Graph the function \( y=-3 \sin \left(\frac{\pi}{2} x\right)-4 \). Show at least two cycles. Use the graph to determine the domain and range of the function.

Sure! Let's graph the function \( y = -3 \sin\left(\frac{\pi}{2} x\right) - 4 \) and analyze its domain and range. We'll cover at least two complete cycles of the sine wave. ## 1. Understanding the Function The given function is a transformed sine function. Let's break down its components: - **Amplitude (A):** The coefficient before the sine function is 3 (absolute value). So, the amplitude is \( |A| = 3 \). - **Angular Frequency (B):** Inside the sine function, the coefficient of \( x \) is \( \frac{\pi}{2} \). So, \( B = \frac{\pi}{2} \). - **Period (T):** The period of the sine function is calculated by: \[ T = \frac{2\pi}{B} = \frac{2\pi}{\frac{\pi}{2}} = 4 \] - **Phase Shift (C):** There is no horizontal shift since there's no addition or subtraction inside the sine function. - **Vertical Shift (D):** The function is shifted downward by 4 units, so \( D = -4 \). - **Reflection:** The negative sign before the amplitude indicates a reflection over the **x-axis**. ## 2. Graphing the Function We'll plot the function over two periods, i.e., from \( x = 0 \) to \( x = 8 \) (since each period is 4 units). ### Key Points Calculation Let's compute the y-values at critical points within each period. | \( x \) | \( \frac{\pi}{2}x \) | \( \sin\left(\frac{\pi}{2}x\right) \) | \( y = -3\sin\left(\frac{\pi}{2}x\right) - 4 \) | |--------|---------------------|---------------------------------------|-----------------------------------------------| | 0 | 0 | 0 | \( -3(0) - 4 = -4 \) | | 1 | \( \frac{\pi}{2} \) | 1 | \( -3(1) - 4 = -7 \) | | 2 | \( \pi \) | 0 | \( -3(0) - 4 = -4 \) | | 3 | \( \frac{3\pi}{2} \)| -1 | \( -3(-1) - 4 = -1 \) | | 4 | \( 2\pi \) | 0 | \( -3(0) - 4 = -4 \) | | 5 | \( \frac{5\pi}{2} \)| 1 | \( -3(1) - 4 = -7 \) | | 6 | \( 3\pi \) | 0 | \( -3(0) - 4 = -4 \) | | 7 | \( \frac{7\pi}{2} \)| -1 | \( -3(-1) - 4 = -1 \) | | 8 | \( 4\pi \) | 0 | \( -3(0) - 4 = -4 \) | ### Plotting the Points Here's a table of key points: | \( x \) | \( y \) | |--------|--------| | 0 | -4 | | 1 | -7 | | 2 | -4 | | 3 | -1 | | 4 | -4 | | 5 | -7 | | 6 | -4 | | 7 | -1 | | 8 | -4 | ### Sketch of the Graph Below is a rough sketch of the graph over \( x = 0 \) to \( x = 8 \): ``` y | -1 | * * | / \ / \ -2 | / \ / \ -3 | / \ / \ -4 |------*------*-------*---------*-------*------*------ x -5 | / \ / \ -6 | / \ / \ -7 | * * * * | ``` **Explanation of the Sketch:** - **Start at (0, -4):** The graph begins at \( y = -4 \). - **First Trough at (1, -7):** Reaches the minimum y-value of -7. - **Back to Midline at (2, -4):** Returns to the midline. - **First Peak at (3, -1):** Reaches the maximum y-value of -1. - **Back to Midline at (4, -4):** Completes the first cycle. - **Second Cycle:** Repeats the pattern from \( x = 4 \) to \( x = 8 \). ## 3. Determining the Domain and Range - **Domain:** The domain of the sine function is all real numbers. \[ \text{Domain}: \quad (-\infty, \infty) \] - **Range:** The sine function has a range of \([-1, 1]\). After the transformations: - **Amplitude Scaling:** Multiplied by 3 → \([-3, 3]\). - **Reflection:** Still \([-3, 3]\) since reflection over the x-axis inverts the sine wave but doesn't change the range. - **Vertical Shift:** Subtracted by 4 → \([-7, -1]\). \[ \text{Range}: \quad [-7, -1] \] ## 4. Summary - **Graph Features:** - **Amplitude:** 3 - **Period:** 4 units - **Maximum Value:** -1 - **Minimum Value:** -7 - **Vertical Shift:** Down by 4 units - **Reflection:** Over the x-axis - **Domain:** All real numbers \((-\infty, \infty)\) - **Range:** \([-7, -1]\) This analysis should help you visualize and understand the behavior of the function \( y = -3 \sin\left(\frac{\pi}{2} x\right) - 4 \). If you have access to graphing tools like Desmos or a graphing calculator, plotting the function there can provide a precise visual representation.