Graph a sine function whose amplitude is 5, period is 6π , midline is y=−2 , and y-intercept is (0, −2) . The graph is not a reflection of the parent function over the x-axis.
im not so sure how i would graph this :/
To graph a sine function with the given characteristics:
1. Start by plotting the y-intercept, (0, -2), on the coordinate plane.
2. Since the midline is y = -2, this means that the average value of the function is -2. The amplitude of the function is 5, so the maximum value is 5 units above the midline and the minimum value is 5 units below the midline.
3. Draw a dotted line to represent the midline y = -2.
4. From the y-intercept, go up 5 units and plot a point. This represents the maximum value.
5. From the maximum point, go down 5 units and plot another point. This represents the minimum value.
6. The period of the function is 6π, which means that it completes one full cycle in 6π units. Divide this into four equal sections, each measuring π/2.
7. Mark the points π/2, π, 3π/2, and 2π on the x-axis.
8. The function is not reflected over the x-axis, so the shape of the graph will start at the midline, increase to the maximum value, decrease to the minimum value, and then return to the midline in a smooth curve.
9. Use these points as a guide to sketch the curve of the sine function, starting from the midline, reaching the maximum value, then going down to the minimum value, and finally returning to the midline.
10. Repeat the pattern for each subsequent cycle until the desired portion of the graph is drawn.
Remember to label the axes and indicate the amplitude, period, and midline on the graph, if necessary.
To graph the sine function with the given conditions, follow these steps:
Step 1: Determine the equation of the sine function.
Since the amplitude is 5, the midline is y = -2, and the y-intercept is (0, -2), the equation of the function can be written as:
y = A*sin(Bx) + C
where A represents the amplitude, B represents the frequency or period, and C represents the midline.
In this case, A = 5, C = -2, and the period is 6π, which tells us that B = (2π/period) = (2π/(6π)) = 1/3.
So, the equation of the function becomes:
y = 5*sin((1/3)x) - 2
Step 2: Determine the key points.
Identify the key points that will help us plot the graph. For a sine function, these points are the maximum, minimum, and x-intercepts.
The maximum and minimum values are determined by the amplitude. In this case, the maximum value is A = 5, and the minimum value is -A = -5.
The x-intercepts can be found by setting the sine function equal to zero:
5*sin((1/3)x) - 2 = 0
sin((1/3)x) = 2/5
To find the x values, we can take the arcsin of 2/5:
(1/3)x = arcsin(2/5)
x = 3 * arcsin(2/5)
Step 3: Plot the graph.
Using the key points obtained in the previous step, plot the graph on a coordinate plane. Make sure to include enough x-values to cover at least one period (6π) and label the key points.
The graph will oscillate between the maximum value of 5 and the minimum value of -5 along the y-axis. The x-intercepts will occur where the function crosses the midline (y = -2).
Remember that the graph should not be a reflection of the parent function over the x-axis, so make sure the positive and negative regions of the graph maintain the same shape but with different y-values.
I hope this explanation helps you in graphing the sine function with the given conditions!
start with
y = 5sin (?x)
period = 2π/k
6π = 2π/k
6πk = 2π
k = 1/3
so far: y = 5 sin ( (1/3)x )
midline is y=−2 , so we have to drop our curve down by 2 units
so far: y = 5 sin ( (1/3)x ) - 2
Testing this for x = 0
y = 5(0) - 2 = -2
y = 5sin (x/3) - 2 will do it
check:
http://www.wolframalpha.com/input/?i=y+%3D+5sin(x%2F3)+-+2+,+y+%3D+-2,+for+0+%E2%89%A4+x+%E2%89%A4+26