Answer:
y = 20sin(4x)
Step-by-step explanation:
Solving the Problem: Understanding the Graph
The Origin
The graph crosses the origin, meaning that if we plug 0 into the function's equation, it produces 0.
In context of trigonometry, sin(0) = 0 whereas cos(0) = 1, so we can eliminate y = 20cos(4x).
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The Period
At x = π/2, the function's y-value returns to 0, thus the function's period is π/2.
This means that if we plug in π/2 as our x-value into the function's equation it should simplify to 0.
Plugging x = π/2 into each of the two remaining options we get,
[tex]y=20\sin\left(4\left(\dfrac{\pi}{2}\right)\right)=20\sin(2\pi)=20 \cdot 0=0[/tex]
[tex]y=20\sin\left(\dfrac{\dfrac{\pi}{4} }{2}\right)=20\sin\left(\dfrac{1}{4}\left(\dfrac{\pi}{2}\right)\right)=20\sin\left(\dfrac{\pi}{8}\right) \approx7.654[/tex].
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Sense Check
The graph crosses the x-axis more often than its parent function's graph of sin(x).
Here, it crosses the x-axis at x = π/4 and x = π/2 instead at x = π and x = 2π. From this, we can tell that the graph is "moving faster".
Looking at the coefficients on the x term, a coefficient greater than 1 will produce a value greater than the inputted x value. Knowing this, a small x value can make the sine function, and the overall function, produce a big value.
A bigger coefficient makes the function reach values "faster".
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So, our answer must be y = 20sin(4x), as it shares the graph's points at the origin and at (π/2, 0).
