Extending Beyond Polynomials — Visual Lab
Module 4. Dividing and rooting break the polynomial mold. Rational functions add asymptotes and holes; radical functions add domain restrictions. Drive the controls below and watch the structure appear in real time.
This is a working drafting table, not a picture. Move a slider and the curve, its asymptotes, its holes, and its domain & range redraw instantly — with a plain-English readout naming every feature. Switch between the Rational and Radical modes with the tabs.
Rational & Radical Lab
Two function families, one frame. Tab between them, drag the sliders, and read what each control does to the graph.
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What You're Seeing & What To Try
A quick orientation, then four experiments to run on the lab above.
What you're seeing
Rational mode graphs \(f(x)=\dfrac{a(x-z)}{(x-p)(x-q)}\). The red dashed lines are vertical asymptotes at the denominator's zeros \(x=p\) and \(x=q\) — the function shoots toward \(\pm\infty\) there. The blue dashed line is the horizontal asymptote, which the curve flattens toward far left and right. Turn on "Make a hole" and the factor \((x-p)\) cancels: that asymptote collapses into a single hole (an open circle) — the graph is undefined at one point but no longer blows up.
Radical mode graphs \(y = a\sqrt{x-h}+k\). The faint dashed curve is the parent \(\sqrt{x}\); the solid curve is your transform. The shaded band on the right is the domain (\(x \ge h\)) and the shaded band on the \(y\)-side is the range.
Try this
- In Rational mode, drag \(p\) and \(q\) together until they meet. What happens to the two vertical asymptotes when \(p=q\)?
- Turn on "Make a hole." Watch the asymptote at \(x=p\) become an open circle. Where exactly is the hole — and why isn't the function just zero there?
- Set the numerator zero \(z\) equal to the asymptote \(q\). Where does the \(x\)-intercept go, and what does that tell you about reading intercepts off a graph?
- Switch to Radical mode. Make \(a\) negative and watch the curve flip. How do the domain and the range each respond to changing \(h\) versus \(k\)?
Key Vocabulary & Standards
The three words that drive this module — and the TEKS this lab is built to make visible.
- Vertical asymptote — a vertical line \(x=c\) the graph approaches but never touches, occurring at a denominator zero that does not cancel.
- Horizontal asymptote — a horizontal line \(y=c\) the graph levels off toward as \(x\to\pm\infty\); found by comparing the degrees of numerator and denominator.
- Hole (removable discontinuity) — a single missing point where a common factor cancels from numerator and denominator.
- Domain — the set of allowed inputs. For \(\sqrt{x-h}\), the inside must be \(\ge 0\), so \(x \ge h\).
- Range — the set of output values; for \(a\sqrt{x-h}+k\) it is \(y\ge k\) (or \(y\le k\) when \(a<0\)).
This lab targets the rational- and radical-function strands of Algebra II, with the transformation and attribute standards woven through.
2A.6 rational & radical functions and their attributes · 2A.2 domain, range, and attributes of parent functions · 2A.4 transformations \(a\,f(x-h)+k\).
Back to the course
Return to the full Algebra II course page — all five modules, worked examples, materials, and resources.
← Back to Algebra IIPlan & pacing
See where Module 4 sits in the year, and the policies behind the course.
Syllabus Pacing Guide