The equation of a curve is $y = 3 - 4\sin 2x$.

1. Identify the range of the sine function

We know that $-1 \le \sin 2x \le 1$.

2. Calculate the minimum and maximum values of y

Maximum value of $y$ occurs when $\sin 2x = -1$:
$y_{\max} = 3 - 4(-1) = 7$

Minimum value of $y$ occurs when $\sin 2x = 1$:
$y_{\min} = 3 - 4(1) = -1$

1. Determine the amplitude, period, and principal axis of the function

For $y = 3 - 4\sin 2x$:

• Principal axis: $y = 3$
• Amplitude: $4$
• Period: $\frac{2\pi}{2} = \pi$

2. Identify the key coordinates for the first period ($0 \le x \le \pi$)

The curve is an inverted sine wave (due to the negative sign) shifted up by 3.
Key points for the first period:

• Start: $(0, 3)$
• Minimum: $(\frac{\pi}{4}, 3 - 4) = (\frac{\pi}{4}, -1)$
• Midpoint: $(\frac{\pi}{2}, 3)$
• Maximum: $(\frac{3\pi}{4}, 3 + 4) = (\frac{3\pi}{4}, 7)$
• End of first period: $(\pi, 3)$

3. Identify the key coordinates for the second period and sketch the full graph

The pattern repeats for $\pi \le x \le 2\pi$:

• Minimum: $(\frac{5\pi}{4}, -1)$
• Midpoint: $(\frac{3\pi}{2}, 3)$
• Maximum: $(\frac{7\pi}{4}, 7)$
• End: $(2\pi, 3)$
  Draw a smooth curve through all these points.