Sketch one full cycle of the graph of y=3sin(0.5x)-2.Give the amplitude and the period and the indicate the turning points and intercepts clearly.
amplitude = 3
period = 2π/0.5 = 4π
center line: y = -2
To sketch the graph of y = 3sin(0.5x) - 2 and determine the amplitude, period, turning points, and intercepts, follow these steps:
Step 1: Amplitude
The amplitude of a sine function is the distance between the maximum and minimum values of the graph. In this case, the amplitude is 3, as indicated by the coefficient in front of sin(0.5x).
Step 2: Period
The period of a sine function is the distance it takes for one complete cycle to occur. The standard formula for the period is T = (2π) / |B|, where B is the coefficient of x. In this case, B = 0.5, so T = (2π) / |0.5| = 4π.
Step 3: Turning Points
The turning points of a sine function occur where the function changes direction. These points are at the maximum and minimum values of the graph. To find the turning points of this function, set y = 3sin(0.5x) - 2 equal to the maximum and minimum values (±3). Solve for x to find the corresponding values. Plug these x-values back into the original equation to find the y-values.
Step 4: Intercepts
The intercepts of a graph occur where the function intersects the x-axis (y = 0) or the y-axis (x = 0). To find the x-intercepts, set y = 3sin(0.5x) - 2 equal to 0 and solve for x. To find the y-intercept, substitute x = 0 into the equation y = 3sin(0.5x) - 2 and solve for y.
Now, let's apply these steps:
Step 1: Amplitude
The amplitude is 3.
Step 2: Period
The period is 4π.
Step 3: Turning Points
To find the turning points, set y = 3sin(0.5x) - 2 equal to the maximum and minimum values, which are ±3.
To find the maximum turning point:
3sin(0.5x) - 2 = 3
sin(0.5x) = 5/3
0.5x = sin^(-1)(5/3)
x = 2sin^(-1)(5/3)
To find the minimum turning point:
3sin(0.5x) - 2 = -3
sin(0.5x) = -1/3
0.5x = sin^(-1)(-1/3)
x = 2sin^(-1)(-1/3)
Plug these x-values back into the original equation to find the y-values.
Step 4: Intercepts
To find the x-intercepts, set y = 3sin(0.5x) - 2 equal to 0 and solve for x:
3sin(0.5x) - 2 = 0
3sin(0.5x) = 2
sin(0.5x) = 2/3
0.5x = sin^(-1)(2/3)
x = 2sin^(-1)(2/3)
To find the y-intercept, substitute x = 0 into the equation and solve for y:
y = 3sin(0.5(0)) - 2
y = 3sin(0) - 2
y = 0 - 2
y = -2
Now that we have all the necessary information, we can plot the graph and label the key points. The graph will consist of one complete cycle within the period:
- The amplitude is 3, so the maximum and minimum points will be at y = 3 and y = -3, respectively.
- The period is 4π, so one complete cycle will be from x = 0 to x = 4π.
- The turning points will occur at x = 2sin^(-1)(5/3) and x = 2sin^(-1)(-1/3).
- The x-intercepts will occur at x = 2sin^(-1)(2/3).
- The y-intercept is y = -2.
Once you have marked these points on the graph, you can connect them smoothly with a sine curve to complete the sketch of the full cycle.