To graph the function $y = -3 \cot \left( x + \frac{3\pi}{4} \right)$, here are the steps you need to follow:

Step 1: Identify the Basic Properties of the Cotangent Function

For the cotangent function $y = a \cot(bx + c)$, the period is given by $\frac{\pi}{|b|}$. In this case:

• Amplitude: The amplitude is affected by the coefficient $-3$, but for cotangent, we usually focus on vertical scaling, not traditional amplitude.
• Phase Shift: The horizontal shift is calculated by setting $bx + c = 0$. Here, we have: $x + \frac{3\pi}{4} = 0 \Rightarrow x = -\frac{3\pi}{4}$ So, the graph is shifted left by $\frac{3\pi}{4}$.
• Period: Since $b = 1$, the period of the cotangent function is $\pi$.

Step 2: Draw the Asymptotes

For the cotangent function, asymptotes occur at the beginning and end of each period. Given the phase shift:

• First Asymptote: At $x = -\frac{3\pi}{4}$
• Next Asymptote: One period later, at $x = -\frac{3\pi}{4} + \pi = \frac{\pi}{4}$

Step 3: Identify Key Points between the Asymptotes

1. X-Intercept (Midpoint): The midpoint between the asymptotes will be where the function crosses the x-axis. The midpoint between $-\frac{3\pi}{4}$ and $\frac{\pi}{4}$ is: $x = -\frac{\pi}{4}$
2. Points Left and Right of the Midpoint:
   • Choose a point to the left, say $x = -\frac{\pi}{2}$.
   • Choose a point to the right, say $x = 0$.

Calculate the function values at these points:

• At $x = -\frac{\pi}{4}$: $y = 0$ (x-intercept).
• At $x = -\frac{\pi}{2}$: Substitute into the function to get a positive value (the function will be decreasing as we move to the right).
• At $x = 0$: Substitute into the function to get a negative value.

Step 4: Plot the Points and Draw the Curve

1. Place the two vertical asymptotes at $x = -\frac{3\pi}{4}$ and $x = \frac{\pi}{4}$.
2. Mark the x-intercept at $x = -\frac{\pi}{4}$.
3. Plot the points you calculated on either side of the x-intercept.
4. Sketch the cotangent curve, decreasing from left to right, between the asymptotes.

This approach should help you accurately graph $y = -3 \cot \left( x + \frac{3\pi}{4} \right)$ on the provided grid.

Would you like more details on any of these steps, or do you have further questions?

Related Questions

1. What is the impact of the coefficient $-3$ on the cotangent graph?
2. How does shifting affect the position of asymptotes in trigonometric functions?
3. How is the period of a trigonometric function calculated, especially for functions other than sine and cosine?
4. What are typical transformations applied to cotangent and tangent functions?
5. How would the graph change if the function were $y = 3 \cot(x + \frac{3\pi}{4})$?

Tip

When graphing functions with phase shifts and transformations, always start by identifying the asymptotes and intercepts for better accuracy in plotting.