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Concept module

Rational Functions / Asymptotes and Behavior

Vary one shifted reciprocal family so domain breaks, vertical and horizontal asymptotes, intercepts, and removable-hole behavior stay tied to the same graph.

Interactive lab

Keep the stage, immediate control feedback, and linked study tools in one working bench.

Rational functions

Drag either probe marker or use the controls to inspect the forbidden x-value.

Controls

Adjust the live parameters and watch the bench respond.

Vertical asymptote x-value

h

Move the forbidden x-value left or right.

Horizontal asymptote y-value

k

-1

Lift or lower the long-run level y = k.

Branch scale

a

Flip which branch sits above y = k and change how strongly the graph bends near x = h.

Probe distance

d

0.55

Move the sample points closer to or farther from the vertical asymptote.

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Show

Show removable holeAdd one canceled-factor hole without moving the true asymptotes.

Hole x-value

p

1.6

Place the removable hole when the canceled-factor view is turned on.

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

Graph readingPrompt 1 of 3

Shrink the probe distance d. The branch values shoot away much faster near x = h, which is why the vertical asymptote is not just another missing point.

Graphs

Switch graph views without breaking the live stage and time link.

Near-asymptote response

Shows how the left and right branch values change as the probe points move toward the vertical asymptote.

distance from x = h: 0.25 to 2.2f(x): -10 to 10

Left of x = hRight of x = hHorizontal asymptote

Hover or scrub to link the graph back to the stage.distance from x = h / f(x)

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

h

Vertical asymptote x-value

Moves the forbidden x-value left or right, which shifts the whole family horizontally without changing the horizontal asymptote level.

Graph: Near-asymptote responseOverlay: AsymptotesOverlay: Near-asymptote probes

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Use the live prompts to keep domain breaks, asymptotes, and removable holes on the same graph story.

Graph readingPrompt 1 of 3

Graph: Near-asymptote response

Shrink the probe distance d. The branch values shoot away much faster near x = h, which is why the vertical asymptote is not just another missing point.

Control: Probe distanceControl: Branch scaleGraph: Near-asymptote responseOverlay: Near-asymptote probesOverlay: AsymptotesEquation

d

Equation

a

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

4 visible

Overlay focus

Asymptotes

Show the vertical and horizontal asymptote guides.

What to notice

Moving h shifts the vertical asymptote left or right, while moving k lifts or lowers the horizontal one.

Why it matters

It keeps the long-run and forbidden-value behavior visible on the same graph.

Control: Vertical asymptote x-valueControl: Horizontal asymptote y-valueGraph: Near-asymptote responseEquation

h

Equation

k

Challenge mode

Use the asymptote graph as a domain-break bench. The goal is to separate the true asymptote from a removable hole while the branch orientation still matches the long-run level.

0/1 solved

ConditionCore

4 of 8 checks

Domain-break checkpoint

Build a reciprocal family where the true vertical asymptote sits near

x = - 1

, the removable hole sits on positive

x

, and the right branch stays below the horizontal asymptote.

Graph-linkedGuided start2 hints

Suggested start

Start from the clean shifted reciprocal, then turn the hole on and move the family until the real asymptote and the removable break tell different graph stories.

Matched

Open the Near-asymptote response graph.

Near-asymptote response

Matched

Keep the Asymptotes visible.

Matched

Keep the Near-asymptote probes visible.

Matched

Keep the Removable hole marker visible.

Pending

Move the true vertical asymptote into the left-side band between

- 1.1

and

- 0.9

Pending

Keep the horizontal asymptote above the x-axis, between

0.7

and

0.95

-1

Pending

Flip the branch scale negative, between

- 2.1

and

- 1.5

, so the right branch stays below

y = k

Pending

Place the removable hole on positive x, between

1.15

and

1.45

1.6

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

The reciprocal family has vertical asymptote x = 1 and horizontal asymptote y = -1. At distance d = 0.55, the left branch is -4.64 and the right branch is 2.64. To the right of the vertical asymptote the branch sits above the horizontal asymptote, while the left branch sits below it. The domain breaks at 1, with x-intercept near 3, y-intercept -3.

Equation detailsDeeper interpretation, notes, and worked variable context.

Shifted reciprocal family

Keeps one vertical asymptote at x = h and one horizontal asymptote at y = k while a sets the branch orientation and bend strength.

f (x) = k + \frac{a}{x - h}

h

Vertical asymptote x-value 1

k

Horizontal asymptote y-value -1

a

Branch scale 2

Domain breaks

Marks the x-values that the graph cannot use because the denominator or canceled-factor seam removes them.

x \neq = h (and x \neq = p when a removable hole is shown)

h

Vertical asymptote x-value 1

d

Probe distance 0.55

p

Hole x-value 1.6

Horizontal asymptote

Shows that the reciprocal part fades far from the vertical break, so the graph settles toward the level y = k.

x \to \pm \infty lim f (x) = k

k

Horizontal asymptote y-value -1

a

Branch scale 2

Progress

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Let the live model runChange one real controlOpen What to notice

Try this setup

Jump to a named bench state or copy the one you are looking at now. Shared links reopen the same controls, graph, overlays, and compare context.

