Question

ASAP Write the equation of the sine function graphed below. There is no horizontal shift.

Answer 1

`y=3 \sin(3x)+1`

Explanation:

The standard form of a sine function is:

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

where:

• A is the amplitude (height from the mid-line to the peak).
• 2π/B is the period (horizontal length of one cycle of the curve).
• C is the phase shift (horizontal shift - positive is to the left).
• D is the vertical shift.

The amplitude of a sinusoidal function is one-half of the positive difference between the maximum and minimum values of a function.

From observation of the given graph, the maximum value is 4 and the minimum value is -2. Therefore:

`\sf A=(|4-(-2)|)/(2)=3`

The period is the horizontal length of one cycle of the curve. From observation of the given graph, the period is 2π/3. Therefore:

`\sf (2\pi)/(B)=(2\pi)/(3)\implies B=3`

The phase shift is the horizontal shift. Since we have been told that there is no horizontal shift, then:

`\sf C=0`

The midline is the horizontal line located at the y-value that is midway between the maximum and minimum y-values of the function. Since the maximum y-value is 4 and the minimum y-value is -2, the midline of the graphed function is:

`\sf Midline:\quad y=(4-2)/(2)=1`

The midline of the parent sine function y = sin(x) is the x-axis (y = 0), so the graphed function has undergone a vertical shift upwards of 1 unit:

`\sf D=1`

Substitute the found values of A, B, C and D into the formula:

`y=3 \sin(3(x+0))+1`

`y=3 \sin(3x)+1`

Therefore, the equation of the graphed function is:

`\large\boxed{\boxed{y=3 \sin(3x)+1}}`