---
title: "Vertical Dilation — AP Precalc Definition & Exam Guide"
description: "Vertical dilation stretches or compresses a graph by a factor of |a| in g(θ) = a tan θ. Learn how AP Precalc tests it on the tangent function in Topic 3.8."
canonical: "https://fiveable.me/ap-pre-calc/key-terms/vertical-dilation"
type: "key-term"
subject: "AP Pre-Calculus"
unit: "Unit 3"
---

# Vertical Dilation — AP Precalc Definition & Exam Guide

## Definition

Vertical dilation is the transformation g(θ) = a·tan θ that stretches (|a| > 1) or compresses (|a| < 1) the graph of tan θ vertically by a factor of |a|; if a is negative, the graph is also reflected over the x-axis (AP Precalc essential knowledge 3.8.C.3).

## What It Is

A vertical dilation multiplies every [output](/ap-pre-calc/unit-1/change-tandem/study-guide/eQFiTo22fpkDFsnj "fv-autolink") of a function by a constant. For the tangent function, that means going from f(θ) = tan θ to g(θ) = a tan θ. Every y-value on the graph gets scaled by a factor of |a|. If |a| > 1, the graph stretches taller (it gets steeper between [asymptotes](/ap-pre-calc/key-terms/asymptote "fv-autolink")). If 0 < |a| < 1, the graph compresses toward the x-axis. And if a is negative, the dilation comes with a bonus: a reflection over the x-axis, so the tangent curve flips from increasing to decreasing between consecutive asymptotes.

Here's the part that trips people up. A vertical dilation only touches outputs, never inputs. So for a tan θ, the period is still π, the [vertical asymptotes](/ap-pre-calc/unit-3/secant-cosecant-cotangent-functions/study-guide/nhIcN0Whx8hECmPVKlvK "fv-autolink") are still at θ = π/2 + kπ, and the x-intercepts don't move (anything times zero is still zero). The asymptotes and zeros are anchored; everything in between gets pulled taller or squashed shorter.

## Why It Matters

Vertical dilation lives in Topic 3.8 (The Tangent Function) in [Unit 3](/ap-pre-calc/unit-3 "fv-autolink"): Trigonometric and Polar Functions, under learning objective 3.8.C, which asks you to describe additive and multiplicative transformations of tangent. Essential knowledge 3.8.C.3 is the exact statement: the graph of g(θ) = a tan θ is a vertical dilation of tan θ by a factor of |a|, with a reflection over the x-axis when a < 0. It matters because tangent breaks the pattern you learned with [sine](/ap-pre-calc/key-terms/sine-function "fv-autolink") and cosine. Sinusoids have an amplitude, so multiplying by a changes a bounded wave's height. Tangent is unbounded, so there's no amplitude to change. You have to describe the effect as a dilation, in transformation language, and that's precisely the vocabulary the exam expects.

## Connections

### Tangent Function (Unit 3)

Vertical dilation is one of the standard transformations applied to f(θ) = tan θ. The [dilation](/ap-pre-calc/key-terms/dilation "fv-autolink") scales the slope-style steepness of the curve between asymptotes, but the periodic asymptotic behavior from 3.8.B stays exactly where it was.

### [Amplitude (Unit 3)](/ap-pre-calc/key-terms/amplitude)

For sine and [cosine](/ap-pre-calc/key-terms/cosine-function "fv-autolink"), multiplying by a changes the amplitude. Tangent has no amplitude because it's unbounded, so the same multiplication gets a different name. Vertical dilation is the umbrella term that works for every function, amplitude is the special case for bounded sinusoids.

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

[Phase shift](/ap-pre-calc/key-terms/phase-shift "fv-autolink") (the c in tan(θ + c)) moves the graph sideways and changes inputs, while vertical dilation scales outputs. The exam loves stacking these together, so you need to read a tan(b(θ + c)) + d and assign each constant to its own job.

