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 \( \qquad \) \( p \) represents. \( \qquad \) (1) (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) \( f(x)=b \tan (\boldsymbol{p} x) \), Period \( = \) \( \qquad \) 1.3.3 Write down the equations of the asymptotes of the functions below a) \( g(x)=3 \tan (2 x) \) where \( x \in\left[-90^{\circ} ; 90^{\circ}\right] \) b) Describe the transformation that the graph of \( \boldsymbol{f} \) has to undergo to form the graph of \( g \). (2) [35]

Certainly! Let's address each part of the question systematically. ### **1.3.1 Identifying \( a \) and \( p \) in \( m(x) = a \sin(p x) \)** - **\( a \) represents** the **amplitude** of the function. *The amplitude determines the vertical stretch or compression of the sine wave.* - **\( p \) represents** the **angular frequency** (or **frequency factor**) of the function. *The angular frequency affects the number of cycles the function completes within a given interval, thereby influencing the period of the function.* --- ### **1.3.2 Determining the Period of Trigonometric Functions** The **period** of a function is the length of the smallest interval over which the function completes one full cycle. #### **a) For \( g(x) = a \sin(p x) \):** - **Period Formula:** \[ \text{Period} = \frac{2\pi}{|p|} \] *This formula shows that the period decreases as the angular frequency \( p \) increases.* #### **b) For \( f(x) = b \tan(p x) \):** - **Period Formula:** \[ \text{Period} = \frac{\pi}{|p|} \] *The period of the tangent function is \( \pi \) divided by the absolute value of the angular frequency \( p \).* --- ### **1.3.3 Equations of the Asymptotes and Transformation Description** #### **a) Asymptotes of \( g(x) = 3 \tan(2x) \) within \( x \in [-90^\circ, 90^\circ] \):** The **asymptotes** of the tangent function occur where the function is undefined, which happens at: \[ 2x = \pm \frac{\pi}{2} + n\pi \quad \text{for any integer } n \] Converting to degrees and solving within the given interval: \[ 2x = \pm 90^\circ \Rightarrow x = \pm 45^\circ \] - **Equations of the Asymptotes:** \[ x = -45^\circ \quad \text{and} \quad x = 45^\circ \] #### **b) Transformation from \( f(x) \) to \( g(x) = 3 \tan(2x) \):** Assuming \( f(x) = \tan(x) \), the graph undergoes the following transformations to become \( g(x) \): 1. **Vertical Stretch:** - **Factor:** 3 - **Effect:** The graph is stretched vertically by a factor of 3, making it steeper. 2. **Horizontal Compression:** - **Factor:** \( \frac{1}{2} \) (since the coefficient of \( x \) is 2) - **Effect:** The graph is compressed horizontally, reducing the period from \( \pi \) to \( \frac{\pi}{2} \). These transformations result in the graph of \( g(x) = 3 \tan(2x) \) having steeper slopes and more frequent cycles within the same interval compared to \( f(x) = \tan(x) \). --- **Summary of Answers:** 1. **1.3.1** - \( a \) represents **amplitude**. - \( p \) represents **angular frequency**. 2. **1.3.2** - **a)** Period of \( g(x) = a \sin(p x) \) is \( \frac{2\pi}{p} \). - **b)** Period of \( f(x) = b \tan(p x) \) is \( \frac{\pi}{p} \). 3. **1.3.3** - **a)** Asymptotes of \( g(x) = 3 \tan(2x) \) are \( x = -45^\circ \) and \( x = 45^\circ \). - **b)** The graph of \( f(x) = \tan(x) \) undergoes a **vertical stretch by a factor of 3** and a **horizontal compression by a factor of \( \frac{1}{2} \)** to form the graph of \( g(x) = 3 \tan(2x) \).