Question

Below is the graph of a trigonometric function. It intersects its midline at (3.7, 5) and again at (5.9, 5).

What is the period of the fiction? Give an Exact Value

Answer 1

4.4

Explanation:

The parent function of this graph is: y = sin(x)

The sine function is periodic, meaning it repeats forever.

Standard form of a sine function:

`\sf f(x)= A \sin(B(x+C))+D`

where:

• A = amplitude (height from the mid-line to the peak)
• 2π/B = period (horizontal distance between consecutive peaks)
• C = phase shift (horizontal shift - positive is to the left)
• D = vertical shift

The period is the horizontal distance between consecutive peaks, which is the same as twice the horizontal distance between the intersection of the curve and the mid-line.

Given consecutive points of intersection between the curve and the mid-line:

• (3.7, 5)
• (5.9, 5)

Therefore, the horizontal distance between these two points is:

5.9 - 3.7 = 2.2

⇒ Period = 2.2 × 2 = 4.4

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To create the equation for the function.

From inspection of the given graph:

• Amplitude (A) = 6
• Mid-line is y = 5
• Vertical shift (D) = +5

Period = 2π/B = 4.4 ⇒ B = 5π/11

Phase Shift (C) = -3.7

Substituting the values into the standard form:

`\implies \sf f(x)= 6 \sin\left((5)/(11)\pi(x-3.7)\right)+5`

`\implies \sf f(x)= 6 \sin\left((5)/(11) \pi x-(37)/(22)\pi \right)+5`