Graph one complete period of the given sine or cosine curve. (Check your answer with your graphing calculatorl) \[ \begin{array}{l}f(x)=-2+\sin x\end{array} \] \[ \begin{array}{l}\text { Amplitude }=\square\end{array} \text { Phase Shift }=\square \] (Hint this is foumd by taking the period and dividing by 4)

To graph the function \( f(x) = -2 + \sin x \) and determine its amplitude and phase shift, let's break it down step by step. ### 1. **Understanding the Function** The given function is: \[ f(x) = -2 + \sin x \] This is a sine function with a vertical shift. Let's rewrite it in the standard sine form: \[ f(x) = A \sin(Bx - C) + D \] Where: - \( A \) is the **amplitude**. - \( B \) affects the **period** of the function. - \( C \) is the **phase shift**. - \( D \) is the **vertical shift**. Comparing the given function to the standard form: - \( A = 1 \) (since the coefficient of \( \sin x \) is 1) - \( B = 1 \) (no coefficient implies 1) - \( C = 0 \) (no horizontal shift) - \( D = -2 \) (the function is shifted down by 2 units) ### 2. **Determining Amplitude and Phase Shift** - **Amplitude (\( A \))**: \[ \text{Amplitude} = |A| = |1| = 1 \] - **Phase Shift (\( C \))**: \[ \text{Phase Shift} = \frac{C}{B} = \frac{0}{1} = 0 \] (There is no horizontal shift.) ### 3. **Graphing the Function** Here's how to graph \( f(x) = -2 + \sin x \): 1. **Start with the basic sine curve** \( \sin x \): - **Amplitude**: 1 - **Period**: \( 2\pi \) - **Phase Shift**: 0 - **Vertical Shift**: 0 2. **Apply the vertical shift**: - The entire sine curve is shifted downward by 2 units because of the "+ −2". 3. **Final Graph Characteristics**: - **Amplitude**: 1 (the wave oscillates 1 unit above and below the midline) - **Midline**: \( y = -2 \) - **Maximum Point**: \( y = -2 + 1 = -1 \) - **Minimum Point**: \( y = -2 - 1 = -3 \) - **Phase Shift**: 0 (no horizontal shift) ### 4. **Graph Sketch** Since I can't provide an actual graph here, I'll describe it: - **Midline**: Draw a horizontal line at \( y = -2 \). - **Wave Oscillation**: The sine wave will oscillate between \( y = -1 \) and \( y = -3 \). - **Starting Point**: Begins at the midline \( y = -2 \) when \( x = 0 \). - **Period**: Completes one full cycle from \( x = 0 \) to \( x = 2\pi \). ### 5. **Verification with a Graphing Calculator** You can verify the graph by entering the function \( f(x) = -2 + \sin x \) into a graphing calculator or an online graphing tool like [Desmos](https://www.desmos.com/calculator). The graph should match the description provided above. ### 6. **Summary** \[ \begin{array}{l} \text{Amplitude} = 1 \\ \text{Phase Shift} = 0 \end{array} \] Feel free to reach out if you need further assistance with graphing or understanding trigonometric functions!