Question

Write the equation of the trigonometric graph.

Answer 1

Answer:

$f(x)= \sin(6x)+3$

Step-by-step 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:

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

• 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 parent function y = sin(x) has the following:

• Amplitude (A) = 1
• Period = 2π
• Phase shift (C) = 0
• Vertical shift (D) = 0
• Mid-line:  y = 0

From inspection of the given graph:

• Amplitude (A) = 1
• $\sf Period=\dfrac{\pi}{12}-\dfrac{-\pi}{4}=\dfrac{\pi}{3}$
• Phase shift (C) = 0
• Vertical shift (D) = +3  (as mid-line is y = 3)

$\sf If\:Period=\dfrac{\pi}{3} \implies \dfrac{2 \pi}{B}=\dfrac{\pi}{3}\implies B=6$

Substituting the values into the standard form:

$\implies f(x) =1 \sin (6(x + 0)) + 3$

$\implies f(x) = \sin (6x) + 3$

Therefore, the equation of the given trigonometric graph is:

$f(x)= \sin(6x)+3$