Worksheet on exponential growth and decay functions with multiple-choice questions

Let’s solve each problem one by one. We’ll find:

- The amplitude (how tall the wave is from center to peak)
- The period (how long it takes for one full wave cycle)
- The midline (the horizontal line in the middle of the wave)

We use these rules for equations like:
y = A sin(Bx + C) + D or y = A cos(Bx + C) + D

→ Amplitude = |A|
→ Period = 2π / |B|
→ Midline = y = D

---

Problem 1: y = -3/5 cos(6x) - 0.38

This is cosine form.

- A = -3/5 → amplitude = |-3/5| = 3/5
- B = 6 → period = 2π / 6 = π/3
- D = -0.38 → midline = y = -0.38

✔ Final Answer for #1:
Amplitude: 3/5, Period: π/3, Midline: y = -0.38

---

Problem 2: y = 4/9 cos(-8/7 x) - 48/47

Cosine again.

Note: cos(-θ) = cos(θ), so negative inside doesn’t change anything.

- A = 4/9 → amplitude = 4/9
- B = -8/7 → but we take absolute value → |B| = 8/7 → period = 2π / (8/7) = 2π * 7/8 = 14π/8 = 7π/4
- D = -48/47 → midline = y = -48/47

✔ Final Answer for #2:
Amplitude: 4/9, Period: 7π/4, Midline: y = -48/47

---

Problem 3: y = 9/10 cos(-11/4 x)

No vertical shift → D = 0

- A = 9/10 → amplitude = 9/10
- B = -11/4 → |B| = 11/4 → period = 2π / (11/4) = 2π * 4/11 = 8π/11
- D = 0 → midline = y = 0

✔ Final Answer for #3:
Amplitude: 9/10, Period: 8π/11, Midline: y = 0

---

Problem 4: y = -7/11 sin(-2x) + 6

Sine function.

- A = -7/11 → amplitude = |-7/11| = 7/11
- B = -2 → |B| = 2 → period = 2π / 2 = π
- D = 6 → midline = y = 6

✔ Final Answer for #4:
Amplitude: 7/11, Period: π, Midline: y = 6

---

Problem 5: y = -1/2 cos(-4/3 x) - 6/13

Cosine.

- A = -1/2 → amplitude = 1/2
- B = -4/3 → |B| = 4/3 → period = 2π / (4/3) = 2π * 3/4 = 6π/4 = 3π/2
- D = -6/13 → midline = y = -6/13

✔ Final Answer for #5:
Amplitude: 1/2, Period: 3π/2, Midline: y = -6/13

---

Problem 6: y = 8/13 cos(x) + 9/10

Cosine.

- A = 8/13 → amplitude = 8/13
- B = 1 → period = 2π / 1 = 2π
- D = 9/10 → midline = y = 9/10

✔ Final Answer for #6:
Amplitude: 8/13, Period: 2π, Midline: y = 9/10

---

Problem 7: y = -12/25 sin(-2x) + 2/11

Sine.

- A = -12/25 → amplitude = 12/25
- B = -2 → |B| = 2 → period = 2π / 2 = π
- D = 2/11 → midline = y = 2/11

✔ Final Answer for #7:
Amplitude: 12/25, Period: π, Midline: y = 2/11

---

Problem 8: y = 2/5 sin(-1/2 x) + 4/41

Sine.

- A = 2/5 → amplitude = 2/5
- B = -1/2 → |B| = 1/2 → period = 2π / (1/2) = 2π * 2 = 4π
- D = 4/41 → midline = y = 4/41

✔ Final Answer for #8:
Amplitude: 2/5, Period: 4π, Midline: y = 4/41

---

Final Answer:
1. Amplitude: 3/5, Period: π/3, Midline: y = -0.38
2. Amplitude: 4/9, Period: 7π/4, Midline: y = -48/47
3. Amplitude: 9/10, Period: 8π/11, Midline: y = 0
4. Amplitude: 7/11, Period: π, Midline: y = 6
5. Amplitude: 1/2, Period: 3π/2, Midline: y = -6/13
6. Amplitude: 8/13, Period: 2π, Midline: y = 9/10
7. Amplitude: 12/25, Period: π, Midline: y = 2/11
8. Amplitude: 2/5, Period: 4π, Midline: y = 4/41