Write the equation of the trigonometric function shown in the graph.
Check the final answer first, then review the worked steps.
Problem
Write the equation of the trigonometric function shown in the graph.
Answer
\(y = \sin(\frac{\pi}{50}x) + 4\)
Step-by-step solution
- Identify the midline and amplitude: The maximum value of the function is $y = 5$ and the minimum value is $y = 3$. The midline $k$ is the average of the maximum and minimum: $k = \frac{5 + 3}{2} = 4$. The amplitude $a$ is the distance from the midline to the maximum: $a = 5 - 4 = 1$.
- Determine the period: The graph shows a minimum at $x = -25$ and the next minimum at $x = 75$. The period $P$ is the distance between these two points: $P = 75 - (-25) = 100$. The frequency coefficient $b$ is calculated as $b = \frac{2\pi}{P} = \frac{2\pi}{100} = \frac{\pi}{50}$.
- Determine the phase shift: A standard cosine function $y = \cos(x)$ starts at a maximum at $x=0$. This graph has a maximum at $x = 25$. Therefore, the function is shifted to the right by $25$ units. Alternatively, we can use a reflected cosine function ($-a \cos(b(x-h)) + k$) which starts at a minimum at $x = -25$. Using the standard cosine form $y = a \cos(b(x - h)) + k$, we have $y = 1 \cos(\frac{\pi}{50}(x - 25)) + 4$.
- Final Equation: Combining these components, the equation is $y = \cos(\frac{\pi}{50}(x - 25)) + 4$. This can also be written as $y = \cos(\frac{\pi}{50}x - \frac{\pi}{2}) + 4$, which simplifies to $y = \sin(\frac{\pi}{50}x) + 4$ using the identity $\cos(\theta - \frac{\pi}{2}) = \sin(\theta)$.