This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions for each part of your homework: a.i) For the function y = - 1 Step 1: Determine the coordinates at E. The point E represents the minimum value of the function. The minimum value of is -1, which occurs at = 180^. Substituting this into the function: y = (180^) - 1 = -1 - 1 = -2 The coordinates at E are (180^, -2). Step 2: Determine the interval where the function is increasing. The function y = - 1 is a vertical shift of y = . The increasing intervals for are from 180^ to 360^. The function is increasing for (180^, 360^). Step 3: Determine the range of the function. The range of is [-1, 1]. For y = - 1, we subtract 1 from these values. The minimum value is -1 - 1 = -2. The maximum value is 1 - 1 = 0. The range of the function is [-2, 0]. a.ii) For the function y = + 1 Step 1: Determine the interval where the function is increasing. The function y = + 1 is a vertical shift of y = . The increasing intervals for are from 0^ to 90^ and from 270^ to 360^. The function is increasing for (0^, 90^) (270^, 360^). Step 2: Determine the range of the function. The range of is [-1, 1]. For y = + 1, we add 1 to these values. The minimum value is -1 + 1 = 0. The maximum value is 1 + 1 = 2. The range of the function is [0, 2]. b.i) For the function y = + 2 Step 1: Determine the coordinates at E. The point E represents the minimum value of the function. The minimum value of is -1, which occurs at = 270^. Substituting this into the function: y = (270^) + 2 = -1 + 2 = 1 The coordinates at E are (270^, 1). Step 2: Determine the interval where the function is increasing. The function y = + 2 is a vertical shift of y = . The increasing intervals for are from 0^ to 90^ and from 270^ to 360^. The function is increasing for (0^, 90^) (270^, 360^). Step 3: Determine the range of the function. The range of is [-1, 1]. For y = + 2, we add 2 to these values. The minimum value is -1 + 2 = 1. The maximum value is 1 + 2 = 3. The range of the function is [1, 3]. b.ii) For the function y = Step 1: Determine the interval where the function is increasing. The standard cosine function y = increases from 180^ to 360^. The function is increasing for (180^, 360^). Step 2: Determine the range of the function. The range of the standard cosine function y = is [-1, 1]. c.i) For the function y = 2 + (5)/(2) Step 1: Determine the coordinates at E. The point E represents the minimum value of the function. The minimum value of is -1, which occurs at = 180^. Substituting this into the function: y = 2(180^) + (5)/(2) = 2(-1) + (5)/(2) = -2 + (5)/(2) = (-4+5)/(2) = (1)/(2) The coordinates at E are (180^, (1)/(2)). Step 2: Determine the interval where the function is increasing. The function y = 2 + (5)/(2) is a vertical stretch and shift of y = . The increasing intervals for are from 180^ to 360^. The function is increasing for (180^, 360^). Step 3: Determine the range of the function. The range of is [-1, 1]. For y = 2, the range is [2(-1), 2(1)] = [-2, 2]. For y = 2 + (5)/(2), we add (5)/(2) to these values. The minimum value is -2 + (5)/(2) = (-4+5)/(2) = (1)/(2). The maximum value is 2 + (5)/(2) = (4+5)/(2) = (9)/(2). The range of the function is [(1)/(2), (9)/(2)]. Drop the next question!