Question

Sketch one cycle of a sinusoidal curve with the given amplitude and period and passing through the given point. a) amplitude \(2,\) period \(180^{\circ},\) point (0,0) b) amplitude \(1.5,\) period \(540^{\circ},\) point (0,0)

Short answer

(a) \(y = 2\sin{(2x)}\), (b) \(y = 1.5\sin{\left(\frac{2}{3}x\right)}\)

Step-by-step solution

- Determine the Equation Formula

The general form of a sinusoidal curve is given by the equation: \[y = A \times \text{sin}(B(x - C)) + D\]where: - A is the amplitude, - B is related to the period such that \ B = \frac{360^{\circ}}{\text{period}}, - C is the horizontal shift, - D is the vertical shift.

- Apply Parameters for Case (a)

Given the amplitude is 2, period is 180°, and the point is (0,0): - Amplitude (A) = 2 - Period = 180° - Since the function passes through (0,0) and we usually take the sine function which starts at (0,0): A = 2, B = \frac{360^{\circ}}{180^{\circ}} = 2, C = 0, D = 0. Hence, the equation is: \[y = 2 \times \text{sin}(2x)\]

- Plot the Sinusoidal Curve for Case (a)

Sketch the graph of \( y = 2 \times \text{sin}(2x) \) over one period of 180°: - From 0° to 180°, - The graph starts at (0,0), - Reaches the amplitude 2 at 45° (since it is the peak), - Returns to 0 at 90°, - Reaches -2 at 135° (since it is the trough), - Ends back at 0 at 180°.

- Apply Parameters for Case (b)

Given the amplitude is 1.5, period is 540°, and the point is (0,0): - Amplitude (A) = 1.5 - Period = 540° - Since the function passes through (0,0) and we usually take the sine function which starts at (0,0): A = 1.5, B = \frac{360^{\circ}}{540^{\circ}} = \frac{2}{3}, C = 0, D = 0. Hence, the equation is: \[y = 1.5 \times \text{sin}\left(\frac{2}{3}x\right)\]

- Plot the Sinusoidal Curve for Case (b)

Sketch the graph of \( y = 1.5 \times \text{sin}\left(\frac{2}{3}x\right) \) over one period of 540°: - From 0° to 540°, - The graph starts at (0,0) - Reaches the amplitude 1.5 at 135° (since it is the peak), - Returns to 0 at 270°, - Reaches -1.5 at 405° (since it is the trough), - Ends back at 0 at 540°.

Key concepts

Amplitude

Amplitude is a key feature of sinusoidal curves. It represents the maximum vertical distance from the central axis (or equilibrium position) to the peak of the curve. In simpler terms, it tells you how 'tall' the waves are. For the sine function, amplitude is the coefficient in front of the sine. For instance, in the equation \( y = 2 \times \text{sin}(2x) \), the amplitude is 2. This means the highest points (peaks) on the graph are 2 units above the middle line, and the lowest points (troughs) are 2 units below. Understanding amplitude helps in predicting the behavior of the wave and is crucial for graphing.

Period

The period of a sinusoidal curve is the horizontal length it takes for the curve to complete one full cycle. Imagine tracing a smooth and repetitive wave; the period tells you how far you need to go horizontally before the wave pattern repeats. For the sine function, the period is related to the coefficient inside the sine function. In the equation \(y = 2 \times \text{sin}(2x)\), to find the period, we divide 360° by the coefficient of x. Here, it is \(B = 2\), so the period is \( \frac{360°}{2} = 180°\). This means it takes 180° for the sine wave to go from the starting point, through the peak, back to the trough, and to the end of one cycle. Knowing the period is essential for accurately sketching the wave.

Sin Function

The sine function is the backbone of sinusoidal curves. It's a special type of mathematical function that creates smooth and repetitive wave patterns. The basic sin function \( y = \text{sin}(x) \) starts at the origin (0,0), rises to a peak at 90°, falls back to 0 at 180°, drops to a trough at 270°, and returns to 0 at 360°. This cycle then repeats. To customize a sine function to different amplitudes and periods, we use the general form \( y = A \times \text{sin}(Bx) \). Here, A determines the amplitude, and B influences the period. For example, \(y = \text{sin}(x) \), \(y = 2 \times \text{sin}(2x) \), and \(y = 1.5 \times \text{sin}\left(\frac{2}{3}x\right)\) are different forms showing various amplitudes and periods.

Graphing

Graphing sinusoidal curves is all about understanding the pattern and parameters like amplitude and period. Start by marking important points: the starting point, peak, trough, and cycle endpoint. For the equation \( y = 2 \times \text{sin}(2x) \), plot as follows: 0° (start at 0), 45° (peak at 2), 90° (back to 0), 135° (trough at -2), and 180° (back to 0). Repeat these points to extend the wave. For \( y = 1.5 \times \text{sin}( \frac{2}{3}x) \), note the period changes to 540°, and the peak and trough are at 1.5 and -1.5, respectively. Plot: 0° (0), 135° (1.5), 270° (0), 405° (-1.5), and 540° (0). Always use the amplitude and period to guide your graphing for accuracy.