The equation given is:

$y = -\frac{1}{2} \cos \left( \frac{3}{4} x \right)$

Key points to graph within one cycle:

1. Amplitude: The coefficient of the cosine function, $-\frac{1}{2}$, gives the amplitude. The graph will oscillate between $y = -\frac{1}{2}$ and $y = \frac{1}{2}$. The negative sign reflects the graph vertically.

2. Period: The period of the function is calculated using the formula: $\text{Period} = \frac{2\pi}{\frac{3}{4}} = \frac{8\pi}{3}$ So, one full cycle occurs from $x = 0$ to $x = \frac{8\pi}{3}$.

3. Phase shifts: Since there are no added constants inside or outside the cosine, there are no phase shifts or vertical shifts.

4. X-intercepts, Minima, and Maxima:

   • X-intercepts: Cosine crosses zero at $x = \frac{4\pi}{3}$ and $x = \frac{8\pi}{3}$.
   • Maxima: The maximum value occurs when $\cos \left( \frac{3}{4} x \right) = 1$, at $x = 0$.
   • Minima: The minimum value occurs when $\cos \left( \frac{3}{4} x \right) = -1$, at $x = \frac{4\pi}{3}$.

Points to plot:

• Maximum: $(0, -\frac{1}{2})$
• X-intercepts: $(\frac{4\pi}{3}, 0)$, $(\frac{8\pi}{3}, 0)$
• Minimum: $(\frac{4\pi}{3}, \frac{1}{2})$

This information should guide you in plotting the graph on the grid. Let me know if you'd like further details on any step!

5 Related Questions:

1. How do you calculate the period for other trigonometric functions like sine or tangent?
2. What happens to the graph if we change the amplitude or period?
3. How can phase shifts affect the appearance of trigonometric graphs?
4. How would the graph differ if the coefficient of $x$ inside the cosine were different?
5. What real-world applications use cosine functions?

Tip: Always check if the trigonometric function has any phase or vertical shifts, as they significantly impact the graph's shape and position.