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Teacher Guide: Vertical Shift

Learn how the constant D in y = A sin(x) + D moves a sine or cosine wave up or down, sets its midline and decides its maximum, minimum and range.

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Learning Objectives
  • Describe how the constant D in y=Asin⁡(x)+Dy = A\sin(x) + D and y=Acos⁡(x)+Dy = A\cos(x) + D moves the graph
  • Identify the midline y=Dy = D from an equation or a graph
  • Find the maximum, minimum and range of a shifted sine or cosine function
  • Calculate D and |A| from given maximum and minimum values
  • Build simple sinusoidal models for tides, Ferris wheels and daily temperature
Prerequisites
  • • Graphing y=sin⁡(x)y = \sin(x) (lesson Graphing the Sine Function)
  • • Graphing y=cos⁡(x)y = \cos(x) (lesson Graphing the Cosine Function)
  • • Amplitude ∣A∣|A| and period 2π∣B∣\frac{2\pi}{|B|} (lesson Amplitude and Period)
  • • Absolute value and interval notation
Discussion Starters
  • 1. Why is the midline of a Ferris wheel's height function the height of its center?
  • 2. If you only know the highest and lowest temperature of a day, what can you say about the model, and what is still missing?
  • 3. Which changes the range more: doubling A or adding 2 to D? Explain with an example.
  • 4. Can two different equations of the form y=Asin⁡(x)+Dy = A\sin(x) + D have the same range? What do they have in common?
Common Misconceptions

The amplitude is the maximum value (in y=2sin⁡(x)+5y = 2\sin(x) + 5 the amplitude would be 7)

Remediation: Draw the midline first, then measure the amplitude from it with a ruler. Compare y=2sin⁡(x)y = 2\sin(x) and y=2sin⁡(x)+5y = 2\sin(x) + 5: same height of the wave, different position.

A negative A changes the maximum to D+AD + A

Remediation: Graph y=−3sin⁡(x)+1y = -3\sin(x) + 1 next to y=3sin⁡(x)+1y = 3\sin(x) + 1 and ask students to read both maxima from the picture. They are equal, so the formula must use ∣A∣|A|.

sin⁡(x+2)\sin(x + 2) and sin⁡(x)+2\sin(x) + 2 are the same shift

Remediation: Make a table of values for both at x=0,π2,πx = 0, \frac{\pi}{2}, \pi. Only sin⁡(x)+2\sin(x) + 2 adds 2 to every output; the other moves the wave sideways.

Differentiation Ideas

For Struggling Students:

  • • Start with A=1A = 1 and only change D, using the target mode to see the whole wave slide
  • • Always draw the midline as a dashed line before marking the maximum and minimum
  • • Use a two-row table: midline, then midline plus and minus the amplitude

For On-Level Students:

  • • Find A and D from maximum and minimum values, then check with the graph
  • • Model a Ferris wheel with given diameter and platform height
  • • Write the range of shifted functions in interval notation

For Advanced Students:

  • • Build a full model y=Asin⁡(Bt)+Dy = A\sin(Bt) + D from a real tide table, including the period
  • • Find all values of D for which y=3sin⁡(x)+Dy = 3\sin(x) + D never touches the x-axis
  • • Compare a sine model and a cosine model for the same Ferris wheel
Standards Alignment
  • HSF.TF.B.5 (CCSS.MATH.CONTENT.HSF.TF.B.5)

    Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline

  • HSF.BF.B.3 (CCSS.MATH.CONTENT.HSF.BF.B.3)

    Identify the effect on the graph of replacing f(x) by f(x) + k for specific values of k

  • HSF.IF.C.7e (CCSS.MATH.CONTENT.HSF.IF.C.7e)

    Graph trigonometric functions, showing period, midline, and amplitude

Lesson Resources
  • visualMatch the Target Wave

    Students move the A and D sliders until their sine curve covers a dashed target curve

  • activityTide Table Detective

    Students take a real local tide table, read high and low water and build a model with A and D

  • worksheetMidline, Maximum, Minimum

    Twelve equations: find the midline, extremes and range, then sketch the graph

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

Adding a constant DD to a sine or cosine function moves the whole wave up or down. This is called a vertical shift:
y=Asin⁡(x)+Dy=Acos⁡(x)+Dy = A\sin(x) + D \qquad y = A\cos(x) + D
  • If D>0D > 0, the graph moves up by DD units.
  • If D<0D < 0, the graph moves down by ∣D∣|D| units.
  • The shape, the amplitude and the period do not change.

