For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 1 + 2 sin ¼ x

Verified step by step guidance

1. Identify the general form of the sine function: \(y = A \sin(B(x - C)) + D\), where \(A\) is the amplitude, \(\frac{2\pi}{B}\) is the period, \(C\) is the phase shift, and \(D\) is the vertical translation.
2. Compare the given function \(y = 1 + 2 \sin \frac{1}{4} x\) to the general form. Rewrite it as \(y = 2 \sin \left( \frac{1}{4} x \right) + 1\) to clearly see the components.
3. Determine the amplitude \(A\) by looking at the coefficient in front of the sine function. Here, \(A = 2\).
4. Calculate the period using the formula \(\text{Period} = \frac{2\pi}{B}\), where \(B = \frac{1}{4}\). So, the period is \(\frac{2\pi}{\frac{1}{4}}\).
5. Identify the vertical translation \(D\) as the constant added outside the sine function, which is \(1\). Since there is no subtraction or addition inside the sine argument, the phase shift \(C\) is \(0\).

Key Concepts

Amplitude of a Trigonometric Function

Amplitude measures the maximum distance a sine or cosine function's graph reaches from its midline. It is the absolute value of the coefficient before the sine or cosine term. For example, in y = 2 sin(¼x), the amplitude is 2, indicating the wave oscillates 2 units above and below its central axis.

Period of a Trigonometric Function

The period is the length of one complete cycle of the sine or cosine function. It is calculated by dividing 2π by the absolute value of the coefficient of x inside the function. For y = 2 sin(¼x), the period is 2π ÷ ¼ = 8π, meaning the function repeats every 8π units along the x-axis.

Vertical Translation and Phase Shift

Vertical translation shifts the graph up or down and is determined by the constant added outside the sine or cosine function, here +1. Phase shift moves the graph left or right and depends on horizontal shifts inside the function's argument. In y = 1 + 2 sin(¼x), the vertical translation is +1, and there is no phase shift since there is no horizontal addition or subtraction inside the sine.