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Parabola Grapher

Graph any quadratic function y=ax²+bx+c. See roots, vertex, axis of symmetry, and y-intercept on an interactive coordinate plane.

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Quick Tips

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Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Standard Form

Classic quadratic x² − 5x + 6 = 0 with two integer roots (x = 2 and x = 3)

Key values: a = 1 · b = −5 · c = 6

Physics Problem

Projectile height equation −4.9t² + 20t + 1.5 = 0 (when does the ball hit the ground?)

Key values: a = −4.9 · b = 20 · c = 1.5

Complex Roots

Equation x² + 2x + 5 = 0 with no real solutions (discriminant < 0)

Key values: a = 1 · b = 2 · c = 5

Documentation

Key Features of a Parabola

To graph $y = ax^2 + bx + c$ , find these features:

Feature	Formula	What it tells you
Vertex	$\left(-\frac{b}{2a},\; f\!\left(-\frac{b}{2a}\right)\right)$	Turning point (min or max)
Axis of symmetry	$x = -\frac{b}{2a}$	Mirror line
y-intercept	$(0, c)$	Where it crosses the y-axis
x-intercepts	Quadratic formula roots	Where it crosses the x-axis (if real)
Direction	$a > 0$ : up, $a < 0$ : down	Opens upward or downward

Focus and Directrix

A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).

For $y = a(x - h)^2 + k$ :

\text{Focus: } \left(h,\; k + \frac{1}{4a}\right) \qquad \text{Directrix: } y = k - \frac{1}{4a}

Physical meaning: Parabolic mirrors and satellite dishes use this property — parallel rays hitting the parabola all reflect to the focus. This is why satellite dishes and car headlights are parabolic.

Three Forms of a Quadratic

Standard

y = ax^2 + bx + c

Shows the y-intercept directly.

Vertex

y = a(x - h)^2 + k

Shows the vertex directly.

Factored

y = a(x - r_1)(x - r_2)

Shows the x-intercepts directly.

Graphing Steps

Find the vertex and plot it
Draw the axis of symmetry (dashed vertical line through vertex)
Find the y-intercept (plug in x = 0) and plot it
Plot the mirror point of the y-intercept across the axis
Find x-intercepts if they exist (discriminant ≥ 0)
Plot 1–2 additional points for accuracy and draw a smooth curve

Frequently Asked Questions

How do I graph a parabola from the equation $y = ax^2 + bx + c$ ?

Find the vertex at $(-b/2a, f(-b/2a))$ , draw the axis of symmetry $x = -b/2a$ , plot the y-intercept $(0, c)$ , mirror it across the axis, find x-intercepts if the discriminant is non-negative, then draw a smooth curve through the points.

What determines whether a parabola opens upward or downward?

The sign of the coefficient $a$ determines the direction. If $a > 0$ , the parabola opens upward and has a minimum at the vertex. If $a < 0$ , it opens downward and has a maximum at the vertex.

What are the focus and directrix of a parabola?

The focus is a point at $(h, k + 1/4a)$ and the directrix is the line $y = k - 1/4a$ for a parabola in vertex form $y = a(x - h)^2 + k$ . Every point on the parabola is equidistant from the focus and directrix. This property is used in satellite dishes and headlight reflectors.

What is the axis of symmetry?

The axis of symmetry is the vertical line $x = -b/(2a)$ that passes through the vertex. The parabola is a mirror image on both sides of this line. Any point on one side has a corresponding point at the same height on the other side.

How do the three forms of a quadratic relate to graphing?

Standard form $y = ax^2 + bx + c$ shows the y-intercept directly. Vertex form $y = a(x - h)^2 + k$ shows the vertex directly. Factored form $y = a(x - r_1)(x - r_2)$ shows the x-intercepts directly. Each form highlights different features of the graph.

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