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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.

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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.

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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

Why it behaves this way

Explanation

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.

Frozen walkthrough

Step through the frozen example

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

Premium unlocks saved study tools, exact-state sharing, and the richer review surfaces that support this guided flow.

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Frozen valuesUsing frozen parameters

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

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.

Make a prediction before you reveal the next step.

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 against the live bench.

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)?

Use the live bench to test the result before moving on.

Accessibility

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.