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Mathematical Architects · Algebra II

Exploring Quadratic Functions — Visual Lab

The parabola is the first curve. Move the controls and watch one quadratic appear three ways at once — and see what the vertex, the axis, and the roots really mean.

This is the Module 2 lab bench. Every quadratic can be written three equivalent ways — standard \(ax^2+bx+c\), vertex \(a(x-h)^2+k\), and factored \(a(x-r_1)(x-r_2)\). They look different, but they describe the same parabola. Drag \(a\), \(h\), and \(k\); the graph, the vertex, the axis of symmetry, the roots, and all three equations update together — and when the curve lifts off the x-axis, watch the roots turn complex.

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The Quadratic Forms Explorer

Three sliders. One parabola. Three equivalent equations — standard, vertex, and factored — all kept in sync.

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How to Read the Lab

Two minutes of orientation, then four moves to try. Everything updates live as you drag.

What you're seeing

  • The violet curve is your parabola \(f(x)=a(x-h)^2+k\). The a slider stretches it and flips it open-up or open-down; h slides the vertex left and right; k slides it up and down.
  • The filled violet dot is the vertex \((h,\,k)\) — the turning point — and the dashed vertical line is the axis of symmetry \(x = h\).
  • The amber dots are the roots (x-intercepts): where the parabola crosses the x-axis. If the curve never touches the axis, there are no real roots and the dots disappear.
  • The three cards below show the standard, vertex, and factored forms of the very same function — click a card to spotlight it.
  • The readout names the vertex, axis, direction of opening, min/max, y-intercept, and the discriminant \(b^2-4ac\) that decides how many real roots there are.

Try this

  1. Leave \(a=1\) and drag k up above \(0\). The vertex lifts off the x-axis, the amber roots vanish, and the factored card warns the roots are now complex — that's a negative discriminant.
  2. Set \(k=0\) exactly. The two roots collapse into one repeated root at the vertex — a perfect square. Read it off the factored card.
  3. Hold \(h\) and \(k\) fixed and flip a from positive to negative. The parabola opens the other way and the vertex switches from a minimum to a maximum.
  4. Move only h. Watch the axis of symmetry \(x=h\) track the vertex, and notice the standard form's middle term \(b=-2ah\) change while the vertex form just shifts.
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Key Vocabulary & Standards

The words that unlock Module 2, and the TEKS this lab is built to teach.

The vocabulary of a parabola
  • Parabola The U-shaped graph of a quadratic function. It opens up when \(a>0\) and down when \(a<0\).
  • Vertex  (h, k) The turning point of the parabola — its minimum (if it opens up) or maximum (if it opens down).
  • Axis of symmetry The vertical line \(x = h\) that splits the parabola into two mirror-image halves.
  • Roots / zeros The x-intercepts — the inputs that make \(f(x)=0\). A quadratic has \(0\), \(1\), or \(2\) real roots.
  • Discriminant The value \(b^2-4ac\). Positive → two real roots; zero → one repeated root; negative → no real roots (a complex conjugate pair).
  • Standard · Vertex · Factored form Three equivalent ways to write the same quadratic: \(ax^2+bx+c\), \(a(x-h)^2+k\), and \(a(x-r_1)(x-r_2)\).
TEKS in this lab
2A.4A2A.4B 2A.4D2A.4F 2A.4H 2A.7A2A.7B 2A.8A2A.8B2A.8C

2A.4 — Connect the standard, vertex, and factored forms of a quadratic; identify the vertex, axis of symmetry, and intercepts; describe how \(a\), \(h\), and \(k\) transform the parent \(y=x^2\).

2A.7 — Use the discriminant \(b^2-4ac\) to determine the number and type of solutions — including when the roots are complex.

2A.8 — Solve quadratic equations and interpret the vertex as a maximum or minimum in a modeling context.

Process standards 2A.1A–2A.1G — tools, representations, and reasoning — run through every move in this lab.

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Where to Go Next

Back to the course, or deeper into the plan for Module 2.

Back to the Course

The full Algebra II itinerary — all five modules, worked examples, and the Parent Function Explorer.

← Algebra II home

Course Syllabus

Policies, the studio learning environment, grading, and the full itinerary by grading period.

Read the Syllabus

Pacing Guide

Module 2 mapped to the Beta calendar — the 24 days of quadratic functions, day by day.

View the Pacing Guide

Instructor: Dr. Goodluck Ijezie-Desbois, PharmD · Beta Academy · Room: TBA
This lab is built on the TEA Bluebonnet Learning — Secondary Mathematics open curriculum, licensed CC BY-NC 4.0. Non-commercial classroom use.