# Quadratic Function: Graph, Vertex, Derivative and Integral

> The quadratic function f(x) = ax² + bx + c: parabola graph, vertex formula, roots, derivative 2ax + b, integral and worked calculus examples.

Source: https://practicecalculus.com/functions/quadratic-function
Updated: 2026-11-03

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

## At a glance

- **Domain:** $(-\infty, \infty)$
- **Range:** $\left[f\left(-\frac{b}{2a}\right), \infty\right)$ if $a > 0$
- **Derivative:** $2ax + b$
- **Integral:** $\frac{a}{3}x^3 + \frac{b}{2}x^2 + cx + C$
- **Vertex:** $x = -\frac{b}{2a}$
- **Roots:** $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

## Key takeaways

- The graph of $f(x) = ax^2 + bx + c$ is a parabola that opens up if $a > 0$ and down if $a < 0$.
- The derivative $f'(x) = 2ax + b$ is zero at the vertex $x = -\frac{b}{2a}$.
- The second derivative is the constant $2a$, so a quadratic is concave up everywhere or concave down everywhere.
- Its integral is a cubic: $\frac{a}{3}x^3 + \frac{b}{2}x^2 + cx + C$.

A **quadratic function** is a polynomial of degree two. Its graph is a **parabola**, a symmetric U-shaped curve with a single turning point called the vertex. Quadratics model projectile height, area and profit, and they are the first curves where derivatives reveal something you cannot see from a line.

## Formula and forms

$$f(x) = ax^2 + bx + c, \qquad a \ne 0$$

| Form | Equation | Shows |
| --- | --- | --- |
| Standard | $ax^2 + bx + c$ | $y$-intercept $c$ |
| Vertex | $a(x - h)^2 + k$ | vertex $(h, k)$ |
| Factored | $a(x - r_1)(x - r_2)$ | roots $r_1, r_2$ |

## Graph of a quadratic function

```graph
title: 'Graph of the parabola y = x² - 2x - 3 and its derivative y = 2x - 2'
caption: 'The vertex is at $(1, -4)$, exactly where the derivative crosses zero. The roots are $x = -1$ and $x = 3$.'
x: [-3, 5]
y: [-6, 8]
functions:
  - expr: x^2 - 2x - 3
    label: y = x² - 2x - 3
  - expr: 2x - 2
    label: y = 2x - 2 (derivative)
    dashed: true
points:
  - [1, -4, 'vertex (1, -4)']
  - [-1, 0, '(-1, 0)']
  - [3, 0, '(3, 0)']
```

Key features:

- **Direction.** Opens up if $a > 0$ (vertex is a minimum), down if $a < 0$ (vertex is a maximum).
- **Axis of symmetry.** The vertical line $x = -\frac{b}{2a}$.
- **Roots.** From the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. The discriminant $b^2 - 4ac$ tells you whether there are two, one or no real roots.
- **Range.** $\left[k, \infty\right)$ if $a > 0$ and $\left(-\infty, k\right]$ if $a < 0$, where $k$ is the vertex height.

## Derivative of a quadratic

By the [power rule](/rules/power-rule) and the [sum rule](/rules/sum-and-difference-rule):

$$f'(x) = 2ax + b, \qquad f''(x) = 2a$$

The derivative of a quadratic is a [linear function](/functions/linear-function). It is zero at exactly one point, $x = -\frac{b}{2a}$, which is the vertex. The second derivative is the constant $2a$, so the parabola is concave up everywhere when $a > 0$ and concave down everywhere when $a < 0$. That is why the vertex is always the absolute extremum.

## Integral of a quadratic

$$\int (ax^2 + bx + c)\,dx = \frac{a}{3}x^3 + \frac{b}{2}x^2 + cx + C$$

The lowercase $c$ is the quadratic's constant term; the uppercase $C$ is the constant of integration.

## Worked examples

**Example 1: vertex with calculus.** For $f(x) = x^2 - 2x - 3$, $f'(x) = 2x - 2 = 0$ gives $x = 1$, and $f(1) = -4$. The vertex is $(1, -4)$, a minimum because $a = 1 > 0$.

**Example 2: maximum height.** A ball's height is $h(t) = -5t^2 + 20t + 1$ meters. Then $h'(t) = -10t + 20 = 0$ at $t = 2$, and the maximum height is $h(2) = -20 + 40 + 1 = 21$ meters.

**Example 3: area under a parabola.** Evaluate $\int_0^3 (x^2 + 1)\,dx$:

$$\left[\tfrac{x^3}{3} + x\right]_0^3 = (9 + 3) - 0 = 12$$

**Example 4: tangent line.** For $f(x) = x^2 - 2x - 3$ at $x = 3$: $f(3) = 0$ and $f'(3) = 4$, so the tangent line is $y = 4(x - 3) = 4x - 12$.

## Common mistakes

1. **Forgetting the minus sign in $-\frac{b}{2a}$.** For $x^2 - 2x - 3$, $b = -2$, so the vertex is at $x = -\frac{-2}{2} = 1$.
2. **Differentiating $ax^2$ as $ax$.** The exponent comes down: $\frac{d}{dx}3x^2 = 6x$.
3. **Integrating $x^2$ as $\frac{x^3}{2}$.** Divide by the new exponent, $3$.
4. **Expanding $(x - 2)^2$ as $x^2 + 4$.** The correct expansion is $x^2 - 4x + 4$.

## How it connects

Quadratics are the simplest [polynomial functions](/functions/polynomial-functions) with curvature, so they are the standard first examples for [optimization problems](/blog/optimization-problems), [curve sketching](/blog/curve-sketching-with-derivatives) and [area between curves](/blog/area-between-curves). The second-degree Taylor polynomial of any smooth function is a quadratic, which is why parabolas approximate so many curves near a point.

## Frequently asked questions

### What is the derivative of a quadratic function?

The derivative of $ax^2 + bx + c$ is $2ax + b$, found with the power rule term by term. For example, the derivative of $3x^2 - 5x + 2$ is $6x - 5$.

### How do you find the vertex of a parabola with calculus?

Set the derivative to zero: $2ax + b = 0$ gives $x = -\frac{b}{2a}$. Substitute that $x$ into $f$ to get the $y$-coordinate of the vertex.

### How do you integrate a quadratic?

Use the reverse power rule on each term: $\int (ax^2 + bx + c)\,dx = \frac{a}{3}x^3 + \frac{b}{2}x^2 + cx + C$.

### How do you tell if a parabola opens up or down?

Look at the sign of $a$, the coefficient of $x^2$. If $a > 0$ it opens up and the vertex is a minimum; if $a < 0$ it opens down and the vertex is a maximum.