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Current bench

Shifted reciprocal preset

This bench still matches one named preset, so the copied link will reopen that same starting point along with the current graph, overlays, and inspect context.

Open default bench

Featured setups

Shifted reciprocal

Open a clean asymptote case with both intercept markers visible.

Removable hole

Open the canceled-factor case and compare the extra domain break with the same asymptote family.

Saved setups

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Checking saved setup access.

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Stable links

Starter track

Step 2 of 60 / 6 complete

Functions and Change

Earlier steps still set up Rational Functions / Asymptotes and Behavior.

1. Graph Transformations2. Rational Functions / Asymptotes and Behavior3. Exponential Change / Growth, Decay, and Logarithms4. Derivative as Slope / Local Rate of Change+2 more steps

Previous step: Graph Transformations.

Start from track beginning

Short explanation

What the system is doing

Rational functions become easier to trust when one shifted reciprocal family stays on the same graph while the asymptotes, probe points, intercepts, and optional removable hole all move together. This bench keeps the family bounded on purpose: one vertical break at x = h, one horizontal level at y = k, one branch scale a, and one optional canceled-factor hole at x = p.

The goal is not to turn this page into a symbolic simplifier. The goal is to make the graph behavior honest. Learners should be able to see which x-values are forbidden, which side of the horizontal asymptote each branch lives on, how the graph blows up near the vertical asymptote, and what a removable hole changes without pretending the whole family has been transformed into a giant algebra engine.

Key ideas

01The denominator decides the forbidden x-value x = h, so the graph breaks there and the domain excludes that point.

02The added constant k sets the horizontal asymptote y = k, so the graph settles toward that level far to the left and far to the right.

03The sign of a decides which side of the horizontal asymptote each branch occupies, and the magnitude of a controls how strongly the graph bends near the vertical asymptote.

Worked example

Read the full frozen walkthrough.

Frozen walkthrough

Use two fixed cases to keep asymptotes, domain breaks, and removable holes tied to one visible reciprocal family.

Live worked examples are available on Premium. You can still read the full frozen walkthrough on the free tier.

View plans

Frozen valuesUsing frozen parameters

For the preset f(x) = -1 + 2/(x - 1), what should the graph show first?

E

Example case

Shifted reciprocal

1. Read the forbidden denominator value

The denominator vanishes at x = 1, so the graph has a vertical asymptote there and the domain excludes x = 1.

2. Read the long-run level

Far from the asymptote, the reciprocal part fades and the graph settles toward y = -1, so that is the horizontal asymptote.

3. Check the intercepts on the same graph

Setting y = 0 gives x = 3, and plugging in x = 0 gives y = -3, so the graph crosses at (3, 0) and (0, -3).

Visible landmarks

VA: x = 1, HA: y = -1, x-int: (3, 0), y-int: (0, -3)

This is the clean shifted-reciprocal story: one forbidden x-value, one horizontal level, and intercepts that still belong to the same family.

Common misconception

A removable hole should behave exactly like a vertical asymptote because both come from the denominator.

A vertical asymptote is where the graph grows without bound as x approaches the forbidden value.

A removable hole is different: the nearby curve still approaches one finite height, but the function is left undefined at that one x-value.

Mini challenge

Set the family so the vertical asymptote is left of the y-axis, the right branch sits below the horizontal asymptote, and a removable hole appears on positive x.

Prediction prompt

Decide which control changes the asymptote position, which one flips the branch orientation, and which one only adds a removable break before you test it.

Check your reasoning

You need a negative h-value, a negative branch scale, and the removable-hole toggle turned on with the hole placed to the right of the y-axis.

The asymptote position comes from h, the branch orientation comes from the sign of a, and the removable hole is a second domain break that does not move the asymptotes.

Quick test

Graph reading

Question 1 of 3

Answer from the graph behavior, not from a detached list of algebra rules.

What is the cleanest reason x = h is excluded from the domain of k + a/(x - h)?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

The simulation shows one rational-function graph on a coordinate plane with dashed asymptote guides, two movable probe markers near the vertical asymptote, visible intercept markers when they exist, and an optional open-circle hole marker.

Sliders move the vertical asymptote, the horizontal asymptote, the branch scale, the probe distance, and the optional hole location so the family stays tied to one compact graph.

Graph summary

The graph tab plots the left-hand and right-hand branch values against distance from the vertical asymptote, together with a dashed horizontal-asymptote guide.

That response view makes it easier to compare blow-up near the forbidden x-value with the long-run level the family approaches away from it.

Keep The Function Branch Coherent

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Compare a second non-polynomial family without leaving the graph-first functions branchFunctions

Rational Functions / Asymptotes and Behavior

See each variable before you move it.

Asymptotes

Domain-break checkpoint

Functions and Change

What the system is doing

Read the full frozen walkthrough.

For the preset f(x) = -1 + 2/(x - 1), what should the graph show first?

What is the cleanest reason x = h is excluded from the domain of k + a/(x - h)?

Keep moving through the concept map.

Exponential Change / Growth, Decay, and Logarithms

Limits and Continuity / Approaching a Value

Graph Transformations