Find Amplitude, Period, and Phase Shift y=5sin(2x-pi/3)+1
The problem presented is asking for the identification of certain characteristics of a trigonometric function, specifically a sine function that has been transformed. The sine function in question is \( y = 5\sin(2x - \pi/3) + 1 \).
The problem asks to determine:
The Amplitude: This is the coefficient in front of the sine function which determines the height of the wave from the center line to the peak. In a standard sine function \( y = A\sin(Bx - C) + D \), the amplitude is represented by |A|.
The Period: This is the length of one full cycle of the sine wave. It is derived from the coefficient B in front of the \( x \) within the sine function, and the standard period of a sine function is \( 2\pi \). The period can be calculated using the formula \( \text{Period} = \frac{2\pi}{|B|} \).
The Phase Shift: This is the horizontal shift of the function along the x-axis and determines where the sine wave starts. It is indicated by the C value in \( Bx - C \) and calculated as \( \frac{C}{B} \) assuming the standard sine function form mentioned above, and if \( C/B \) is positive, the shift is to the right and if negative, to the left.
The problem does not ask for the vertical shift, but that is typically represented by D in the standard form of the transformed sine function.
$y = 5 sin \left(\right. 2 x - \frac{\pi}{3} \left.\right) + 1$
Answer
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:
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 $a$.
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 $T = \frac{2\pi}{|b|}$.
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.
Vertical Shift ($d$): This is the constant added to the sine function, which moves the graph up or down along the y-axis.
Absolute Value: When calculating the amplitude and period, we take the absolute value of $a$ and $b$, respectively, because these properties are always positive.
Simplifying Fractions: When simplifying fractions, we cancel out common factors in the numerator and denominator to find the simplest form.
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.
