Quadratic Function: Interactive Visualization

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Quadratic functions in this visualization

A quadratic function is a function $f(x) = ax^2 + bx + c$ with $a \ne 0$ . Its graph is a parabola, opening up when $a > 0$ and down when $a < 0$ . This page writes one quadratic in three equivalent forms: the mode dial at the top of the panel picks the one you turn, and all three stay in step while you drag the graph or turn the controls.

Standard · vertex · factored form

y = ax^2 + bx + c = a(x - h)^2 + k = a(x - x_1)(x - x_2)

Each form shows one feature of the graph. In the standard form, $c$ gives the y-intercept: the graph crosses the y-axis at the hollow point $(0, c)$ ; its mirror image $(2h, c)$ shows the symmetry. The vertex form shows the vertex $(h, k)$ and the axis of symmetry $x = h$ . The factored form shows the roots, where the graph meets the x-axis; over the real numbers it exists only when the discriminant is not negative.

Vertex and roots from a, b, c

h = -\frac{b}{2a},\quad k = c - \frac{b^2}{4a},\qquad \Delta = b^2 - 4ac,\quad r_{1,2} = \frac{-b \pm \sqrt{\Delta}}{2a}

Completing the square: from standard to vertex form

\begin{aligned} ax^2 + bx + c &= a\left(x^2 + \frac{b}{a}x\right) + c \\ &= a\left(x^2 + \frac{b}{a}x + \frac{b^2}{4a^2}\right) - \frac{b^2}{4a} + c \\ &= a\left(x + \frac{b}{2a}\right)^2 + c - \frac{b^2}{4a} \end{aligned}

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Things to notice

Completing the square is a translation: the three steps above turn the standard form into the vertex form, and $a(x - h)^2 + k$ is $y = ax^2$ moved $h$ along the x-axis and $k$ up or down. Turning $h$ or $k$ in the vertex form, or dragging the vertex, draws $y = ax^2$ and the two moves.
$\Delta > 0$ : two real roots; $\Delta = 0$ : one double root, the vertex on the axis; $\Delta < 0$ : no real roots — the equation $ax^2 + bx + c = 0$ then has two complex conjugate roots, which are not points of this graph.
$a$ is the same number in all three forms: it sets how wide the parabola is and which way it opens. Turning $b$ alone does not slide the graph sideways; the vertex runs along the parabola $y = c - ax^2$ . Turning $a$ alone holds something different in each form: in the standard form $(0, c)$ and the tangent $y = bx + c$ there stay put — every parabola touches that line at that point — and the vertex runs along the line $y = c + \tfrac{b}{2}x$ ; in the vertex form the vertex stays; in the factored form the roots stay and the vertex rides the axis of symmetry.
The page computes with exact fractions, so the forms show exact decimals, fractions or square roots. A number too long to show is rounded, and its equation is then written with ≈. The two axes share one scale, except in the thrown-ball highlight; a note on the graph says so whenever they do not.

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Controls

The mode dial picks the form you turn: the standard form sets $a$ , $b$ and $c$ (knobs), the vertex form $a$ and the vertex $(h, k)$ , the factored form $a$ and the two roots (thumbwheels for fine steps); with no real roots the factored position is blocked. Open Conversion details to see the other two forms' numbers with the textbook formulas.
Grabbing a point on the graph switches to its form. Drag the vertex to move the parabola, a root along the x-axis ( $a$ and the other root stay; an irrational other root is first written to six decimal places, a change too small to see; a double root moves the whole parabola sideways), or the y-intercept along the y-axis (only $c$ changes); drag elsewhere to pan. Click any number to type it, as a decimal or a fraction such as 3/4.
Presentation mode opens a full-screen view for projectors (the arrow keys move the parabola; H hides the answers: only the numbers of the form you turn stay, the other two forms and the conversions show question marks).

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Quadratic functions in this visualization

Things to notice

Controls

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