How do you sketch #y = 3 sin 2 (x-1)#?

Question

Caroline Adams · Accepted Answer

graph{3sin(2(x-1)) [-10, 10, -5, 5]}

If we consider #Asin[B(x+C)]#, the first term A is increasing the amplitude of the sin graph. So if we make A = 3 we would get the following graph. graph{3sinx [-10, 10, -5, 5]} We will look at C next, this is the movement of the graph left or right, where a negative C value moves the graph to the right. So we move the whole graph 1 to the right in this case. #3sin(1(x-1))# give the following graph. graph{3sin(x-1) [-10, 10, -5, 5]} Finally B is stretching the graph parallel to the x axis by a factor of #1/B xx 2Pi# So in your case B = 2, so #1/2 xx 2Pi = Pi# radians. This gives us the new period for your graph, this means a complete cycle occurs every #Pi# rads instead of every #2Pi# rads. Then graphing this: #3sin(2(x-1))# graph{3sin(2(x-1)) [-10, 10, -5, 5]}

Adam Aguirre · Answer

undefined To sketch the graph of $ y = 3\sin(2(x-1)) $, follow these steps:

1. Identify the key components of the function:
   - Amplitude: $ |3| = 3 $
   - Period: $ \frac{2\pi}{2} = \pi $
   - Phase shift: $ 1 $ unit to the right

2. Plot the key points:
   - Starting with the origin, mark the points where the function crosses the x-axis.
   - Mark the peak and trough points, considering the amplitude and phase shift.

3. Sketch the graph:
   - Use smooth curves to connect the key points, keeping in mind the shape of the sine function.

Remember, the amplitude determines the height of the peaks and troughs, the period determines the distance between consecutive peaks (or troughs), and the phase shift shifts the entire graph horizontally.