\( h(x) \) relion to the given equation \( g(x)=2 \sin x ; g(x)=3 \cos x ; h(x)=\sin (2 x) \) and \( h(x)=\cos 3 x \) which are drawn above. if: \( m(x)=a \sin (p x) \). Corocily complete the statements below. 1.3 .1 a represents \( \qquad \) \( p \) represents. \( \qquad \) (1) 1.3 .2 Hence or otherwise deduce the general formula for calculating the period of the following functions: a) \( g(x)=a \sin (p x) \) \[ \begin{array}{l} \text { Period }= \\ \text { Period }= \end{array} \] b) \( f(x)=b \tan (p x) \), (I) Write down the equations of the asymptotes of the functions below \( g(x)=3 \tan (2 x) \) where \( x \in\left[-90^{\circ} ; 90^{\circ}\right] \) 1.3.3 a) \( \qquad \) b) Describe the transformation that the graph of \( f \) has to undergo to form the \( g \) of \( g \).

Let's break down the problem step by step. ### 1.3.1 We have the function \( m(x) = a \sin(px) \). - **a represents**: The amplitude of the sine function. It determines how high and low the graph of the sine function will go. Therefore, \( a \) represents the amplitude. - **p represents**: The frequency of the sine function. It affects how many cycles the sine function completes in a given interval. Therefore, \( p \) represents the frequency. ### 1.3.2 To deduce the general formula for calculating the period of the functions: a) For \( g(x) = a \sin(px) \): - The period \( T \) of the sine function is given by the formula: \[ \text{Period} = \frac{2\pi}{p} \] b) For \( f(x) = b \tan(px) \): - The period \( T \) of the tangent function is given by the formula: \[ \text{Period} = \frac{\pi}{p} \] ### 1.3.3 For the function \( g(x) = 3 \tan(2x) \): - **(I) Write down the equations of the asymptotes**: The asymptotes of the tangent function occur at: \[ x = \frac{\pi}{2p} + \frac{k\pi}{p} \quad \text{for } k \in \mathbb{Z} \] For \( g(x) = 3 \tan(2x) \), we have \( p = 2 \). Thus, the asymptotes are: \[ x = \frac{\pi}{4} + \frac{k\pi}{2} \quad \text{for } k \in \mathbb{Z} \] Within the interval \( x \in [-90^\circ, 90^\circ] \) or \( x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), the asymptotes are: - \( x = -\frac{\pi}{4} \) - \( x = \frac{\pi}{4} \) - **b) Describe the transformation that the graph of \( f \) has to undergo to form \( g \)**: The transformation from \( f(x) = b \tan(px) \) to \( g(x) = 3 \tan(2x) \) involves: 1. **Vertical Stretch**: The graph is stretched vertically by a factor of 3 (since \( b = 3 \)). 2. **Horizontal Compression**: The graph is compressed horizontally by a factor of \( \frac{1}{2} \) (since \( p = 2 \)). In summary: - \( a \) represents the amplitude. - \( p \) represents the frequency. - The period of \( g(x) \) is \( \frac{2\pi}{p} \) and for \( f(x) \) is \( \frac{\pi}{p} \). - The asymptotes of \( g(x) = 3 \tan(2x) \) are \( x = -\frac{\pi}{4} \) and \( x = \frac{\pi}{4} \). - The transformation involves a vertical stretch by 3 and a horizontal compression by a factor of \( \frac{1}{2} \).