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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) \( f(x)=\boldsymbol{b} \tan (\boldsymbol{p} x) \), Period \( = \) \( \qquad \) 1.3.3 Write down the equations of the asymptotes of the functions below a) \( \quad g(x)=3 \tan (2 x) \) 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 \( \boldsymbol{g} \). [35]

Ask by Reyes Donnelly. in South Africa
Mar 18,2025

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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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Beyond the Answer

In trigonometric functions, \( a \) represents the amplitude, which dictates how high or low the wave reaches relative to its midline. Amplitude is crucial for understanding the height of sine and cosine graphs, while \( p \) is related to the frequency of the wave, affecting how many cycles occur within a given interval. To deduce the period of the functions, you can use the formula \( \text{Period} = \frac{2\pi}{|p|} \) for sine and cosine functions, implying that the greater the value of \( p \), the shorter the period of the wave. For tangent functions, the period is given by \( \text{Period} = \frac{\pi}{|p|} \), showcasing how tangent functions repeat their values at a different rate compared to sine and cosine. For the asymptotes of \( g(x) = 3 \tan(2x) \), we know that tangent functions have vertical asymptotes where the function approaches infinity. They occur at \( x = \frac{\pi}{2p} + n\frac{\pi}{p} \) for integers \( n \). In this case, the asymptotes can be found at \( x = -\frac{\pi}{4}, \frac{\pi}{4} \). As for the transformation from \( f(x) \) to \( g(x) \), you must compress the graph horizontally by a factor of \( \frac{1}{2} \) and then stretch it vertically by a factor of \( 3 \). This transforms the wave into a steeper, more pronounced tangent function!

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