Reading the Midline, Maximum and Minimum

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

Identify A and D: Compare with $y = A\sin(x) + D$ = $A = 3$, $D = 2$

Write the midline: The midline is $y = D$ = $y = 2$

Find the maximum and minimum: $D + |A| = 2 + 3$ and $D - |A| = 2 - 3$ = Maximum $5$, minimum $-1$

Write the range: All y-values from the minimum to the maximum = $[-1, 5]$

Answer: Midline $y = 2$, maximum $5$, minimum $-1$, range $[-1, 5]$.

A Shift Downward

Describe the graph of $y = \cos(x) - 4$ compared with $y = \cos(x)$.

Identify D: $y = \cos(x) + (-4)$ = $D = -4$

Decide the direction: $D < 0$, so the graph moves down = Down by $4$ units

Find the new midline: The x-axis $y = 0$ moves to $y = -4$ = Midline $y = -4$

Find the new maximum and minimum: $-4 + 1 = -3$ and $-4 - 1 = -5$ = Maximum $-3$, minimum $-5$

Answer: The cosine wave moves 4 units down. It keeps its shape and swings between $-5$ and $-3$ around the midline $y = -4$.

A Negative A with a Shift

For $y = -2\cos(x) + 1$, find the maximum and the minimum and where they occur on $[0, 2\pi]$.

Identify A and D: Compare with $y = A\cos(x) + D$ = $A = -2$, $D = 1$

Use the amplitude, not A itself: $|A| = |-2| = 2$ = Amplitude $2$

Find the maximum and minimum values: $1 + 2 = 3$ and $1 - 2 = -1$ = Maximum $3$, minimum $-1$

Find where they occur: At $x = 0$: $-2\cos(0) + 1 = -1$. At $x = \pi$: $-2\cos(\pi) + 1 = 3$ = Minimum at $x = 0$ and $x = 2\pi$, maximum at $x = \pi$

Answer: Maximum $3$ at $x = \pi$, minimum $-1$ at $x = 0$ and $x = 2\pi$. The negative $A$ flips the wave, but the midline $y = 1$ and the range $[-1, 3]$ stay the same.

Finding A and D from the Maximum and Minimum

A sine wave starts on its midline, rises first, and swings between $9$ and $3$. Write its equation in the form $y = A\sin(x) + D$.

Find the midline: $D = \frac{9 + 3}{2} = \frac{12}{2}$ = $D = 6$

Find the amplitude: $|A| = \frac{9 - 3}{2} = \frac{6}{2}$ = $|A| = 3$

Choose the sign of A: The wave starts on the midline and goes up, like $\sin(x)$ = $A = 3$

Write the equation: Substitute $A = 3$ and $D = 6$ = $y = 3\sin(x) + 6$

Answer: $y = 3\sin(x) + 6$. Check: $6 + 3 = 9$ and $6 - 3 = 3$.

Modeling a Harbor Tide

The water in a harbor is $11$ m deep at high tide and $3$ m deep at low tide. One full cycle takes $12$ hours, and at $t = 0$ the water is at its average depth and rising. Write a model $d(t) = A\sin(Bt) + D$ and find the depth at $t = 3$ hours.

Find the midline (average depth): $D = \frac{11 + 3}{2}$ = $D = 7$

Find the amplitude: $A = \frac{11 - 3}{2}$ (rising at the start, so positive) = $A = 4$

Find B from the period: $B = \frac{2\pi}{12}$ = $B = \frac{\pi}{6}$

Write the model: Substitute $A$, $B$ and $D$ = $d(t) = 4\sin\left(\frac{\pi}{6}t\right) + 7$

Evaluate at t = 3: $4\sin\left(\frac{\pi}{2}\right) + 7 = 4 \cdot 1 + 7$ = $d(3) = 11$

Answer: $d(t) = 4\sin\left(\frac{\pi}{6}t\right) + 7$. After 3 hours the water is $11$ m deep, which is high tide.

Mistake: Writing the maximum of $y = -3\sin(x) + 1$ as $1 + (-3) = -2$

Why: The wave reaches the amplitude above and below the midline, and the amplitude is always positive. The sign of $A$ only flips the wave.

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

Mistake: Reading $y = \sin(x) - 2$ as a shift up by 2

Why: In $y = A\sin(x) + D$ the shift is $D$ with its sign. Here $D = -2$.

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

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

Why: 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 $|A| = 2$. The maximum is $5 + 2 = 7$.

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

Why: $\text{max} - \text{min}$ is the full height from the lowest to the highest point, which is two amplitudes.

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

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.

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

Tides

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

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

Temperature over a Day

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

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

Q: Does the vertical shift change the period or the amplitude?

A: No. $D$ 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.

Q: Is the vertical shift the same as the midline?

A: 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 ($y = 3$).

Q: Why does $\sin(x) + 2$ shift up, but $\sin(x + 2)$ does not?

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

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

A: Yes, when $D > |A|$. Then the minimum $D - |A|$ is positive. For example $y = 2\sin(x) + 5$ never goes below $3$. Ferris wheel heights and tide depths behave like this.