomial function of degree 2, written in the form $f (x) = a x^{2} + b x + c$ , where $a$ , $b$ , $c \in R$ and $a \neq = 0$ . The condition $a \neq = 0$ is essential — if $a = 0$ , the function becomes linear, not quadratic.

The graph of any quadratic function is called a parabola — a smooth, symmetric, U-shaped (or ∩-shaped) curve.

The sign of $a$ tells you the overall shape:

If $a > 0$ : the parabola opens upward (concave up, like a cup ∪)
If $a < 0$ : the parabola opens downward (concave down, like a cap ∩)

The magnitude of $∣ a ∣$ controls the steepness: a larger $∣ a ∣$ gives a narrower parabola, while a smaller $∣ a ∣$ gives a wider one.

What is a Quadratic Function?

Analogy

Think of $a > 0$ as a bowl that can hold water, and $a < 0$ as an upside-down bowl that can't. The bigger the value of $∣ a ∣$ , the steeper the sides of the bowl.

Standard Form: $f(x) = ax^2 + bx + c$

The standard form of a quadratic is $f (x) = a x^{2} + b x + c$ . Each coefficient plays a distinct role:

Coefficient	Role
$a$	Direction (up/down) and steepness of the parabola
$b$	Influences the position of the axis of symmetry
$c$	The y-intercept — the value of $f (x)$ when $x = 0$

The y-intercept is always at the point $(0, c)$ , since substituting $x = 0$ gives $f (0) = c$ .

Example

For $f (x) = 2 x^{2} - 4 x + 3$ :

$a = 2 > 0$ , so the parabola opens upward
$b = - 4$
$c = 3$ , so the y-intercept is $(0, 3)$

Warning

Don't confuse $c$ with the y-intercept coordinate. The y-intercept is the point $(0, c)$ , not just the number $c$ . In exam questions, always write the full coordinate pair when asked for an intercept.

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What is a Quadratic Function?

Standard Form: $f(x) = ax^2 + bx + c$

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