\( \left. \begin{array} { l }{ y = - a \cdot \sin ( a x + b ) - a , } \\ { x \in [ 0 ; 2 \pi ] } \end{array} \right. \)

The given equation is \[ y = -a \cdot \sin(ax + b) - a \] where \( x \in [0, 2\pi] \). Let's analyze this function step by step. ### 1. Understanding the Function The function is a transformation of the sine function. The general form of a sine function is \( y = A \sin(Bx + C) + D \), where: - \( A \) is the amplitude, - \( B \) affects the period, - \( C \) is the phase shift, - \( D \) is the vertical shift. In your case: - The amplitude is \( |A| = | -a | = a \) (since amplitude is always positive), - The period is given by \( \frac{2\pi}{|B|} = \frac{2\pi}{|a|} \), - The phase shift is \( -\frac{C}{B} = -\frac{b}{a} \), - The vertical shift is \( D = -a \). ### 2. Characteristics of the Function - **Amplitude**: The amplitude of the function is \( a \). - **Period**: The period of the function is \( \frac{2\pi}{|a|} \). - **Vertical Shift**: The entire graph is shifted down by \( a \) units. - **Phase Shift**: The graph is shifted to the left by \( \frac{b}{a} \) units. ### 3. Range of the Function To find the range of the function, we can analyze the sine component: - The sine function \( \sin(ax + b) \) oscillates between -1 and 1. - Therefore, \( -\sin(ax + b) \) oscillates between -1 and 1 as well. - Multiplying by \( -a \) flips the range to \( [-a, a] \). - Finally, subtracting \( a \) shifts the entire range down by \( a \). Thus, the range of \( y \) is: \[ [-a - a, a - a] = [-2a, 0] \] ### 4. Behavior on the Interval \( [0, 2\pi] \) - The function will complete one or more cycles depending on the value of \( a \). - If \( a = 1 \), the function will complete one full cycle in the interval \( [0, 2\pi] \). - If \( a > 1 \), the function will complete \( a \) cycles in the interval. - If \( 0 < a < 1 \), the function will complete \( \frac{1}{a} \) cycles in the interval. ### Conclusion The function \( y = -a \cdot \sin(ax + b) - a \) is a sine wave that has been reflected, scaled, and shifted. Its range is \( [-2a, 0] \), and its behavior depends on the values of \( a \) and \( b \).