Find Amplitude, Period, and Phase Shift y=7cos(3x)
The problem is asking for the identification of three key characteristics of the cosine function given in the form y = A*cos(Bx - C):
Amplitude (A): The amplitude is the coefficient in front of the cosine function, which dictates the maximum value the function reaches above and below its midline (or equilibrium point).
Period (T): The period is the length of one complete cycle of the function, usually measured as the distance between successive peaks or valleys. For a cosine function in the form y = A*cos(Bx - C), the period is calculated using the formula T = 2π / |B|, where B is the coefficient affecting the x-variable inside the cosine.
Phase Shift: The phase shift is the horizontal shift of the function along the x-axis. It's determined by the C-value in the cosine function y = A*cos(Bx - C). The phase shift moves the graph left or right, usually expressed as a value that indicates how much the function’s standard position (starting at x=0) is moved. In this form, the phase shift is calculated by the formula C/B, with its direction depending on the sign of C. If there is no C in the given function, or if C equals zero, there is no phase shift.
The question requires the application of these definitions to the given function to determine its amplitude, period, and phase shift.
$y = 7 cos \left(\right. 3 x \left.\right)$
Answer
Solution:
Step 1: Identify the coefficients in the standard cosine function format
The given function $y=7\cos(3x)$ can be compared to the standard form $a\cos(bx-c)+d$ to determine the amplitude, period, phase shift, and vertical shift.
- Amplitude coefficient ($a$): $7$
- Frequency coefficient ($b$): $3$
- Phase shift coefficient ($c$): $0$
- Vertical shift ($d$): $0$
Step 2: Calculate the amplitude
The amplitude is the absolute value of the amplitude coefficient ($a$).
Amplitude: $|7| = 7$
Step 3: Determine the period of the function
Step 3.1
The period $T$ of a cosine function is found using the formula $T = \frac{2\pi}{|b|}$.
Step 3.2
Substitute the frequency coefficient $b=3$ into the period formula.
Period: $T = \frac{2\pi}{|3|}$
Step 3.3
Since the absolute value of $3$ is $3$, the period is $T = \frac{2\pi}{3}$.
Step 4: Compute the phase shift
Step 4.1
The phase shift is calculated using the formula $\frac{c}{b}$.
Step 4.2
Insert the values for $c=0$ and $b=3$ into the phase shift formula.
Phase Shift: $\frac{0}{3}$
Step 4.3
Dividing $0$ by $3$ yields a phase shift of $0$.
Step 5: Summarize the trigonometric function's characteristics
- Amplitude: $7$
- Period: $\frac{2\pi}{3}$
- Phase Shift: $0$ (None)
- Vertical Shift: $0$ (None)
Knowledge Notes:
The amplitude, period, phase shift, and vertical shift are properties of trigonometric functions that describe their shape and position on a graph.
Amplitude: The amplitude of a trigonometric function is the maximum distance from the midline of the wave to its peak or trough. It is the absolute value of the coefficient $a$ in the standard form $a\cos(bx-c)+d$ or $a\sin(bx-c)+d$.
Period: The period of a trigonometric function is the length of one complete cycle of the wave. It is determined by the coefficient $b$ and can be calculated using the formula $T = \frac{2\pi}{|b|}$ for cosine and sine functions.
Phase Shift: The phase shift of a trigonometric function is the horizontal displacement of the function's graph from its usual position. It is calculated using the formula $\frac{c}{b}$, where $c$ is the phase shift coefficient in the standard form. A positive phase shift moves the graph to the right, while a negative phase shift moves it to the left.
Vertical Shift: The vertical shift is the upward or downward displacement of the graph of the function. It corresponds to the value of $d$ in the standard form. A positive vertical shift moves the graph upwards, and a negative vertical shift moves it downwards.
In the given problem, the trigonometric function $y=7\cos(3x)$ has an amplitude of $7$, a period of $\frac{2\pi}{3}$, no phase shift, and no vertical shift. These properties define the function's graph and can be used to sketch it accurately.