### [Vertical Asymptote (Unit 3)](/ap-pre-calc/key-terms/vertical-asymptote)

Vertical dilation cannot move tangent's asymptotes. The asymptotes exist where cos θ = 0, and multiplying outputs by a doesn't change where the function is undefined. This is a classic MCQ trap.

## On the AP Exam

This shows up almost entirely in multiple-choice form, in a few predictable flavors. One type gives you a function like h(θ) = 2.5 tan θ and asks you to evaluate it at a nice angle (h(π/4) = 2.5 · 1 = 2.5, since tan(π/4) = 1). Another compares p(θ) = -5 tan θ to q(θ) = 5 tan θ and checks whether you know a negative a means the same dilation factor plus a reflection over the x-axis. A third works in reverse: 'for which a does g(θ) = a tan θ have a vertical dilation by a factor of 2.5 and a reflection over the x-axis?' (answer: a = -2.5, because the dilation factor is |a|). The hardest stems combine transformations, asking you to identify a function with a period of π/3, a vertical dilation by 2, and a phase shift of π/6 left, which means decoding every constant in a tan(b(θ + c)). Your job is always the same: match each constant to its graphical effect and remember that |a| is the dilation while the sign of a is the reflection.

## Vertical Dilation vs Amplitude

These describe the same operation (multiplying outputs by a) but they are not interchangeable. Amplitude only applies to bounded functions like sine and cosine, where it measures half the distance between max and min. Tangent has no max or min, so saying '2 tan θ has an amplitude of 2' is wrong on the AP exam. Say 'vertical dilation by a factor of 2' instead. Vertical dilation is the correct, universal term; amplitude is the sinusoid-only special case.

## Key Takeaways

- The graph of g(θ) = a tan θ is a vertical dilation of tan θ by a factor of |a|, per essential knowledge 3.8.C.3.
- If a is negative, the transformation is a vertical dilation by |a| combined with a reflection over the x-axis.
- Vertical dilation scales outputs only, so the period (still π), the vertical asymptotes, and the x-intercepts of tangent do not move.
- Tangent has no amplitude because it's unbounded, so always describe a tan θ using 'vertical dilation,' not 'amplitude.'
- To evaluate a dilated tangent quickly, use tan(π/4) = 1, so something like 2.5 tan(π/4) is just 2.5.
- In a combined transformation like a tan(b(θ + c)) + d, the a controls vertical dilation and reflection while b, c, and d handle period, phase shift, and vertical translation.

## FAQs

### What is a vertical dilation in AP Precalc?

It's the transformation g(θ) = a tan θ (or a·f(x) for any function) that stretches or compresses the graph vertically by a factor of |a|. If |a| > 1 the graph stretches, if |a| < 1 it compresses, and if a < 0 it also reflects over the x-axis.

### Does vertical dilation change the amplitude of tangent?

No, because tangent has no amplitude. Tangent is unbounded between its asymptotes, so multiplying by a is described as a vertical dilation by |a|, never as a change in amplitude. Using 'amplitude' for tangent will cost you on the exam.

### Does vertical dilation move the asymptotes of tan θ?

No. The asymptotes of a tan θ stay at θ = π/2 + kπ because they come from cos θ = 0, and scaling outputs by a doesn't change where the function is undefined. The period stays π too.

### What's the difference between vertical dilation and vertical translation?

Dilation multiplies outputs (a tan θ scales the graph by |a|), while translation adds to outputs (tan θ + d slides the whole graph up or down d units). Multiplication stretches, addition shifts. The CED splits these into 3.8.C.3 and 3.8.C.1 respectively.

### What does a negative value of a do in g(θ) = a tan θ?

It reflects the graph over the x-axis on top of the dilation. So g(θ) = -2.5 tan θ has a vertical dilation by a factor of 2.5 plus a reflection, meaning the curve decreases between asymptotes instead of increasing.

## Related Study Guides

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

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