Question

the graph of y=-3.cos(3x)

Answer 1

Here are the key features and a simple sketch plan for y = −3 cos(3x).

- Amplitude: 3 (so graph ranges 3 units above and below the midline).
- Reflection: the negative sign flips cosine across the x-axis.
- Midline: y = 0.
- Period: 2π / 3 (because period = 2π / |3|).
- Domain: all real numbers. Range: [−3, 3].

Important points over one period (0 ≤ x ≤ 2π/3):
- x = 0: y = −3 (minimum)
- x = π/6: y = 0 (ascending zero)
- x = π/3: y = 3 (maximum)
- x = π/2: y = 0 (descending zero)
- x = 2π/3: y = −3 (minimum, completes the period)

General formulas:
- x-intercepts: x = π/6 + k(π/3), k ∈ Z.
- Maxima (y = 3): x = π/3 + (2kπ)/3.
- Minima (y = −3): x = (2kπ)/3.

To sketch:
1. Draw the midline y = 0.
2. Mark one period length 2π/3 on the x-axis.
3. Plot the five key points above and draw the smooth cosine wave flipped (starts at a minimum).
4. Repeat the pattern left and right.

If you want, I can produce a plotted image or give coordinates for more periods.