1.2 Drawing from your own observations in QUESTION 1.1 and 1.2, answer the following questions. In relation 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 : $$ m(x)=\boldsymbol{a} \sin (\boldsymbol{p} x) $$. Correctly complete the statements below. 1.3.1 $$ a $$ represents. $$ \qquad $$ prepresents. $$ \qquad $$ (1) 1.3.2 Hence or otherwise deduce the general formula for calculating the period of the following functions: a) $$ g(x)=\boldsymbol{a} \sin (\boldsymbol{p} x) $$ Period $$ = $$ $$ \qquad $$ b) $$ f(x)=\boldsymbol{b} an (\boldsymbol{p} x) $$, Period $$ = $$ $$ \qquad $$ 1.3.3 Write down the equations of the asymptotes of the functions below a) $$ \quad g(x)=3 an (2 x) $$ where $$ x \in\left[-90^{\circ} ; 90^{\circ} ight] $$ b) Describe the transformation that the graph of $$ \boldsymbol{f} $$ has to undergo to form the graph of $$ \boldsymbol{g} $$. [35]

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1.2 Drawing from your own observations in QUESTION 1.1 and 1.2, answer the following questions. In relation 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 : $$ m(x)=\boldsymbol{a} \sin (\boldsymbol{p} x) $$. Correctly complete the statements below. 1.3.1 $$ a $$ represents. $$ \qquad $$ prepresents. $$ \qquad $$ (1) 1.3.2 Hence or otherwise deduce the general formula for calculating the period of the following functions: a) $$ g(x)=\boldsymbol{a} \sin (\boldsymbol{p} x) $$ Period $$ = $$ $$ \qquad $$ b) $$ f(x)=\boldsymbol{b} 	an (\boldsymbol{p} x) $$, Period $$ = $$ $$ \qquad $$ 1.3.3 Write down the equations of the asymptotes of the functions below a) $$ \quad g(x)=3 	an (2 x) $$ where $$ x \in\left[-90^{\circ} ; 90^{\circ}ight] $$ b) Describe the transformation that the graph of $$ \boldsymbol{f} $$ has to undergo to form the graph of $$ \boldsymbol{g} $$. [35]

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### 1.3.1 Identifying Parameters

In the function $$ m(x) = a \sin(px) $$:
- $$ a $$ represents the **amplitude** of the sine wave. It determines how far the wave oscillates from its midline (the maximum and minimum values).
- $$ p $$ represents the **frequency** of the sine wave, which affects the number of cycles the wave completes in a given interval.

### 1.3.2 Deducing the General Formula for Period

The period of a trigonometric function is the length of one complete cycle of the wave. The general formula for the period of the functions is derived from the coefficient of $$ x $$ inside the sine or tangent function.

a) For $$ g(x) = a \sin(px) $$:
- The period $$ T $$ is given by the formula:
$$
T = \frac{2\pi}{|p|}
$$

b) For $$ f(x) = b \tan(px) $$:
- The period $$ T $$ is given by the formula:
$$
T = \frac{\pi}{|p|}
$$

### 1.3.3 Asymptotes of the Functions

a) For the function $$ g(x) = 3 \tan(2x) $$:
- The tangent function has vertical asymptotes where the function is undefined, which occurs at:
$$
x = \frac{\pi}{2p} + \frac{k\pi}{p} \quad \text{for } k \in \mathbb{Z}
$$
For $$ p = 2 $$:
$$
x = \frac{\pi}{4} + \frac{k\pi}{2}
$$
Thus, within the interval $$ x \in [-90^\circ, 90^\circ] $$ (or $$ x \in [-\frac{\pi}{2}, \frac{\pi}{2}] $$), the asymptotes are at:
$$
x = -\frac{\pi}{4}, \frac{\pi}{4}
$$
or in degrees:
$$
x = -45^\circ, 45^\circ
$$

b) To describe the transformation that the graph of $$ f $$ has to undergo to form the graph of $$ g $$:
- The transformation involves a vertical stretch by a factor of 3 (due to the coefficient of 3 in front of the tangent function) and a horizontal compression by a factor of $$ \frac{1}{2} $$ (due to the coefficient of 2 in the argument of the tangent function).

### Summary of Answers

1.3.1 
- $$ a $$ represents: **amplitude**
- $$ p $$ represents: **frequency**

1.3.2 
- Period of $$ g(x) = a \sin(px) $$: $$ T = \frac{2\pi}{|p|} $$
- Period of $$ f(x) = b \tan(px) $$: $$ T = \frac{\pi}{|p|} $$

1.3.3 
- Asymptotes of $$ g(x) = 3 \tan(2x) $$: $$ x = -45^\circ, 45^\circ $$
- Transformation from $$ f $$ to $$ g $$: **Vertical stretch by a factor of 3 and horizontal compression by a factor of $$ \frac{1}{2} $$**.
1.3.1 - $$ a $$ represents: **amplitude** - $$ p $$ represents: **frequency** 1.3.2 - Period of $$ g(x) = a \sin(px) $$: $$ T = \frac{2\pi}{|p|} $$ - Period of $$ f(x) = b \tan(px) $$: $$ T = \frac{\pi}{|p|} $$ 1.3.3 - Asymptotes of $$ g(x) = 3 \tan(2x) $$: $$ x = -45^\circ, 45^\circ $$ - Transformation from $$ f $$ to $$ g $$: **Vertical stretch by a factor of 3 and horizontal compression by a factor of $$ \frac{1}{2} $$**.

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