# Write the equation of the trigonometric function shown in the graph

**URL:** <https://sorumatik.co/t/write-the-equation-of-the-trigonometric-function-shown-in-the-graph/690229>\
**Category:** Maths Question\
**Created:** [Ağustos 16, 2026, 5:49ös UTC](https://sorumatik.co/t/write-the-equation-of-the-trigonometric-function-shown-in-the-graph/690229 "2026-08-16T17:49:02Z")\
**Posts on this page:** 2\
**Page:** 1

<div class="post-metadata">

**Author:** ![Jose\_Rosario](https://sorumatik.co/letter_avatar_proxy/v4/letter/j/67e7ee/32.png) [@Jose\_Rosario](https://sorumatik.co/u/Jose_Rosario)\
**Post date:** [Ağustos 16, 2026, 5:49ös UTC](https://sorumatik.co/t/write-the-equation-of-the-trigonometric-function-shown-in-the-graph/690229/1 "2026-08-16T17:49:02Z")

</div>

-121 -101  
-81  
-61  
-41  
-0T

Write the equation of the trigonometric function shown in the graph

 ![-121 -101 -81 -61 -41 -0T Write the equation of the trigonometric function shown in the...](https://sorumatik.co/uploads/default/original/3X/9/4/948b937aef122e5b2d5e0beb29955cd4c5282e59.jpeg)

---

<div class="post-metadata">

**Author:** ![Sinav\_Rehberi](https://sorumatik.co/user_avatar/sorumatik.co/sinav_rehberi/32/154383_2.png) [@Sinav\_Rehberi](https://sorumatik.co/u/Sinav_Rehberi)\
**Post date:** [Ağustos 16, 2026, 5:49ös UTC](https://sorumatik.co/t/write-the-equation-of-the-trigonometric-function-shown-in-the-graph/690229/2 "2026-08-16T17:49:15Z")

</div>

## Writing the Equation of the Trigonometric Function from the Graph

💡 **FORMULA USED:**

The standard form of a transformed cosine function is:

y = A \cos(B(x - C)) + D

Where:

- A = \frac{y\_{\max} - y\_{\min}}{2} is the **amplitude**.
- D = \frac{y\_{\max} + y\_{\min}}{2} is the **midline** (vertical shift).
- B = \frac{2\pi}{T} is the **frequency parameter** , where T is the period.
- C is the **horizontal phase shift**.

* * *

🧠 **SOLUTION STEPS:**

**Step 1 — Identify the Maximum and Minimum Values**  
By observing the peaks and troughs on the graph:

- The highest point (peak) of the curve is at y\_{\max} = -3.
- The lowest point (trough) of the curve is at y\_{\min} = -5.

**Step 2 — Calculate the Midline (D) and Amplitude (A)**

- **Midline (D):**
D = \frac{-3 + (-5)}{2} = \frac{-8}{2} = -4
- **Amplitude (A):**
A = \frac{-3 - (-5)}{2} = \frac{2}{2} = 1

**Step 3 — Determine the Period (T) and Frequency Parameter (B)**

- **Period (T):** The distance between two consecutive peaks (from x = 0 to x = 8\pi) is:
T = 8\pi - 0 = 8\pi
- **Frequency Parameter (B):**
B = \frac{2\pi}{T} = \frac{2\pi}{8\pi} = \frac{1}{4}

**Step 4 — Determine Phase Shift (C) and Write the Equation**

- At x = 0, the wave is at its peak value (y = -3).
- Since a standard cosine wave \cos(x) starts at a maximum at x = 0, no horizontal shift is required (C = 0).
- Substituting A = 1, B = \frac{1}{4}, C = 0, and D = -4 into the general equation:
y = 1 \cdot \cos\left(\frac{1}{4}x\right) - 4

━━━━━━━━━━━━━━━━━━━━━━━━━━━  
✅ **ANSWER:** y = \cos\left(\frac{1}{4}x\right) - 4_(or y = \cos\left(\frac{x}{4}\right) - 4)_  
━━━━━━━━━━━━━━━━━━━━━━━━━━━

🎯 **KEY CONCEPTS:**

**1. Midline (Vertical Shift)**

- **Definition:** The horizontal central axis y = D about which the trigonometric function oscillates.
- **In this problem:** The midline is at y = -4, indicating the parent cosine wave shifted down by 4 units.

**2. Period and Frequency**

- **Definition:** The horizontal length required for the function to complete one full cycle.
- **In this problem:** The graph completes one full cycle every 8\pi units, resulting in B = \frac{1}{4}.

⚠ **COMMON MISTAKES:**

**❌ Setting Parameter B Equal to the Period**

- **Wrong:** Writing B = 8\pi directly into the cosine function.
- **Right:** Using the relationship B = \frac{2\pi}{\text{Period}} = \frac{2\pi}{8\pi} = \frac{1}{4}.
- **Why it’s wrong:** The value B represents spatial frequency, which is inversely proportional to the period length T.

**❌ Confusing Peak-to-Trough Distance with Full Period**

- **Wrong:** Assuming the distance from peak to trough (0 to 4\pi) is the full period.
- **Right:** The distance between a peak and the adjacent trough is only **half** a period (\frac{T}{2} = 4\pi \implies T = 8\pi).
- **Why it’s wrong:** A complete period requires the wave to go from peak to peak or from trough to trough.

* * *

Would you like me to show you how to express this exact same graph using a sine function, or would you like a similar practice problem to test your skills?