In Exercises 53–60, use a vertical shift to graph one period of the function. y = sin x + 2

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Identify the base function and its characteristics. Here, the base function is \(y = \sin x\), which has a period of \(2\pi\), an amplitude of 1, and oscillates between -1 and 1.

Understand the effect of the vertical shift. The function \(y = \sin x + 2\) shifts the entire sine curve upward by 2 units, so the midline of the graph moves from \(y=0\) to \(y=2\).

Determine the new range of the function after the vertical shift. Since the original sine function ranges from -1 to 1, adding 2 shifts this range to \(1 \leq y \leq 3\).

Plot one period of the function from \(x=0\) to \(x=2\pi\). At key points: \(x=0\), \(x=\frac{\pi}{2}\), \(x=\pi\), \(x=\frac{3\pi}{2}\), and \(x=2\pi\), calculate the corresponding \(y\) values using \(y = \sin x + 2\).

Sketch the graph using the calculated points, ensuring the wave oscillates smoothly between \(y=1\) and \(y=3\) with the midline at \(y=2\), completing one full period over the interval \([0, 2\pi]\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Vertical Shift in Trigonometric Functions

A vertical shift moves the entire graph of a function up or down without changing its shape. For y = sin x + 2, the '+2' shifts the sine curve two units upward, raising the midline from y = 0 to y = 2.

Period of the Sine Function

The period of the sine function is the length of one complete cycle, which is 2π for y = sin x. Graphing one period means plotting the function from 0 to 2π to capture a full wave of the sine curve.

Graphing Sine Functions

Graphing sine functions involves plotting key points such as maxima, minima, and zeros over one period. For y = sin x + 2, these points are shifted vertically, so the maximum is at y = 3, minimum at y = 1, and midline at y = 2.