Graphing a Cotangent Function
<p>To graph \( y = 2\cot(2x) \), we need to identify the properties of the cotangent function, including its period, phase shift, amplitude, and asymptotes.</p>
<p>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.</p>
<p>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 \( k \).</p>
<p>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.</p>
<p>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.</p>
<p>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 \( \cot(\frac{\pi}{2}) = 0 \).</p>
<p>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.</p>