The midline

The midline is the horizontal line the wave oscillates around. For y=sin⁡(x)y = \sin(x) it is the x-axis (y=0y = 0). After the shift it is
midline: y=D\text{midline: } y = D

Maximum, minimum and range

The wave reaches ∣A∣|A| units above and below the midline:
FeatureFormula
Midliney=Dy = D
Maximum value$D +A
Minimum value$D -A
Range$[D -A,\ D +A]$

Going backwards: A and D from data

If you know the highest and lowest values, the midline lies exactly halfway between them and the amplitude is half the distance:
D=max+min2∣A∣=max−min2D = \frac{\text{max} + \text{min}}{2} \qquad |A| = \frac{\text{max} - \text{min}}{2}
In the general form y=Asin⁡(B(x−C))+Dy = A\sin(B(x - C)) + D, the vertical shift is always the last letter, DD. In this lesson B=1B = 1 and C=0C = 0 unless a real situation needs a different period.

Worked Examples

For y=3sin⁡(x)+2y = 3\sin(x) + 2, find the midline, the maximum, the minimum and the range.
1

Identify A and D

Compare with y=Asin⁡(x)+Dy = A\sin(x) + D → A=3A = 3, D=2D = 2
2

Write the midline

The midline is y=Dy = D → y=2y = 2
3

Find the maximum and minimum

D+∣A∣=2+3D + |A| = 2 + 3 and D−∣A∣=2−3D - |A| = 2 - 3 → Maximum 55, minimum −1-1
4

Write the range

All y-values from the minimum to the maximum → [−1,5][-1, 5]

Common Mistakes

Writing the maximum of y=−3sin⁡(x)+1y = -3\sin(x) + 1 as 1+(−3)=−21 + (-3) = -2

Why it's wrong: The wave reaches the amplitude above and below the midline, and the amplitude is always positive. The sign of AA only flips the wave.

Correct: Maximum =D+∣A∣=1+3=4= D + |A| = 1 + 3 = 4, minimum =D−∣A∣=1−3=−2= D - |A| = 1 - 3 = -2.

Reading y=sin⁡(x)−2y = \sin(x) - 2 as a shift up by 2

Why it's wrong: In y=Asin⁡(x)+Dy = A\sin(x) + D the shift is DD with its sign. Here D=−2D = -2.

Correct: y=sin⁡(x)−2y = \sin(x) - 2 moves the wave down 2 units, the midline is y=−2y = -2.

Calling the maximum the amplitude: in y=2sin⁡(x)+5y = 2\sin(x) + 5 the amplitude is 7

Why it's wrong: The amplitude is measured from the midline, not from the x-axis. The shift adds height but does not make the wave taller.

Correct: The amplitude is ∣A∣=2|A| = 2. The maximum is 5+2=75 + 2 = 7.

Forgetting to divide by 2: amplitude =max−min= \text{max} - \text{min}

Why it's wrong: max−min\text{max} - \text{min} is the full height from the lowest to the highest point, which is two amplitudes.

Correct: ∣A∣=max−min2|A| = \frac{\text{max} - \text{min}}{2} and D=max+min2D = \frac{\text{max} + \text{min}}{2}. For max 9 and min 3: ∣A∣=3|A| = 3, D=6D = 6.

Why It Matters

Most real waves do not swing around zero. They swing around some typical level, and that level is the vertical shift:
  • Tides: the water in a harbor rises and falls around an average depth, for example between 1 m and 5 m around a midline of 3 m.
  • Ferris wheels: a seat goes up and down around the height of the wheel's center, never below the boarding platform.
  • Daily temperature: the temperature swings around the day's average, for example between 12°C and 24°C around 18°C.
  • Blood pressure and heartbeats: pressure pulses between about 80 and 120 around a midline of 100.
  • Daylight hours: day length varies over the year around about 12 hours.
With the midline formula D=max+min2D = \frac{\text{max} + \text{min}}{2} you can turn two measurements into a working model, and with the range you can check at a glance whether a model makes sense.

Real World Applications

Ferris Wheels

The height of a seat on a Ferris wheel goes up and down like a cosine wave. The midline is the height of the wheel's center.

Example:

A wheel with diameter 40 m whose lowest seat is 2 m above the ground can be modeled by h(t)=−20cos⁡(π10t)+22h(t) = -20\cos\left(\frac{\pi}{10}t\right) + 22 (t in minutes): amplitude 20, midline 22, range [2,42][2, 42].

