Question - Graphing a Cotangent Function
Solution:
To graph \( y = 2\cot(2x) \), we need to identify the properties of the cotangent function, including its period, phase shift, amplitude, and asymptotes.
The basic cotangent function has the form \( y = \cot(x) \) with vertical asymptotes at \( x = k\pi \) where \( k \) is an integer, since cotangent is undefined when sine is 0, which happens at these points. The period of the cotangent function is \( \pi \), meaning it repeats every \( \pi \) units.
For \( y = 2\cot(2x) \), the period is \( \frac{\pi}{b} \), where \( b \) is the coefficient of \( x \), which is 2 in this case. Thus, the period of this function is \( \frac{\pi}{2} \). This means that vertical asymptotes occur at points \( x = \frac{k\pi}{2} \) for integer \( k \).
The amplitude, normally affecting the height of the peaks in sine and cosine functions, doesn't apply to the cotangent function as it goes to infinity at the asymptotes.
The graph will oscillate between the asymptotes and will have a point of symmetry at \( x = \frac{k\pi}{2} \) for odd \( k \). You can plot key points by evaluating \( y = 2\cot(2x) \) at various \( x \) values, bearing in mind that cotangent is the reciprocal of tangent.
Some points to consider for one period of the function starting from \( x = 0 \) up to \( x = \frac{\pi}{2} \) would include the undefined points where vertical asymptotes occur (at \( x = 0 \) and \( x = \frac{\pi}{2} \)) and a point of intersection on the x-axis at \( x = \frac{\pi}{4} \) where \( \cot(\frac{\pi}{2}) = 0 \).
Now you can sketch the graph using the asymptotes at \( x = 0 \) and \( x = \frac{\pi}{2} \), the x-axis intersection at \( x = \frac{\pi}{4} \), and the fact that the cotangent function decreases as \( x \) increases within each period.
Tutor Solution:
Firstly, we need to understand the basic shape and properties of the cotangent function. The cotangent function, cot(x), has a period of \( \pi \), meaning it repeats every \( \pi \) units. It has vertical asymptotes where the function is undefined, which occurs when the argument of the cotangent equals \( n\pi \), where \( n \) is an integer, because cotangent is the reciprocal of tangent, which is zero at these points.
The function provided is a transformed cotangent function. The "2" inside the cotangent function's argument means that the period is halved, so the new period is \( \frac{\pi}{2} \), and the "2" outside the cotangent function means that the function is vertically stretched by a factor of 2.
Here are the steps to graph \( y = 2\cot(2x) \):
1. Identify the period of the function.
Since the period of the standard cotangent function is \( \pi \), and we have a multiplier of 2 within the argument, the period of our function is \( \frac{\pi}{2} \). This means the function will repeat every \( \frac{\pi}{2} \) units along the x-axis.
2. Locate the vertical asymptotes.
Vertical asymptotes occur where the function is undefined, which is when the argument of the cotangent equals \( n\pi \), where \( n \) is an integer. Therefore, for \( y = 2\cot(2x) \), the vertical asymptotes will be at \( x = \frac{n\pi}{2} \). Over one period, you'll have asymptotes at \( x = 0 \) and \( x = \frac{\pi}{2} \).
3. Plot key points of the function.
The cotangent function starts from positive infinity at \( x = 0 \) (just to the right of the vertical asymptote), crosses through zero at the middle of the period, and approaches negative infinity at \( x = \frac{\pi}{2} \) (just to the left of the vertical asymptote). With our function being vertically stretched by a factor of 2, these key points will also be scaled accordingly.
4. Sketch the graph between asymptotes.
Connect the points smoothly, remembering that the function falls from infinity to cross the x-axis at \( x = \frac{\pi}{4} \) (since this is the midpoint of the period) and then falls to negative infinity as it approaches the next asymptote.
5. Repeat for additional periods if necessary.
Once you have graphed one period of the function, you can replicate this shape to the left and right to continue the function for the desired range.
Keep in mind that the graph of \( 2\cot(2x) \) will exhibit these properties: - It will have a sinusoidal shape with a distinct sharp turn rather than a smooth wave, unlike sine or cosine functions. - It will be undefined and have vertical asymptotes at multiples of \( x = \frac{n\pi}{2} \). - The range of the function is all real numbers since the values can go from negative to positive infinity.
With these steps, you should be able to plot the function \( y = 2\cot(2x) \) accurately over one or more periods.CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com