Mathematics · 11th Grade · Trigonometric Functions and Periodic Motion · Weeks 10-18

Graphing Sine and Cosine: Phase Shift and Vertical Shift

Students will graph sine and cosine functions, incorporating phase shifts and vertical shifts (midlines).

• Common Core State Standards
• CCSS.Math.Content.HSF.TF.B.5

About This Topic

Phase shifts and vertical shifts complete the full transformation picture for sine and cosine functions. A vertical shift D moves the midline of the wave from y = 0 to y = D, changing the average value of the function but not its amplitude or period. A phase shift C moves the starting position of the wave left or right along the x-axis, which affects where the maximum, minimum, and zero crossings occur but not the shape of the wave.

In the standard form y = A*sin(B(x - C)) + D, the phase shift is C (positive C shifts right) and the vertical shift is D. Students frequently confuse these two parameters because both change where the graph sits, but they operate in different directions. The midline is determined exclusively by D, while the starting point of the cycle is determined by C.

Active learning is especially effective here because phase shifts require students to trace a single point through its transformation, a spatial reasoning task that benefits from physical and visual anchors. Matching activities that require students to identify all four parameters from a graph before writing an equation develop the complete skill set.

Key Questions

1. Differentiate between a phase shift and a vertical shift in trigonometric graphs.
2. Predict the new starting point of a sine or cosine wave after a phase shift.
3. Analyze how the midline of a trigonometric function relates to its average value.

Learning Objectives

• Analyze the effect of a phase shift (C) on the horizontal position of key points (maximum, minimum, zeros) of sine and cosine graphs.
• Calculate the new midline (y = D) of a sine or cosine function given its equation in the form y = A*sin(B(x - C)) + D.
• Compare and contrast the graphical transformations resulting from phase shifts and vertical shifts for sine and cosine functions.
• Create a trigonometric equation in the form y = A*sin(B(x - C)) + D that models a given graph with specified phase and vertical shifts.
• Explain how the parameters C and D in y = A*sin(B(x - C)) + D influence the starting point and average value of the function, respectively.

Before You Start

Graphing Basic Sine and Cosine Functions

Why: Students must be able to graph the parent functions y = sin(x) and y = cos(x) to understand how transformations alter them.

Understanding Function Transformations (Stretches, Compressions, Reflections)

Why: Familiarity with how parameters affect graph shape and position is foundational for understanding phase and vertical shifts.

Key Vocabulary

Phase Shift	A horizontal translation of a periodic function, shifting the graph left or right along the x-axis. It is represented by 'C' in the equation y = A*sin(B(x - C)) + D.
Vertical Shift	A vertical translation of a periodic function, shifting the graph up or down along the y-axis. It is represented by 'D' in the equation y = A*sin(B(x - C)) + D and determines the new midline.
Midline	The horizontal line that runs through the center of a periodic function's graph, representing the average value of the function. For y = A*sin(B(x - C)) + D, the midline is y = D.
Amplitude	Half the distance between the maximum and minimum values of a periodic function. It is represented by 'A' in the equation y = A*sin(B(x - C)) + D and is not affected by phase or vertical shifts.

Watch Out for These Misconceptions

Common Misconception

A phase shift of C in y = sin(x - C) moves the graph to the left.

What to Teach Instead

In y = sin(x - C), a positive value of C shifts the graph to the right. The sign inside the parentheses is subtracted, so students sometimes read a positive C as leftward. Writing y = sin(x - pi/4) and asking 'what x-value makes the argument equal to zero?' helps clarify the direction.

Common Misconception

The midline is always y = 0.

What to Teach Instead

The midline is y = D, where D is the vertical shift. It represents the average value the function oscillates around. If D = 3, the midline is y = 3, and the function oscillates between 3 - A and 3 + A. Students benefit from identifying the midline as the horizontal line exactly halfway between maximum and minimum values on a graph.

Common Misconception

Phase shift and period change affect the graph in similar ways.

What to Teach Instead

A period change compresses or stretches all cycles uniformly. A phase shift moves the entire wave left or right without changing its shape or length. They are independent transformations. Students who confuse them often write incorrect B values when reading a shifted graph.

Active Learning Ideas

Desmos Investigation: Trace the Starting Point

Students adjust the C slider in y = sin(x - C) and track where the first maximum occurs, recording x-coordinates in a table. They compare the location of the maximum to the value of C and write the pattern as a rule, then test it on cosine.

30 min·Pairs

Card Sort: Equation to Key Feature

Provide equation cards like y = 2cos(x - pi/3) + 1 and feature cards listing midline, max, min, and starting point. Students match each equation to its feature card without graphing, explaining their reasoning for each match before verifying on Desmos.

35 min·Small Groups

Think-Pair-Share: What Moved, What Stayed?

Show three graphs of y = sin(x) with one parameter changed at a time (vertical shift, then phase shift, then both). Students identify which parameter changed in each step and describe the effect in precise language, comparing answers with a partner before class discussion.

20 min·Pairs

Gallery Walk: Write the Equation

Post six graphs of sine and cosine functions with all four parameters varied. Groups rotate and write the equation for each graph, including all transformations. Groups then compare equations at each station and resolve discrepancies before a whole-class debrief.

40 min·Small Groups

Real-World Connections

• Electrical engineers use sinusoidal functions to model alternating current (AC) voltage and current, where phase shifts can represent time delays in signal transmission or the timing of power delivery to different parts of a circuit.
• Oceanographers use these functions to model tidal patterns, with phase shifts indicating variations in high and low tide times due to local geography and vertical shifts representing changes in average sea level over time.

Assessment Ideas

Exit Ticket

Provide students with the equation y = 3*sin(2(x - pi/4)) + 1. Ask them to identify the phase shift and vertical shift, and then state the equation of the new midline and the x-coordinate of the first maximum after x=0.

Quick Check

Display two graphs side-by-side: one of y = sin(x) and another of y = sin(x - pi/2) + 2. Ask students to write down the phase shift and vertical shift applied to the first graph to obtain the second, and to describe the changes in words.

Discussion Prompt

Pose the question: 'If you are given a sine wave that starts at its midline and increases, and you apply a phase shift of pi/2 to the right and a vertical shift of 3 units up, where would the new starting point be, and what would be the new midline?' Facilitate a discussion comparing student responses.

Frequently Asked Questions

What is a phase shift in a trigonometric function?

A phase shift is a horizontal translation of the graph. In y = A*sin(B(x - C)) + D, the parameter C is the phase shift. A positive C moves the graph C units to the right; a negative C moves it left. It changes where the cycle begins but not the shape, amplitude, or period.

How do you find the midline of a sine or cosine function?

The midline is the horizontal line y = D in the equation y = A*sin(B(x - C)) + D. On a graph, it sits exactly halfway between the maximum and minimum values: D = (max + min) / 2. The function oscillates symmetrically above and below this line by an amount equal to the amplitude.

How is a phase shift different from a period change?

A phase shift slides the wave horizontally without changing its shape or how long each cycle takes. A period change stretches or compresses every cycle, changing the horizontal spacing of all features. Both affect the x-axis but in fundamentally different ways. On a graph, a phase shift looks like a slide; a period change looks like a squeeze or stretch.

What active learning strategies help students master phase and vertical shifts?

Slider explorations where students change only one parameter at a time while holding others fixed help isolate each transformation's effect. Card sorts requiring students to extract all four parameters from a graph and match them to an equation build the complete reading skill. The act of discussing discrepancies with a partner generates more durable learning than solo practice.