How do I graph a parabola from its equation?

Enter the quadratic in the tool using standard form y = ax² + bx + c or vertex form y = a(x − h)² + k. The precise mode instantly plots a smooth curve and marks the vertex, axis of symmetry, x-intercepts, and y-intercept. Adjust the graph window (x and y range) so the whole curve is visible, then export SVG or PNG.

How do you find the vertex of a parabola?

In standard form y = ax² + bx + c, the vertex x-coordinate is h = −b / (2a). Put that value back into the equation to get the y-coordinate, or use k = c − b² / (4a). In vertex form y = a(x − h)² + k the vertex is simply (h, k). The tool computes and displays the vertex automatically in its readout panel.

What is the axis of symmetry of a parabola?

The axis of symmetry is the vertical line x = −b / (2a) that passes through the vertex. The parabola is a perfect mirror image across this line, so any two points at equal horizontal distances from it have the same y-value. Toggle "Axis of symmetry" in the tool to draw it as a dashed vertical line.

How many x-intercepts does a parabola have?

It depends on the discriminant D = b² − 4ac. If D is positive the parabola has two x-intercepts, if D equals zero it has exactly one (the vertex touches the x-axis), and if D is negative it has none because the curve never crosses the axis. The tool reports "no real roots" in that last case instead of drawing points.

Does a parabola open up or down?

The leading coefficient a decides the direction. When a is positive the parabola opens upward and the vertex is its lowest point; when a is negative it opens downward and the vertex is its highest point. A larger magnitude of a makes the curve narrower, while a value closer to zero makes it wider.

Can I download or print the parabola graph?

Yes. The precise mode exports a clean SVG or a high-resolution PNG (rendered at 2x) that drops neatly into worksheets, slides, and homework. Precise mode runs entirely in your browser and does not use image-generation credits.

Math Tool

Parabola Graph Generator to Graph a Parabola with Vertex & Roots

Graph a parabola from its equation and label every key feature. Enter a quadratic in standard form y = ax² + bx + c or vertex form y = a(x − h)² + k, and the tool plots a smooth curve with the vertex, axis of symmetry, x-intercepts (roots), y-intercept, and optional focus and directrix — then export SVG or PNG. Deterministic precise mode plus an AI sketch mode.

Standard form y = ax² + bx + c or vertex formExact vertex, axis, roots & y-interceptOptional focus & directrixSVG & PNG export

y = a·x² + b·x + c

abc

Graph window

X minX maxY minY max

Readout

Vertex: (0, -4)
Axis: x = 0
Y-intercept: (0, -4)
Roots: x = -2, x = 2
Direction: Opens up

Parabola Graph Examples

Common ways to graph and label a parabola on a coordinate plane

View:

Labeled Parabola Graph

A parabola on a numbered coordinate plane with vertex, axis, and roots labeled.

labeledvertex

Parabola in Vertex Form

A parabola plotted from vertex form y = a(x − h)² + k with the vertex highlighted.

vertex-form

Axis of Symmetry

A parabola with its dashed vertical axis of symmetry running through the vertex.

axissymmetry

Roots and X-Intercepts

A parabola crossing the x-axis at its two roots, marked as labeled points.

rootsx-intercepts

Focus and Directrix

A parabola with its focus point and dashed directrix, vertex midway between them.

focusdirectrix

Standard vs Vertex Form

The same parabola shown as standard form and vertex form side by side.

standardvertex-form

What is a parabola?

A parabola is the U-shaped curve you get when you graph a quadratic function y = ax² + bx + c, where a is not zero. Every parabola is symmetric about a vertical line and has a single turning point called the vertex — a lowest point if the curve opens upward or a highest point if it opens downward. The sign of the leading coefficient a decides the direction: when a is positive the parabola opens up, and when a is negative it opens down. The size of a controls how wide or narrow the curve is: a small value like 0.25 gives a wide, gentle parabola, while a large value gives a steep, narrow one. Parabolas describe the path of a thrown ball, the shape of a satellite dish, and countless other real-world curves.

Standard form vs vertex form

The same parabola can be written two ways. Standard form, y = ax² + bx + c, makes the y-intercept obvious — it is simply c, the point (0, c). Vertex form, y = a(x − h)² + k, makes the vertex obvious — it sits at (h, k).
The two forms are interchangeable. To go from vertex form to standard form, expand: y = a(x − h)² + k becomes y = ax² − 2ahx + (ah² + k), so b = −2ah and c = ah² + k. This tool accepts either form and converts internally, so every computed feature stays consistent no matter which you type.
Use standard form when a problem gives you a, b, and c directly, and vertex form when you already know the turning point and want to shift or stretch it.

