Determine the amplitude, the period, and the phase shift of...

Determine the amplitude, the period, and the phase shift of the function and, without a graphing calculator, sketch the graph of the function by hand. Then check the graph using
a graphing calculator.
$$y=-\sin \left(\frac{1}{4} x\right)+1$$

Key Concepts

Reflection

A reflection in a sine function occurs when the coefficient multiplying the sine term is negative. This negation flips the graph over the horizontal axis, meaning that peaks become troughs and vice versa while preserving the period and amplitude.

Amplitude

The amplitude of a sine function is the absolute value of the coefficient multiplying the sine term. It represents the vertical stretch or compression of the basic sine graph and determines the distance from the midline of the graph to its maximum or minimum point.

Period

The period of a sine function refers to the horizontal length over which the function completes one full cycle. In the general form y = sin(Bx), the period is calculated as 2? divided by the absolute value of B. This indicates how the graph is stretched or compressed horizontally.

Phase Shift

The phase shift of a sine function is the horizontal translation of the graph from its standard position. In the function y = sin(Bx - C), the phase shift is determined by dividing C by B. This shift moves the starting point of the cycle to the left or right.

Vertical Shift

The vertical shift of a function is the upward or downward displacement of the entire graph. In a function of the form y = sin(Bx) + D, the value D moves the midline of the sine curve, altering the overall vertical position without changing its shape.

Transcript

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