Parabola Graphing Calculator

What the parabola graphing calculator shows

Each label on the board is an attribute of the quadratic function you entered. The grapher updates them when a coefficient changes.

• Vertex: the turning point (h, k). It is a minimum when a > 0 and a maximum when a < 0. The tool labels that point on the curve.
• Axis of symmetry: the vertical line x = h through the vertex. The tool draws that dashed line across the coordinate plane.
• Roots (x-intercepts / zeros): where the curve crosses the x-axis. The tool marks each real root; if none exist, the readout says so.
• y-intercept: the value at x = 0, which equals c in standard form. That is where the curve meets the y-axis.
• Direction of opening: upward when a > 0, downward when a < 0. The board shows which way the arms of the parabola point.
• Width / stretch: controlled by |a|. A larger |a| makes a narrower parabola; a smaller |a| makes a wider one.
• Discriminant (b² − 4ac): its sign decides two, one, or zero real roots. When the discriminant is negative, the tool reports no real roots.
• Coefficients a, b, and c: the three slider inputs that define the quadratic on the board.

Forms of a quadratic you can graph

The sliders take standard-form coefficients. Vertex form and factored form still map to the same a, b, and c, so you can enter an equivalent standard form and read the same labels.

• Standard formy = ax² + bx + cSet a, b, and c on the sliders. The board plots the curve and labels vertex, axis, and roots directly.
• Vertex formy = a(x − h)² + kExpand or convert to standard form, then enter a, b, and c. The labeled vertex should match (h, k).
• Factored formy = a(x − r₁)(x − r₂)Expand to standard form and enter a, b, and c. The labeled roots should match r₁ and r₂ when they are real.

For the full family of conic sections on one coordinate plane, see the conics section on the home page.

How to graph a parabola

1. Set a, b, and c with the sliders so the equation matches the parabola you want.
2. Read the vertex on the board and in the readout as the turning point of the curve.
3. Find the roots where the curve crosses the x-axis; they update when coefficients change.
4. Pan or zoom if the vertex or roots sit outside the default window.

For hand methods such as completing the square or deriving the vertex formula, seehow to graph a parabola. This page stays on the quadratic function grapher and live labels for y = ax² + bx + c.

Worked examples

Enter each set of coefficients below. The board should show the listed vertex, axis, roots, and y-intercept.

1. y = x² − 4

   Enter a = 1, b = 0, c = −4.

   The board shows vertex (0, −4), axis x = 0, roots at x = ±2, and y-intercept −4.

2. y = −2x² + 4x + 1

   Enter a = −2, b = 4, c = 1.

   The board shows a downward parabola, vertex (1, 3), axis x = 1, two irrational roots, and y-intercept 1.

3. y = x² + 4

   Enter a = 1, b = 0, c = 4.

   The board shows an upward parabola, vertex (0, 4), no real roots (discriminant = −16), and y-intercept 4.

How the coefficient a changes the parabola

The a slider controls width and direction. Watch the curve reshape as you move it while b and c stay fixed.

• a > 1 narrows the parabola (steeper arms).
• 0 < a < 1 widens the parabola (flatter arms).
• a < 0 opens the parabola downward; the vertex becomes a maximum.
• Larger |a| steepens the curve either way; smaller |a| flattens it.

Reading the vertex, axis, and roots from the graph

The vertex label sits on the turning point. Its coordinates are the same (h, k) values in the readout.

The axis of symmetry is the dashed vertical line through that point. Every point on the curve has a mirror on the other side of that line.

Root labels sit on the x-axis at each real zero. If the discriminant is negative, those points disappear and the readout reports no real roots.

Parabola grapher vs. a handheld calculator

This grapher runs free in the browser and redraws vertex, axis, and roots as you move the sliders. A TI-84 Plus needs Y= entry, a window setup, and separate TRACE or CALC steps to read the same features.

Desmos and GeoGebra are other online graphers with broad tools. This page stays focused on one quadratic and live labels for vertex, axis, and roots.

Key takeaways

• Graphs any vertical parabola from y = ax² + bx + c with three sliders.
• Labels the vertex, axis of symmetry, and real roots on the board.
• Accepts standard form directly; vertex form and factored form map through equivalent a, b, and c.
• Reports no real roots when the discriminant is negative.
• Runs free in the browser with no download or account.