Finding the vertex and axis of symmetry

For a parabola in standard form, the x-coordinate of the vertex is h = −b / (2a). Substitute that back into the equation to get the y-coordinate, or use k = c − b² / (4a) directly.
The axis of symmetry is the vertical line x = h that passes through the vertex. The parabola is a mirror image of itself across this line, which is why two points with x-values equally far from h always share the same height.
For example, y = x² + 2x − 3 has a = 1 and b = 2, so h = −2 / 2 = −1 and k = −3 − 4/4 = −4. The vertex is (−1, −4) and the axis of symmetry is x = −1.

Roots, x-intercepts, and the discriminant

The roots (also called x-intercepts or zeros) are the x-values where the parabola crosses the x-axis, found by solving ax² + bx + c = 0 with the quadratic formula x = (−b ± √(b² − 4ac)) / (2a).
The expression under the square root, D = b² − 4ac, is the discriminant, and its sign tells you how many real roots exist before you finish the calculation. If D is positive there are two distinct real roots; if D equals zero the vertex sits on the x-axis and there is exactly one root; if D is negative the parabola never touches the x-axis and there are no real roots.
For y = x² + 2x − 3, D = 4 − 4(1)(−3) = 16, which is positive, so there are two roots: x = (−2 ± 4) / 2, giving x = −3 and x = 1.

Focus and directrix

Every parabola can also be defined geometrically as the set of points equally distant from a fixed point, the focus, and a fixed line, the directrix. For y = a(x − h)² + k, the focus is at (h, k + 1/(4a)) and the directrix is the horizontal line y = k − 1/(4a).
The vertex always lies exactly halfway between the focus and the directrix. The quantity 1/(4a) is the focal distance: a wide parabola (small a) has a focus far from the vertex, while a narrow parabola has one close to it.
Toggle "Focus & directrix" in the tool to see both drawn on the graph — useful for conic-sections lessons and for showing why satellite dishes and headlights use parabolic shapes to focus signals at a single point.

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Parabola Graph Generator to Graph a Parabola with Vertex & Roots

Standard form y = ax² + bx + c or vertex formExact vertex, axis, roots & y-interceptOptional focus & directrixSVG & PNG export

What is a parabola?

Standard form vs vertex form

The same parabola can be written two ways. Standard form, y = ax² + bx + c, makes the y-intercept obvious — it is simply c, the point (0, c). Vertex form, y = a(x − h)² + k, makes the vertex obvious — it sits at (h, k).

The two forms are interchangeable. To go from vertex form to standard form, expand: y = a(x − h)² + k becomes y = ax² − 2ahx + (ah² + k), so b = −2ah and c = ah² + k. This tool accepts either form and converts internally, so every computed feature stays consistent no matter which you type.

Use standard form when a problem gives you a, b, and c directly, and vertex form when you already know the turning point and want to shift or stretch it.

Finding the vertex and axis of symmetry

For a parabola in standard form, the x-coordinate of the vertex is h = −b / (2a). Substitute that back into the equation to get the y-coordinate, or use k = c − b² / (4a) directly.

The axis of symmetry is the vertical line x = h that passes through the vertex. The parabola is a mirror image of itself across this line, which is why two points with x-values equally far from h always share the same height.

For example, y = x² + 2x − 3 has a = 1 and b = 2, so h = −2 / 2 = −1 and k = −3 − 4/4 = −4. The vertex is (−1, −4) and the axis of symmetry is x = −1.

Roots, x-intercepts, and the discriminant

The roots (also called x-intercepts or zeros) are the x-values where the parabola crosses the x-axis, found by solving ax² + bx + c = 0 with the quadratic formula x = (−b ± √(b² − 4ac)) / (2a).

The expression under the square root, D = b² − 4ac, is the discriminant, and its sign tells you how many real roots exist before you finish the calculation. If D is positive there are two distinct real roots; if D equals zero the vertex sits on the x-axis and there is exactly one root; if D is negative the parabola never touches the x-axis and there are no real roots.

For y = x² + 2x − 3, D = 4 − 4(1)(−3) = 16, which is positive, so there are two roots: x = (−2 ± 4) / 2, giving x = −3 and x = 1.

Focus and directrix

Every parabola can also be defined geometrically as the set of points equally distant from a fixed point, the focus, and a fixed line, the directrix. For y = a(x − h)² + k, the focus is at (h, k + 1/(4a)) and the directrix is the horizontal line y = k − 1/(4a).

The vertex always lies exactly halfway between the focus and the directrix. The quantity 1/(4a) is the focal distance: a wide parabola (small a) has a focus far from the vertex, while a narrow parabola has one close to it.

Toggle "Focus & directrix" in the tool to see both drawn on the graph — useful for conic-sections lessons and for showing why satellite dishes and headlights use parabolic shapes to focus signals at a single point.