Graph and Properties of Quadratic Functions
📐 Definition
A quadratic function is any function that can be written in the form:
$f(x) = ax^2 + bx + c$
where $a$, $b$, and $c$ are real numbers and $a \neq 0$.
This is called the standard form of a quadratic function.
🔑 Three Forms of Quadratic Functions
| Form | Equation | Reveals |
|---|---|---|
| Standard | $f(x) = ax^2 + bx + c$ | $y$-intercept $(0, c)$ |
| Vertex | $f(x) = a(x - h)^2 + k$ | Vertex $(h, k)$ |
| Factored | $f(x) = a(x - r_1)(x - r_2)$ | $x$-intercepts $r_1, r_2$ |
Key Properties of a Parabola
- If $a > 0$: parabola opens upward (smile 😊) — has a minimum
- If $a < 0$: parabola opens downward (frown 😞) — has a maximum
- The larger $|a|$, the narrower the parabola
- The smaller $|a|$, the wider the parabola
The vertex is the highest or lowest point on the parabola.
From standard form, the vertex is at:
$$\left(-\frac{b}{2a},\; f\left(-\frac{b}{2a}\right)\right)$$
From vertex form $a(x-h)^2 + k$, the vertex is simply $(h, k)$.
Every parabola has a vertical line of symmetry passing through the vertex:
$$x = -\frac{b}{2a} = h$$
Points on the parabola are mirror images across this line.
$y$-intercept: Set $x = 0$ → the point $(0, c)$
$x$-intercepts (roots/zeros): Set $f(x) = 0$ and solve using:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
The discriminant $\Delta = b^2 - 4ac$ tells us:
- $\Delta > 0$: Two distinct real roots