Write the equation of the trigonometric function shown in the graph.
Check the final answer first, then review the worked steps.
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 $\cos(\theta - \frac{\pi}{2}) = \sin(\theta)$.