---
title: "Vertical Asymptote — AP Precalculus Definition & Exam Guide"
description: "A vertical asymptote is a vertical line a function approaches as outputs shoot toward infinity. In AP Precalc, tan θ has them wherever cos θ = 0."
canonical: "https://fiveable.me/ap-pre-calc/key-terms/vertical-asymptote"
type: "key-term"
subject: "AP Pre-Calculus"
unit: "Unit 3"
---

# Vertical Asymptote — AP Precalculus Definition & Exam Guide

## Definition

A vertical asymptote is a vertical line x = a that a function's graph approaches without ever touching, because the function is undefined at that input. In AP Precalculus, tan θ = sin θ/cos θ has vertical asymptotes wherever cos θ = 0, which is at θ = π/2 plus any integer multiple of π.

## What It Is

A vertical asymptote is a vertical line the graph gets arbitrarily close to while the [function](/ap-pre-calc/unit-1/change-tandem/study-guide/eQFiTo22fpkDFsnj "fv-autolink")'s [output values](/ap-pre-calc/unit-1/rates-change/study-guide/P6aTsM1tBCZtaEPy "fv-autolink") blow up toward positive or negative infinity. It happens at input values where the function is undefined, almost always because something is being divided by zero.

In Topic 3.8, the star example is the tangent function. Since tan θ = sin θ/cos θ (EK 3.8.A.2), tangent breaks down at every angle where cos θ = 0. On the [unit circle](/ap-pre-calc/key-terms/unit-circle "fv-autolink"), those are the angles where the terminal ray points straight up or straight down, so the ray's slope is undefined. The result is *periodic asymptotic behavior* (EK 3.8.B.2). Vertical asymptotes appear at θ = π/2 and then repeat every π units, matching tangent's period of π. Between any two consecutive asymptotes, the graph of tan θ increases the whole way and switches from concave down to concave up at the point of inflection in the middle (EK 3.8.B.3).

## Why It Matters

Vertical asymptotes live in **[Unit 3](/ap-pre-calc/unit-3 "fv-autolink") (Trigonometric and Polar Functions), Topic 3.8 (The Tangent Function)**, supporting learning objectives **[AP Pre Calc](/ap-pre-calc "fv-autolink") 3.8.B** (describe key characteristics of the tangent function) and **AP Pre Calc 3.8.C** (describe transformations of tangent). The asymptotes ARE the skeleton of the tangent graph. If you can locate them, you can sketch the whole function, because each branch of tangent just repeats between consecutive asymptotes. Transformations test whether you understand that skeleton. A phase shift slides the asymptotes left or right, a horizontal compression like tan(2x) packs them closer together, and a vertical dilation leaves them exactly where they were. The exam loves asking which transformations move asymptotes and which don't.

## Connections

### Tangent Function (Unit 3)

This is the home base. Tangent gives the slope of the [terminal ray](/ap-pre-calc/key-terms/terminal-ray "fv-autolink") on the unit circle, and slope is undefined when the ray is vertical. Every vertical asymptote of tan θ is just the graph admitting 'a vertical line has no slope.'

### [Phase Shift (Unit 3)](/ap-pre-calc/key-terms/phase-shift)

A [phase shift](/ap-pre-calc/key-terms/phase-shift "fv-autolink") moves vertical asymptotes by the same amount it moves everything else. For g(θ) = tan(θ + c), the asymptote that used to sit at π/2 now sits at π/2 - c. Tracking one asymptote is the fastest way to find c.

### [Domain (Unit 1)](/ap-pre-calc/key-terms/domain)

Vertical asymptotes and domain are two views of the same fact. The asymptote locations of tan θ are exactly the values excluded from its domain. The same logic powers [Unit 1](/ap-pre-calc/unit-1 "fv-autolink")'s rational functions, where a zero of the denominator that doesn't cancel creates a vertical asymptote.

### Limit (Units 1 and 3)

Asymptotic behavior is limit language in disguise. Saying tan θ has a vertical asymptote at π/2 means the output values increase or decrease without bound as θ approaches π/2. That's the unbounded-limit idea AP Precalc uses to describe end behavior and asymptotes alike.

## On the AP Exam

Vertical asymptotes show up in multiple-choice questions that make you locate them after a transformation. Typical stems give you g(x) = tan(2x) and ask where the asymptotes are (the period shrinks to π/2, so asymptotes land at π/4 plus multiples of π/2), or give you tan(θ + c) with a known asymptote location and ask you to solve for c. You should be able to do three things fast. First, find asymptotes of plain tan θ by solving cos θ = 0. Second, apply transformations to those locations (horizontal shifts and compressions move them, vertical dilations and shifts don't). Third, use consecutive asymptotes to describe behavior between them, since tangent increases and flips from concave down to concave up across each branch. No released FRQ has centered on this term verbatim, but asymptote reasoning is exactly the kind of 'describe the function's behavior' justification the free-response section rewards.

## Vertical Asymptote vs Horizontal asymptote

A vertical asymptote is a forbidden input. The function is undefined there, outputs explode to ±∞, and the graph can never touch or cross it. A horizontal asymptote describes end behavior, meaning what the outputs settle toward as the input runs off to ±∞, and a graph absolutely can cross a horizontal asymptote in the middle. Tangent has infinitely many vertical asymptotes and no horizontal ones, since its outputs never settle down.

## Key Takeaways

- A vertical asymptote occurs at an input value where the function is undefined and the outputs increase or decrease without bound near that value.
- Because tan θ = sin θ/cos θ, the tangent function has vertical asymptotes exactly where cos θ = 0, at θ = π/2 plus any integer multiple of π.
- Tangent's asymptotes repeat every π units, matching its period of π, which the CED calls periodic asymptotic behavior.
- Between consecutive asymptotes, tan θ always increases and changes from concave down to concave up at its point of inflection.
- Horizontal shifts and horizontal compressions move tangent's vertical asymptotes, but vertical dilations and vertical shifts leave them in place.
- A function can never cross a vertical asymptote, because the function literally has no value at that input.

## FAQs

### What is a vertical asymptote in AP Precalculus?

It's a vertical line x = a that a graph approaches but never reaches, occurring where the function is undefined and outputs run off to ±∞. In Unit 3, tan θ has vertical asymptotes at θ = π/2 + kπ for every integer k, because cos θ = 0 at those angles.

### Can a function ever cross a vertical asymptote?

No. The function has no output at all at a vertical asymptote, so the graph can't touch or cross it. (Horizontal asymptotes are different; graphs can cross those.)

### Why does the tangent function have vertical asymptotes?

Tangent is the slope of the terminal ray on the unit circle, and tan θ = sin θ/cos θ. When the terminal ray is vertical, like at θ = π/2, cos θ = 0 and the slope is undefined, so the graph has a vertical asymptote there.

### How is a vertical asymptote different from a horizontal asymptote?

A vertical asymptote is about a single bad input where outputs blow up to infinity. A horizontal asymptote is about end behavior, the value outputs approach as the input goes to ±∞. Tangent has infinitely many vertical asymptotes and zero horizontal ones.

### Do transformations move the vertical asymptotes of tan θ?

Horizontal ones do. The function tan(θ + c) shifts every asymptote left by c, and tan(2x) compresses them so they sit at π/4 plus multiples of π/2. Vertical dilations like 5 tan θ or -5 tan θ stretch or reflect the branches but leave the asymptotes exactly where they were.

## Related Study Guides

- [3.8 The Tangent Function](/ap-pre-calc/unit-3/tangent-function/study-guide/MmhWdpovNDRCpyBgCc0x)

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