$$ h(x) $$ 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 $$ g $$.

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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 $$ \frac{1}{2} $$.
- **a represents**: The amplitude of the sine function. - **p represents**: The frequency of the sine function. **General Period Formulas:** - For $$ g(x) = a \sin(px) $$, the period is $$ \frac{2\pi}{p} $$. - For $$ f(x) = b \tan(px) $$, the period is $$ \frac{\pi}{p} $$. **Asymptotes for $$ g(x) = 3 \tan(2x) $$:** - $$ x = -\frac{\pi}{4} $$ and $$ x = \frac{\pi}{4} $$. **Transformation from $$ f(x) $$ to $$ g(x) $$:** - Vertically stretch by 3. - Horizontally compress by a factor of $$ \frac{1}{2} $$.

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