# Write a cosine function that has an amplitude of 4, a midline of y=3 and a period of 1 Submit Answer

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**Created:** [Ağustos 16, 2026, 4:50öö UTC](https://sorumatik.co/t/write-a-cosine-function-that-has-an-amplitude-of-4-a-midline-of-y-3-and-a-period-of-1-submit-answer/690202 "2026-08-16T04:50:48Z")\
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**Post date:** [Ağustos 16, 2026, 4:50öö UTC](https://sorumatik.co/t/write-a-cosine-function-that-has-an-amplitude-of-4-a-midline-of-y-3-and-a-period-of-1-submit-answer/690202/1 "2026-08-16T04:50:48Z")

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Write a cosine function that has  
an amplitude of 4, a midline of  
y =3 and a period of1  
Answer  
f(z) =  
Submit Answer

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**Post date:** [Ağustos 16, 2026, 4:50öö UTC](https://sorumatik.co/t/write-a-cosine-function-that-has-an-amplitude-of-4-a-midline-of-y-3-and-a-period-of-1-submit-answer/690202/2 "2026-08-16T04:50:59Z")

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## Finding the Equation of a Cosine Function

💡 **Formula Used:**

\text{General form of a cosine function: } f(x) = A \cos(B x) + D

\text{Formula for period } P \text{ and frequency factor } B: B = \frac{2\pi}{P}

🧠 **Solution Steps:**

**Step 1 — Identify Given Parameters**  
From the problem statement, we are given:

- Amplitude A = 4
- Midline y = 3, which gives vertical shift D = 3
- Period P = \frac{5\pi}{3}

**Step 2 — Calculate the Value of B**  
Substitute the given period P = \frac{5\pi}{3} into the formula B = \frac{2\pi}{P}:

B = \frac{2\pi}{\frac{5\pi}{3}}

B = 2\pi \times \frac{3}{5\pi}

B = \frac{6\pi}{5\pi} = \frac{6}{5}

**Step 3 — Write the Final Cosine Function**  
Substitute A = 4, B = \frac{6}{5}, and D = 3 into the general form f(x) = A \cos(B x) + D:

f(x) = 4\cos\left(\frac{6}{5}x\right) + 3

━━━━━━━━━━━━━━━━━━━━━━━━━━━  
✅ **Answer:**

f(x) = 4\cos\left(\frac{6}{5}x\right) + 3

_(Equivalent form: f(x) = 4\cos(1.2x) + 3)_  
━━━━━━━━━━━━━━━━━━━━━━━━━━━

🎯 **Key Concepts:**

**1. Amplitude (A)**

- **Definition:** The height from the center line (midline) to the peak or trough of the wave.
- **In this problem:** Given as 4, setting the vertical stretch factor A = 4.

**2. Midline (D)**

- **Definition:** The horizontal line y = D around which the cosine wave oscillates.
- **In this problem:** Given as y = 3, setting the vertical translation D = 3.

**3. Period (P)**

- **Definition:** The horizontal length of one complete cycle of the cosine graph.
- **In this problem:** Given as \frac{5\pi}{3}, which determines the horizontal stretch/compression factor B = \frac{6}{5}.

⚠ **Common Mistakes:**

**❌ Direct Placement of Period into the Function**

- **Wrong:** Writing f(x) = 4\cos\left(\frac{5\pi}{3}x\right) + 3
- **Right:** Writing f(x) = 4\cos\left(\frac{6}{5}x\right) + 3
- **Why it’s wrong:** The value multiplying x inside the cosine function is the frequency coefficient B, not the period P itself. You must always use B = \frac{2\pi}{P} to calculate the coefficient.

Would you like me to generate a similar practice problem on trigonometric transformations to test your understanding?