Find Amplitude, Period, and Phase Shift y=-2cos(x+pi/6)
The problem provided is asking for three specific characteristics of the cosine function presented:
Amplitude: This refers to the vertical stretch of the wave, which dictates how high and how low the wave goes. It is determined by the coefficient in front of the cosine function, which indicates the maximum value that the function's output can reach from the equilibrium position (central axis).
Period: The period of a trigonometric function like cosine is the length of one complete cycle of the wave. It is the horizontal length along the x-axis required for the function to start repeating itself. The period of the cosine function can be affected by the coefficient inside the function, typically in front of the variable (in this case, x).
Phase Shift: This refers to the horizontal shift of the wave along the x-axis. If the graph of the cosine function is moved to the left or to the right, this is called the phase shift. It is determined by the horizontal displacement from the standard position of the cosine wave, usually caused by a constant being added or subtracted inside the function with the variable.
The question presents an equation of a transformed cosine function and asks to find these three characteristics to describe and understand how the graph of the function relates to the basic cosine graph y = cos(x).
$y = - 2 cos \left(\right. x + \frac{\pi}{6} \left.\right)$
Answer
Solution:
Step 1: Identify the coefficients for amplitude, period, phase shift, and vertical shift
Given the function $y = -2\cos(x + \frac{\pi}{6})$, we can determine:
- Amplitude coefficient ($a$): $-2$
- Period coefficient ($b$): $1$
- Phase shift coefficient ($c$): $\frac{\pi}{6}$ (Note the negative sign inside the cosine function)
- Vertical shift ($d$): $0$
Step 2: Calculate the amplitude
The amplitude is the absolute value of $a$.
Amplitude: $|-2| = 2$
Step 3: Determine the period of the cosine function
Step 3.1
The period is found using the formula $\frac{2\pi}{|b|}$.
Step 3.2
Substitute $b = 1$ into the period formula.
Period: $\frac{2\pi}{|1|}$
Step 3.3
Since the absolute value of $1$ is $1$, we have:
Period: $\frac{2\pi}{1}$
Step 3.4
Calculating the period gives us $2\pi$.
Period: $2\pi$
Step 4: Compute the phase shift
Step 4.1
The phase shift is given by $\frac{c}{b}$.
Step 4.2
Insert the values for $c$ and $b$ into the phase shift formula.
Phase Shift: $\frac{-\frac{\pi}{6}}{1}$
Step 4.3
Simplifying the fraction gives us the phase shift.
Phase Shift: $-\frac{\pi}{6}$
Step 5: Summarize the properties of the cosine function
- Amplitude: $2$
- Period: $2\pi$
- Phase Shift: $-\frac{\pi}{6}$ (which is a shift to the left by $\frac{\pi}{6}$)
- Vertical Shift: None (since $d = 0$)
Step 6: Conclusion
The amplitude, period, and phase shift of the function $y = -2\cos(x + \frac{\pi}{6})$ are $2$, $2\pi$, and $-\frac{\pi}{6}$, respectively, with no vertical shift.
Knowledge Notes:
To analyze the function $y = a\cos(bx - c) + d$ and find its amplitude, period, phase shift, and vertical shift, we use the following knowledge:
Amplitude: The amplitude of a trigonometric function is the absolute value of the coefficient $a$. It represents the maximum value the function reaches above or below its midline, which is the vertical shift $d$.
Period: The period of a trigonometric function is the length of one complete cycle of the function. For a cosine function, the period is calculated as $\frac{2\pi}{|b|}$, where $b$ is the coefficient in front of $x$.
Phase Shift: The phase shift is the horizontal shift of the function along the x-axis. It is calculated as $\frac{c}{b}$, where $c$ is the constant added or subtracted inside the function's argument. A positive phase shift indicates a shift to the right, while a negative phase shift indicates a shift to the left.
Vertical Shift: The vertical shift is the value $d$, which moves the function up or down along the y-axis.
Absolute Value: The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative.
By applying these concepts, we can analyze the given trigonometric function and determine its key characteristics.
