Quadratic graph explorer

What the explorer shows

A quadratic function makes a U-shaped curve called a parabola. In Standard form you set a, b and c in y = ax² + bx + c; a can be −3, −2, −1, −1/2, 1/2, 1, 2 or 3 (never 0, which would make a straight line), and b and c run from −10 to 10. In Vertex form you set a, h and k in y = a(x − h)² + k instead.

The graph marks the vertex in yellow, the axis of symmetry as a dashed line, the roots in red where the curve crosses the x-axis, and the y-intercept in blue. The readout writes the other form of the equation, the factored form when the roots are whole numbers or fractions, and whether the parabola has two roots, one or none.

Why explore parabolas this way

Each number in a quadratic changes the graph in its own way, and the forms show different things at a glance. In standard form, c is the y-intercept. In vertex form, h and k are the vertex. In factored form, the roots can be read off. Changing a number and seeing every feature update helps students connect the algebra (completing the square, factoring) to the picture, so the forms stop being separate tricks.

Common mix-ups

• Reading the vertex of y = (x − 3)² as (−3, 0). The vertex is at x = 3: the sign inside the bracket flips.
• Thinking b moves the parabola left and right on its own. Changing b moves the vertex along a curved path, both sideways and up or down.
• Thinking every parabola crosses the x-axis. Some have one root and some have none.
• Thinking a bigger a makes a wider parabola. A bigger a (in size) makes it narrower.

A 10-minute quadratics lesson

• Start with y = x² − 4. Ask for the roots and vertex before reading them.
• Change c to 0, then to 4, and ask what happened to the roots.
• Switch to Vertex form and set h = 2 and k = −1. Ask students to expand it by hand and compare with the readout.
• Try a roots goal from New goal.
• Finish with Match this parabola.

Where it fits by grade

• Algebra 1: graph quadratic functions and show intercepts, maxima and minima (HSF-IF.C.7a), and factor to reveal the zeros (HSA-SSE.B.3a).
• Algebra 1 and 2: complete the square to reveal the maximum or minimum (HSA-SSE.B.3b, HSF-IF.C.8a) and see how adding k or replacing x with x − h shifts a graph (HSF-BF.B.3).
• England: GCSE (Years 10 and 11) asks students to identify roots, intercepts and turning points of quadratic graphs, and higher tier finds turning points by completing the square. A level continues the work.

Learning more

Exponents, negative numbers and solving equations are skills on Eurekami's skill map, and Daily Practice brings them back when they start to fade. Exam Prep includes GCSE Maths practice papers, and the AI tutor asks guiding questions when a quadratic problem is tricky.