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Write the equation of the trigonometric function shown in the graph
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
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ANSWER: y = \cos\left(\frac{1}{4}x\right) - 4 (or y = \cos\left(\frac{x}{4}\right) - 4)
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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?