Math · Functions

Rational Functions / Asymptotes and Behavior

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Wrap-up

What you learned

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Key takeaway

The denominator x - h creates the true vertical asymptote at x = h, so that x-value is excluded from the domain.
The constant k sets the horizontal asymptote y = k, so the graph settles toward that level far to the left and far to the right.

Common misconception

A removable hole is basically another vertical asymptote because both come from the denominator.

Near a vertical asymptote, the function values grow without bound as x approaches the forbidden value.

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

Exponential Change / Growth, Decay, and Logarithms

Change one starting value, one rate, and one target so growth, decay, doubling or half-life, and logarithmic target time all stay tied to the same live curve.

Turn visible asymptote and hole behavior into one-sided limit languageCalculus

Limits and Continuity / Approaching a Value

Move one target-distance slider inward from both sides, compare the finite-limit guide with the actual point, and separate continuous, removable-hole, jump, and blow-up behavior on one graph-first bench.

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Reference and support

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Equation snapshotShifted reciprocal family · Domain breaksShow 2 equations

Shifted reciprocal family
$f (x) = k + \frac{a}{x - h}$
Shows one reciprocal family with vertical asymptote x = h, horizontal asymptote y = k, and branch behavior controlled by a.
Domain breaks
$x \neq = h (and x \neq = p when a removable hole is shown)$
Lists the x-values the function cannot use: x = h for the true asymptote, and x = p as well when a removable hole is shown.

Why it behaves this way

Explanation

This bench treats a rational function as one shifted reciprocal family you can read directly from the graph. As you change h, k, and a, the vertical asymptote x = h, the horizontal asymptote y = k, the branches, and the intercepts move together on one plane. You can also turn on one removable hole at x = p without changing the main asymptote structure.

Read the graph in this order: first find the forbidden x-value and the long-run y-level, then check which side of y = k each branch occupies, then decide whether an extra missing point is a true asymptote or only a removable hole. A vertical asymptote makes nearby values grow without bound. A removable hole leaves a finite nearby height but deletes the point itself.

Key ideas

01The denominator x - h creates the true vertical asymptote at x = h, so that x-value is excluded from the domain.

02The 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 y = k each branch occupies, and the magnitude of a controls how sharply the graph bends near x = h.

04A removable hole adds another excluded x-value without creating blow-up there, so it must be read differently from a vertical asymptote.

Worked examples

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Frozen walkthrough

Step through the frozen example

Frozen walkthrough

Use one plain shifted reciprocal and one hole case. In each example, identify the forbidden x-value, the long-run y-level, and any extra missing point before worrying about smaller details.

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Example 1 of 2

Frozen valuesUsing frozen parameters

For the preset $f (x) = - 1 + 2/ (x - 1)$ , what are the first landmarks to identify on the graph?

E

Vertical asymptote x-value

Shifted reciprocal

1. Find the forbidden x-value

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

2. Find the long-run y-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).

Main graph 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.

Quick test

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Accessibility

Open the text-first descriptions when you need the simulation and graph translated into words.

The simulation shows one rational-function graph 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 same reciprocal family can be read from several landmarks at once.

Graph summary

The response graph shows the left-hand and right-hand values against distance from the vertical asymptote, together with the horizontal-asymptote level.

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

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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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Functions and Change

Rational Functions / Asymptotes and Behavior appears later in this track, so it is cleaner to start from the beginning first.

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