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Vertical Shift

A free Trigonometry lesson from the “Graphs of Trig Functions” unit, with a worked example and practice problems including step-by-step solutions.

A vertical shift moves the midline. In y = a sin(bx) + d, the midline is y = d. In this lesson, the goal is to find the midline after a vertical shift. Prerequisite check: Algebra II or College Algebra foundations. Example 1: y = 3sin(x) has amplitude 3 and midline y = 0. Example 2: y = cos(2x) has period pi because 2pi divided by 2 is pi. A common mistake is treating amplitude as the full height from minimum to maximum; the safer habit is to start with the parent graph, then apply vertical and horizontal changes.

What you'll learn

Why it matters: Trig graphs model repeated behavior such as tides, daylight, wheels, heartbeats, sound, and alternating current.

Worked example

Problem. Example 1 Foundation: What does the period of a trig graph tell you?

  1. Worked Example: First identify exactly what the question is asking: Example 1 Foundation: What does the period of a trig graph tell you?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Period is horizontal cycle length.
  4. After one period, the pattern repeats.

Answer: how far one full cycle travels horizontally

Practice problems

1. Practice 1 Foundation: What is the amplitude of y = 2cos(2x)?

Choices: 2 · 1 · 2pi · pi

Show solution
  1. Amplitude is the distance from the midline to a maximum.
  2. For y = 2cos(2x), the amplitude is 2.
  3. It is not the full height from max to min.

Answer: 2

2. Practice 2 Setup: What is the period of y = sin(x) + 4?

Choices: 2pi · pi/2 · 4pi · not enough information

Show solution
  1. The period is the horizontal length of one cycle.
  2. For y = sin(x) + 4, the period is 2pi.
  3. Check the coefficient inside the trig function.

Answer: 2pi

3. Practice 3 Meaning: What is the midline of y = cos(x - pi/3)?

Choices: y = 0 · y = 1 · x = 0 · y = 2pi

Show solution
  1. The midline is the horizontal center line of the wave.
  2. For y = cos(x - pi/3), the midline is y = 0.
  3. Vertical shifts move the midline.

Answer: y = 0

4. Practice 4 Method: What phase or vertical shift is visible in y = 3sin(x)?

Choices: none · left pi/3 · down 4 · period pi

Show solution
  1. Core Practice: First identify exactly what the question is asking: Practice 4 Method: What phase or vertical shift is visible in y = 3sin(x)?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. A shift changes where the graph sits or starts.
  4. This equation shows none.
  5. Do not confuse shift with amplitude.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: none

5. Practice 5 Reasoning: Which x-value is a vertical asymptote of y = tan(x)?

Choices: x = pi/2 · x = 0 · x = pi · x = 2pi

Show solution
  1. Core Practice: First identify exactly what the question is asking: Practice 5 Reasoning: Which x-value is a vertical asymptote of y = tan(x)?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Tangent is undefined when cosine is 0.
  4. cos(pi/2) = 0.
  5. So x = pi/2 is an asymptote.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x = pi/2

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