Quadratic Grapher — Step-by-Step Solutions

Q: How do I find the vertex of a parabola?

Use x = -b / (2a) for the x-coordinate, then put that x into the equation to get y. The Vertex Form Conversion mode shows both coordinates.

Q: What does the sign of a tell me?

A positive a means the parabola opens upward with a minimum at the vertex; a negative a means it opens downward with a maximum.

Q: What does the discriminant mean?

It is b^2 - 4ac. Positive means two x-intercepts, zero means the vertex touches the x-axis at one point, and negative means the parabola never reaches the x-axis.

Q: How do I find the y-intercept?

Set x = 0. The y-intercept is the point (0, c).

Q: How is this different from the Quadratic Formula Calculator?

This tool focuses on the graph and the shape of the parabola: vertex, axis, intercepts and discriminant. The Quadratic Formula Calculator focuses on solving the equation step by step.

Q: What if a is zero?

Then the equation is linear, not quadratic, and has no parabola or vertex. Use a non-zero a, or use the Slope Calculator for a straight line.

The Quadratic Grapher plots quadratic functions of the form y = ax² + bx + c and visualises the parabola, vertex, axis of symmetry, and roots. It helps you understand how the coefficients a, b, and c affect the shape and position of the graph.

Enter your coefficients to see an interactive graph with key features highlighted. Switch to vertex form or analyse the discriminant to understand the nature of the roots.

Enter coefficients for y = ax² + bx + c

Parabola Graph

Vertex Roots Y-intercept Axis of symmetry

Vertex

(1, -4)

Axis of Symmetry

x = 1

Y-Intercept

(0, -3)

Discriminant

Roots

x₁ = 3, x₂ = -1

Opens

Upward

Your calculations will appear here

How to Use the Quadratic Grapher

Enter the coefficients a, b and c of y = ax^2 + bx + c. The fields start with an example (1, -2, -3); replace them with your own values, and enter 0 for a missing term.
Graph Parabola plots the curve and lists the vertex, axis of symmetry, y-intercept, discriminant, the roots and the direction it opens.
Vertex Form Conversion rewrites the equation as y = a(x - h)^2 + k and states whether the vertex is a minimum or a maximum.
Discriminant Analysis shows b^2 - 4ac, whether the parabola crosses the x-axis, and the nature of the roots with the working.

Formula and Theory

The graph of y = ax^2 + bx + c is a parabola. If a is positive it opens upward and the vertex is the lowest point; if a is negative it opens downward and the vertex is the highest point. The size of a controls the width: a larger |a| makes a narrower parabola.

The vertex is at x = -b / (2a), and its y-value is found by substituting that x. The axis of symmetry is the vertical line through the vertex, x = -b / (2a). The y-intercept is the point (0, c), where the curve crosses the y-axis.

The discriminant D = b^2 - 4ac decides the roots, the points where the parabola crosses the x-axis. If D is positive there are two real roots, if D = 0 there is one repeated root and the vertex touches the axis, and if D is negative the parabola does not cross the x-axis and the roots are a pair of complex conjugates. The roots themselves come from x = (-b plus or minus sqrt(D)) / (2a).

Worked Examples

Example 1: Parabola with two real roots

Problem: Analyse y = x^2 - 4x + 3.

Solution: Vertex x = -(-4) / (2 x 1) = 2, and y = 4 - 8 + 3 = -1, so the vertex is (2, -1). D = 16 - 12 = 4 > 0. Roots: x = (4 plus or minus 2) / 2 = 3 and 1. The y-intercept is (0, 3). Vertex form: y = (x - 2)^2 - 1.

Answer: Vertex (2, -1), roots x = 1 and x = 3, y-intercept 3

Example 2: Parabola opening downward

Problem: Analyse y = -2x^2 + 4x + 1.

Solution: Vertex x = -4 / (2 x (-2)) = 1, and y = -2 + 4 + 1 = 3, so the vertex is (1, 3), a maximum because a is negative. D = 16 - 4(-2)(1) = 24. Roots: x = (-4 plus or minus 4.899) / (-4) = 2.2247 and -0.2247. Vertex form: y = -2(x - 1)^2 + 3.

Answer: Maximum at (1, 3); roots x = 2.2247 and x = -0.2247

Example 3: A repeated root

Problem: Analyse y = x^2 - 6x + 9.

Solution: D = 36 - 36 = 0, so there is one repeated root. Vertex x = 6 / 2 = 3, y = 9 - 18 + 9 = 0, so the vertex is (3, 0) and sits on the x-axis. The equation is y = (x - 3)^2.

Answer: Vertex (3, 0), repeated root x = 3

Example 4: No real roots

Problem: Analyse y = 2x^2 + 4x + 5.

Solution: D = 16 - 40 = -24 < 0, so the parabola does not cross the x-axis. Vertex x = -4 / 4 = -1, y = 2 - 4 + 5 = 3, so the vertex is (-1, 3), a minimum above the axis. Complex roots: x = -1 plus or minus (sqrt(24) / 4) i = -1 plus or minus 1.2247i.

Answer: Vertex (-1, 3); complex roots -1 plus or minus 1.2247i

Frequently Asked Questions

How do I find the vertex of a parabola?

Use x = -b / (2a) for the x-coordinate, then put that x into the equation to get y. The Vertex Form Conversion mode shows both coordinates.

What does the sign of a tell me?

A positive a means the parabola opens upward with a minimum at the vertex; a negative a means it opens downward with a maximum.

What does the discriminant mean?

It is b^2 - 4ac. Positive means two x-intercepts, zero means the vertex touches the x-axis at one point, and negative means the parabola never reaches the x-axis.

How do I find the y-intercept?

Set x = 0. The y-intercept is the point (0, c).

How is this different from the Quadratic Formula Calculator?

This tool focuses on the graph and the shape of the parabola: vertex, axis, intercepts and discriminant. The Quadratic Formula Calculator focuses on solving the equation step by step.

What if a is zero?

Then the equation is linear, not quadratic, and has no parabola or vertex. Use a non-zero a, or use the Slope Calculator for a straight line.