y= 2sin(2x-pi/4) find amplitude, period, phase shift, assymptote and graph
change to
y = 2sin 2(x - π/8)
amplitude: 2
period = 2π/2 = π
phase shift : π/8 to the right
no asymptote in a sine curve
To find the amplitude, period, phase shift, asymptote, and graph of the given function y = 2sin(2x - π/4), let's break it down step by step.
Step 1: Amplitude
The amplitude of a sine function is the absolute value of the coefficient in front of the sine term. In this case, the coefficient is 2. Therefore, the amplitude is |2|, which is 2.
Step 2: Period
The period of a sine function is given by the formula 2π/b, where b is the coefficient of x in the sine term. In this case, the coefficient of x is 2. So, the period is 2π/2, which simplifies to just π.
Step 3: Phase Shift
The phase shift of a sine function is given by the formula c/b, where c represents any horizontal shift of the graph. In this case, the phase shift is -π/4 divided by 2, which simplifies to -π/8.
Step 4: Asymptote
The sine function does not have a horizontal asymptote. However, the vertical asymptote is determined by the domain of the function. Since there are no restrictions on the domain for this function, the vertical asymptote does not exist.
Step 5: Graph
To graph the function, plot a few points on the graph using the amplitude, period, and phase shift we calculated.
- Start with the phase shift: x = -π/8
- Calculate the y-coordinate using the sine function at this x-value: y = 2sin(2*(-π/8) - π/4) ≈ -1.414
So, one point on the graph is (-π/8, -1.414).
- To find additional points, you can use the calculated period.
Use the period of π to find the next point. Add π to the x-coordinate and evaluate the sine function at that point.
For example, if you add π to -π/8, you get -π/8 + π = 7π/8.
Calculate the y-coordinate at this point: y = 2sin(2*(7π/8) - π/4) ≈ 0.586
So, another point on the graph is (7π/8, 0.586).
- Continue this process to find more points on the graph if needed.
By plotting these points, you can sketch the graph of the function. The graph will have an amplitude of 2, a period of π, and a phase shift of -π/8. For a more accurate graph, you can add more points or use technology to plot the graph.
To find the amplitude, period, phase shift, asymptote, and graph of the function y = 2sin(2x - π/4), let's break down each component:
Amplitude:
The amplitude of a sine function represents the maximum distance the graph reaches from the midline. In this case, the coefficient of sine is 2, so the amplitude is |2| = 2.
Period:
The period of a sine function represents the length of one complete cycle of the graph. The general period formula for the sine function is T = (2π) / b, where b is the coefficient of x. In this case, the coefficient of x is 2, so the period is (2π) / 2 = π.
Phase Shift:
The phase shift of a sine function represents the horizontal shift of the graph. The general formula for phase shift is given as c / b, where c is a constant. In this case, the constant is (-π/4), so the phase shift is (-π/4) / 2 = -π/8.
Asymptote:
Sine functions do not have asymptotes.
Graph:
To graph the function, it is helpful to identify some key points:
- The midline is y = 0.
- The amplitude is 2, so the maximum and minimum values are 2 and -2, respectively.
- The period is π.
Using this information, you can now plot the graph. Start by plotting the midline (y = 0). Then, for each quarter of the cycle, mark the key points:
- At 0, the sine function starts at 0.
- At π/4, the sine function reaches its maximum value of 2.
- At π/2, the sine function returns to 0.
- At 3π/4, the sine function reaches its minimum value of -2.
- At π, the sine function returns to 0.
Repeat these points for each additional period.
Here is a sketch of the graph:
```
|
2 | /\
| / \
| / \
| / \
| / \
|/ \
-2 -------------
π/4 π/2
```
Please note that the sketch is not perfectly to scale, but it gives you an idea of how the graph looks.