Find Amplitude, Period, and Phase Shift y=5sin(2x-pi/3)+1

Solution:

Step 1: Identify the parameters from the standard sine function form $a\sin (bx-c)+d$.

• Amplitude ($a$): $5$
• Frequency factor ($b$): $2$
• Phase shift constant ($c$): $\frac{\pi }{3}$
• Vertical shift ($d$): $1$

Step 2: Determine the amplitude, which is the absolute value of $a$.

Amplitude: $|5|=5$

Step 3: Calculate the period using the formula $T=\frac{2\pi }{|b|}$.

Step 3.1: Compute the period for the sine function $5\sin (2x-\frac{\pi }{3})$.

• Apply the formula: $T=\frac{2\pi }{|2|}$
• Simplify the absolute value: $T=\frac{2\pi }{2}$
• Reduce the fraction: $T=\frac{\cancel{2}\pi }{\cancel{2}}=\pi$

Step 4: Compute the phase shift using the formula $\text{Phase Shift}=\frac{c}{b}$.

• Insert the constants: $\text{Phase Shift}=\frac{\frac{\pi }{3}}{2}$
• Simplify by multiplying the numerator by the reciprocal of the denominator: $\text{Phase Shift}=\frac{\pi }{3}\cdot \frac{1}{2}$
• Perform the multiplication: $\text{Phase Shift}=\frac{\pi }{3\cdot 2}=\frac{\pi }{6}$

Step 5: Summarize the properties of the sine function.

• Amplitude: $5$
• Period: $\pi$
• Phase Shift: $\frac{\pi }{6}$ (to the right)
• Vertical Shift: $1$

Knowledge Notes:

To analyze the trigonometric function $y=a\sin (bx-c)+d$, we identify the following properties:

1. Amplitude ($a$): This is the coefficient in front of the sine function, which determines the height of the wave's peaks and troughs. The amplitude is the absolute value of $a$.

2. Period ($T$): The period is the length of one complete cycle of the sine wave. It is inversely proportional to the frequency factor $b$ and is calculated using the formula $T=\frac{2\pi }{|b|}$.

3. Phase Shift: This is the horizontal shift of the sine wave along the x-axis. It is determined by the constant $c$ and the frequency factor $b$, with the formula $\text{Phase Shift}=\frac{c}{b}$. A positive phase shift means the graph shifts to the right, while a negative shift means it moves to the left.

4. Vertical Shift ($d$): This is the constant added to the sine function, which moves the graph up or down along the y-axis.

5. Absolute Value: When calculating the amplitude and period, we take the absolute value of $a$ and $b$, respectively, because these properties are always positive.

6. Simplifying Fractions: When simplifying fractions, we cancel out common factors in the numerator and denominator to find the simplest form.

7. Multiplying Fractions: To multiply fractions, we multiply the numerators together and the denominators together. If one of the numbers is a whole number, we can write it as a fraction with a denominator of 1 to perform the multiplication.

Understanding these properties allows us to graph the sine function accurately and predict its behavior based on the equation parameters.