Math · Interactive
Quadratic Function
Every quadratic function y = ax² + bx + c draws a parabola. Drag a, b, and c below to reshape it and watch the vertex, axis of symmetry, roots, and discriminant update live. Free to use, and exportable into your slides.
Drag the a, b, and c sliders to reshape the parabola. Open fullscreen ↗
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What is a quadratic function?
A quadratic function is a function of the form y = ax^2 + bx + c, where a is not zero. Its graph is a smooth U-shaped curve called a parabola. The coefficient a controls how wide the parabola is and which way it opens: when a is positive the parabola opens upward, and when a is negative it opens downward. Drag the sliders in the grapher above to see how a, b, and c reshape the curve.
The parts of a parabola
Every parabola has a vertex, the highest or lowest point, and a vertical axis of symmetry through it. Where the curve crosses the x-axis are the roots (also called x-intercepts or zeros), and where it crosses the y-axis is the y-intercept, which is simply the value c. The grapher marks the vertex, the roots, and the y-intercept so you can connect the picture to the numbers.
How to find the vertex and axis of symmetry
The axis of symmetry is the vertical line x = -b / (2a), and the x-coordinate of the vertex is the same value. Substitute that x back into the function to get the y-coordinate of the vertex. For example, for y = x^2 - 4x + 3 the axis is x = 2, and the vertex is (2, -1).
The discriminant and the number of roots
The discriminant is b^2 - 4ac. If it is positive the parabola crosses the x-axis at two points (two real roots), if it is exactly zero it touches the x-axis at one point (a double root), and if it is negative the parabola never touches the x-axis (no real roots). The roots themselves come from the quadratic formula, x = (-b plus or minus the square root of the discriminant) / (2a).
Frequently asked questions
What is a quadratic function?
A quadratic function has the form y = ax^2 + bx + c with a not equal to 0. Its graph is a parabola, a symmetric U-shaped curve.
How do you find the vertex of a quadratic function?
The vertex x-coordinate is x = -b / (2a). Substitute that value back into the function to get the y-coordinate. That point is the minimum if a is positive or the maximum if a is negative.
What does the coefficient a do?
The sign of a decides the direction: positive opens upward, negative opens downward. The size of a controls the width: a larger absolute value makes a narrower parabola, a smaller one makes it wider.
How many roots does a quadratic function have?
It depends on the discriminant b^2 - 4ac. Positive gives two real roots, zero gives one repeated root, and negative gives no real roots (the parabola does not cross the x-axis).
How do I use this quadratic grapher?
Drag the a, b, and c sliders to change the equation. The parabola redraws instantly, and the panel shows the vertex, axis of symmetry, roots, and discriminant. You can also export the graph to use in your own slides.