The function \(y = \sin(a(x + b))\) is a sine wave with an amplitude of 1. The period of the sine function is given by \(\frac{2\pi}{a}\). From the graph, the period is \(4\pi\), so:

\(\frac{2\pi}{a} = 4\pi\)

Solving for \(a\), we get:

\(a = \frac{1}{2}\)

The phase shift is determined by \(b\). The graph starts at \(x = -\frac{\pi}{3}\), which corresponds to the phase shift. Therefore, \(b = \frac{\pi}{3}\).

The equation of the curve is given by \(y = p \sin(q\theta) + r\).

(a) The amplitude of the sine function is the coefficient \(p\). From the graph, the maximum value is 1 and the minimum value is -5, giving an amplitude of \(\frac{1 - (-5)}{2} = 3\). Thus, \(p = 3\).

(b) The period of the sine function is given by \(\frac{2\pi}{q}\). From the graph, the period is \(4\pi\), so \(\frac{2\pi}{q} = 4\pi\). Solving for \(q\), we get \(q = \frac{1}{2}\).

(c) The vertical shift \(r\) is the average of the maximum and minimum values of the function. Thus, \(r = \frac{1 + (-5)}{2} = -2\).

(a) The graph of \(y = a \cos(bx) + c\) shows a cosine wave with a maximum value of 8 and a minimum value of -2. The amplitude \(a\) is half the distance between the maximum and minimum values, so \(a = \frac{8 - (-2)}{2} = 5\).

The period of the cosine function is \(\frac{2\pi}{b}\). From the graph, the period is \(\pi\), so \(\frac{2\pi}{b} = \pi\), giving \(b = 2\).

The vertical shift \(c\) is the average of the maximum and minimum values, so \(c = \frac{8 + (-2)}{2} = 3\).

(b)(i) For \(a \cos(bx) + c = \frac{6}{\pi} x\), substitute \(a = 5\), \(b = 2\), \(c = 3\) to get \(5 \cos(2x) + 3 = \frac{6}{\pi} x\). The number of solutions is the number of intersections of the line \(y = \frac{6}{\pi} x\) with the cosine graph in the interval \(0 \leq x \leq 2\pi\), which is 3.

(b)(ii) For \(a \cos(bx) + c = 6 - \frac{6}{\pi} x\), substitute \(a = 5\), \(b = 2\), \(c = 3\) to get \(5 \cos(2x) + 3 = 6 - \frac{6}{\pi} x\). The number of solutions is the number of intersections of the line \(y = 6 - \frac{6}{\pi} x\) with the cosine graph in the interval \(0 \leq x \leq 2\pi\), which is 2.

The graph shown is a transformation of the basic tangent function \(y = \tan x\).

The vertical stretch factor \(a\) is determined by the amplitude of the graph. The graph reaches approximately 3 at \(x = \frac{\pi}{2}\), indicating \(a = 2\).

The horizontal shift \(b\) is determined by the phase shift of the graph. The graph crosses the x-axis at \(x = \frac{\pi}{4}\), so \(b = \frac{\pi}{4}\).

The vertical shift \(c\) is determined by the midline of the graph. The midline appears to be at \(y = 1\), so \(c = 1\).

The graph of \(y = a + b \sin x\) is shown with a maximum value of 1 and a mid value of -2, which indicates \(a = -2\).

The amplitude \(b\) is the distance from the midline to the maximum or minimum. Since the maximum is 1 and the midline is -2, the amplitude is \(1 - (-2) = 3\).

Thus, \(a = -2\) and \(b = 3\).