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.
ScanSolve AI Answer
To sketch the graphs of trigonometric functions like y = (k) or y = (k), we need to determine their amplitude and period. The amplitude is the maximum displacement from the equilibrium position, and the period is the length of one complete cycle of the wave. a) Sketch the graph of y = (3). Step 1: Identify the amplitude and period. The general form of a sine function is y = A (k). Here, A=1, so the amplitude is 1. The period P is given by P = (360^)/(|k|). In this case, k=3. P = (360^)/(3) = 120^ This means one complete cycle of the graph occurs over an interval of 120^. Step 2: Determine key points for one cycle (0^ 120^). We find the values of y at intervals of (P)/(4) = (120^)/(4) = 30^. • At = 0^: y = (3 · 0^) = (0^) = 0 • At = 30^: y = (3 · 30^) = (90^) = 1 (maximum) • At = 60^: y = (3 · 60^) = (180^) = 0 • At = 90^: y = (3 · 90^) = (270^) = -1 (minimum) • At = 120^: y = (3 · 120^) = (360^) = 0 Step 3: Sketch the graph. Draw a coordinate plane with the horizontal axis representing (from 0^ to 120^) and the vertical axis representing y (from -1 to 1). Plot the key points determined in Step 2. Connect these points with a smooth curve. The graph starts at (0^, 0), rises to a maximum of 1 at 30^, crosses the -axis at 60^, falls to a minimum of -1 at 90^, and returns to the -axis at 120^. This completes one cycle. b) Sketch the graph of y = (3). Step 1: Identify the amplitude and period. The general form of a cosine function is y = A (k). Here, A=1, so the amplitude is 1. The period P is given by P = (360^)/(|k|). In this case, k=3. P = (360^)/(3) = 120^ This means one complete cycle of the graph occurs over an interval of 120^. Step 2: Determine key points for one cycle (0^ 120^). We find the values of y at intervals of (P)/(4) = (120^)/(4) = 30^. • At = 0^: y = (3 · 0^) = (0^) = 1 (maximum) • At = 30^: y = (3 · 30^) = (90^) = 0 • At = 60^: y = (3 · 60^) = (180^) = -1 (minimum) • At = 90^: y = (3 · 90^) = (270^) = 0 • At = 120^: y = (3 · 120^) = (360^) = 1 (maximum) Step 3: Sketch the graph. Draw a coordinate plane with the horizontal axis representing (from 0^ to 120^) and the vertical axis representing y (from -1 to 1). Plot the key points determined in Step 2. Connect these points with a smooth curve. The graph starts at (0^, 1), crosses the -axis at 30^, falls to a minimum of -1 at 60^, crosses the -axis again at 90^, and returns to a maximum of 1 at 120^. This completes one cycle. Drop the next question. 📸