Extending Linear Relationships — Visual Lab
Module 1. A single hinge turns a straight line into a V — and a stack of straight lines fences off a region. Drive the controls and watch both ideas come to life on the coordinate plane.
The absolute-value function \(y = a\,|x - h| + k\) is the linear toolkit's first bend: the same shift-and-stretch logic you'll use on every function family, made visible as a moving V. A system of linear inequalities takes those same lines and uses them as fences, trapping a feasible region — the first step toward linear programming. Switch modes below and keep the graph, the algebra, and the plain-English readout in perfect agreement.
Absolute Value & Systems Lab
Pick a mode, then drag the sliders. In Absolute Value, build \(y = a|x - h| + k\) and read off its vertex, axis, and opening. In System of Inequalities, shade the constraints and find the feasible region's corners.
What you're seeing
- The faded dashed V is the parent. It's \(y = |x|\) — the un-transformed shape. The solid V is your function, so the gap between them is the transformation.
- \(h\) and \(k\) move the vertex. The hinge sits exactly at \((h, k)\): \(h\) slides it left/right, \(k\) slides it up/down. The dashed vertical line is the axis of symmetry \(x = h\).
- \(a\) stretches and reflects. \(|a| > 1\) makes the V narrower, \(|a| < 1\) makes it wider, and a negative \(a\) flips it to open downward (a maximum instead of a minimum).
- A system fences off a region. In the second mode, each inequality keeps one side of a line. The shaded polygon — the feasible region — is where every condition is satisfied at once. Its corners are where the boundary lines cross.
- The corner principle (LP). Turn on the objective line: in linear programming the largest value of \(P\) always lands on a corner of the region, never in its interior.
Try this
- Move only \(h\) and \(k\). Keep \(a = 1\). Predict the vertex before you read it — the vertex is always \((h, k)\), no exceptions.
- Make \(a\) negative. Watch the V flip to open downward. Notice how the "Range" readout switches from \(y \ge k\) to \(y \le k\) — the vertex becomes a maximum.
- Push \(|a|\) past 1, then below 1. Find the value of \(a\) that makes your V twice as steep as the parent. (Hint: read the "Width" line.)
- Switch to System mode and turn on the objective line. Slide the objective tilt and watch the "Best corner" jump from one vertex of the region to another. Which corner wins when the tilt is small? When it's large?
Key Vocabulary
The precise words a mathematician uses to describe what the sliders are doing.
A function of the form \(y = a\,|x - h| + k\) whose graph is a V (or inverted V), built from the parent \(y = |x|\).
The hinge of the V, located at \((h, k)\). It is the minimum when \(a > 0\) and the maximum when \(a < 0\).
The vertical line \(x = h\) that splits the V into two mirror-image halves.
A shift (\(h, k\)), stretch/compression, or reflection that reshapes a parent function without changing its family.
Two or more linear inequalities that must all hold at once. Their overlap defines a region of valid points.
The set of all points satisfying every constraint — the shaded polygon. Its corners are the candidates for an optimum in linear programming.
TEKS & Function Key-Features
This lab targets the Absolute Value Functions and Applications of Linear Relationships strands of Module 1, where scholars meet the \(a(x-h)+k\) transformation frame and use systems of linear inequalities to model and constrain real situations.
2A.6C–F analyze and transform absolute-value functions and read their attributes (vertex, axis, domain, range) · 2A.3A–G formulate, graph, and solve systems of two and three linear equations and inequalities, including feasible regions for optimization. Transformations connect back to 2A.2A parent functions.
Ready for the full course map? Head back to Algebra II, read the Syllabus, or check the Pacing Guide to see where Module 1 sits in the year.
Module and topic structure follow the TEA Bluebonnet Learning — Secondary Mathematics open curriculum (Edition 1, adapted from Carnegie Learning), licensed CC BY-NC 4.0. Classroom use is non-commercial.