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Maximizing & Minimizing — Module Hub & Visuals
Every quadratic has one turning point — the highest or lowest value it can reach. Move the dials, and watch where it lands.
Lesson Slides & PDFs
Student-facing lesson slide PDFs are grouped by topic below for review, catch-up, and family support.
📖 Student Edition for this module (PDF) ✍ Skills Practice — Student Edition (PDF)
Graphic Organizers & Activities
Foldable notes and activities for each lesson, with the one to start with listed first. Sign in with your alias and PIN to open them.
Start Here — What Module 5 Is About
What this module is for, and what you will be able to do by the end of it.
Module 05 · Topic 1 · Quadratic Functions
This is the Parabola Lab. The equation \(y = ax^2 + bx + c\) is on a set of three sliders.
Drag any one and the curve redraws instantly — vertex, axis of symmetry, and roots all follow.
The whole point of Module 5 is finding that turning point and reading what it means, so here you can
see it move before you ever solve for it on paper.
Key Vocabulary
The precise words for what you just investigated. The standards this module carries are in the next section.
Parabola
The U-shaped graph of a quadratic function \(y = ax^2 + bx + c\), where \(a \ne 0\).
Vertex
The single turning point — the maximum or minimum value of the function, at \(x = -\tfrac{b}{2a}\).
Axis of Symmetry
The vertical line \(x = -\tfrac{b}{2a}\) that mirrors the parabola onto itself.
Roots / Zeros
The x-values where \(y = 0\) — where the curve meets the x-axis. Also called solutions or x-intercepts.
Discriminant
\(b^2 - 4ac\). Positive → two real roots, zero → one, negative → none.
Three Forms
Standard \(ax^2+bx+c\), vertex \(a(x-h)^2+k\), and factored \(a(x-r_1)(x-r_2)\) — one curve, three views.
Standards in This Module
The Texas Essential Knowledge and Skills this module is built to teach. RDY marks a STAAR readiness standard.
A.6 — graphing quadratic functions · A.7 — attributes of parabolas (vertex, axis, roots, direction) · A.8 — solving quadratic equations. RDY marks STAAR readiness standards.
Make It Make Sense
A quick read on what the lab is showing you, and a few experiments to run.
What you're seeing
- a sets direction and width. Positive opens up (a minimum); negative opens down (a maximum). Bigger \(|a|\) makes a narrower curve.
- c is the y-intercept — the curve always passes through \((0, c)\).
- The vertex sits at \(x = -\tfrac{b}{2a}\); the axis of symmetry is the vertical line through it.
- The roots are where the parabola crosses the x-axis. The discriminant \(b^2 - 4ac\) decides whether there are two, one, or none.
Try this
- Set a = 1, b = 0, c = 0 for the parent \(y = x^2\). Now slide c up and down — the whole curve lifts and drops, vertex and all.
- Make a negative. Watch the min/max banner flip from MINIMUM to MAXIMUM as the parabola turns over.
- Find a quadratic with no real roots (lift it off the x-axis) and confirm the discriminant goes negative.
- Switch to the Factored form and tune the sliders until the two factors show clean whole-number roots.
The Parabola Lab
Drag a, b, and c. Then switch the form panel between standard, vertex, and factored — all three describe the same curve on the screen.
Parabola Lab. The filled point is the vertex — the maximum if the curve opens down, the minimum if it opens up. The dashed line is the axis of symmetry. The points on the x-axis are the real roots (where \(y = 0\)); when the discriminant \(b^2 - 4ac < 0\) the curve lifts off the axis and there are none.
Want to See This in 3D?
Optional, and always second. The lab above is where you measure and calculate; this is the same idea as an everyday object you can turn over and inspect. Open it when the lab already makes sense — or skip it entirely.