1Try It Yourself

A Ferris wheel has a diameter of 40 m. Its lowest point is 2 m above the ground.

At what height is the midline of the height function (the center of the wheel), in meters?

Step 1: Write the mathematical expression

Average the highest and lowest heights:

Tides

Harbor masters and sailors model the water level with a sine or cosine wave whose midline is the mean water level.

Example:

If high tide is 5.2 m and low tide is 0.8 m, the mean water level is 5.2+0.82=3\frac{5.2 + 0.8}{2} = 3 m, so the model has D=3D = 3.

2Try It Yourself

At a port, high tide is 5.2 m and low tide is 0.8 m.

What is the amplitude of the tide model, in meters?

Step 1: Write the mathematical expression

Take half the difference between high and low tide:

Temperature over a Day

Weather scientists model the temperature over 24 hours with a cosine wave around the day's average temperature.

Example:

The model T(t)=−6cos⁡(π12t)+18T(t) = -6\cos\left(\frac{\pi}{12}t\right) + 18, with t in hours after 3:00, is coldest at 3:00 and warmest 12 hours later.

3Try It Yourself

The temperature in a city is modeled by T(t)=−6cos⁡(π12t)+18T(t) = -6\cos\left(\frac{\pi}{12}t\right) + 18 in °C.

What is the highest temperature of the day, in °C?

Step 1: Write the mathematical expression

Add the amplitude to the midline:

Key Takeaways

  • 1In y=Asin⁡(x)+Dy = A\sin(x) + D and y=Acos⁡(x)+Dy = A\cos(x) + D, the constant DD moves the wave up (D>0D > 0) or down (D<0D < 0)
  • 2The midline is the horizontal line y=Dy = D, halfway between the maximum and the minimum
  • 3Maximum =D+∣A∣= D + |A| and minimum =D−∣A∣= D - |A|, always with the absolute value of AA
  • 4The range is [D−∣A∣, D+∣A∣][D - |A|,\ D + |A|]
  • 5From data: D=max+min2D = \frac{\text{max} + \text{min}}{2} and ∣A∣=max−min2|A| = \frac{\text{max} - \text{min}}{2}
  • 6A vertical shift changes neither the amplitude nor the period, only the height of the whole wave

Frequently Asked Questions

Does the vertical shift change the period or the amplitude?

No. DD moves every point of the graph up or down by the same amount, so the wave keeps its height and its width. Only the midline, the maximum, the minimum and the range change.

Is the vertical shift the same as the midline?

They come from the same number. The vertical shift is the movement (for example 3 units up), and the midline is the line where the middle of the wave ends up (y=3y = 3).

Why does sin⁡(x)+2\sin(x) + 2 shift up, but sin⁡(x+2)\sin(x + 2) does not?

A number added outside the function changes the output, so the graph moves vertically. A number added inside, to xx, changes the input, so the graph moves sideways. That sideways move is the phase shift, covered in the lesson Phase Shift.

Can a shifted sine wave stay entirely above the x-axis?

Yes, when D>∣A∣D > |A|. Then the minimum D−∣A∣D - |A| is positive. For example y=2sin⁡(x)+5y = 2\sin(x) + 5 never goes below 33. Ferris wheel heights and tide depths behave like this.

Glossary

Vertical shift
Moving a whole graph up or down by adding a constant DD to the function
Midline
The horizontal line y=Dy = D halfway between the maximum and minimum values of a wave
Maximum value
The highest y-value of the wave, D+∣A∣D + |A|
Minimum value
The lowest y-value of the wave, D−∣A∣D - |A|
Range
All y-values the function takes: [D−∣A∣, D+∣A∣][D - |A|,\ D + |A|] for a shifted sine or cosine
Sinusoidal model
A sine or cosine function used to describe a repeating real situation, such as tides or temperature

Formula Card

Shifted sine and cosine

y=Asin⁡(x)+Dy = A\sin(x) + D, y=Acos⁡(x)+Dy = A\cos(x) + D

D moves the wave up (D > 0) or down (D < 0)

Midline

y=Dy = D

The line the wave oscillates around

Maximum and minimum

D+∣A∣D + |A| and D−∣A∣D - |A|

Highest and lowest values of the wave

Range

[D−∣A∣, D+∣A∣][D - |A|,\ D + |A|]

All output values

From data

D=max+min2D = \frac{\text{max} + \text{min}}{2}, ∣A∣=max−min2|A| = \frac{\text{max} - \text{min}}{2}

Midline and amplitude from the highest and lowest values

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