A sine function has the following key features:
Period = 4
Amplitude = 3
Midline: y=−1
y-intercept: (0, -1)
The function is not a reflection of its parent function over the x-axis.
Use the sine tool to graph the function. The first point must be on the mid-line and the second point must be a maximum or minimum value on the graph closest to the first point.
All of your 5 posts dealing basically with the same type of problem contain the
instruction "Use the sine tool to graph the function". We would have no idea what this.
I can find the equation and you can go from there
basic sine equation:
y = a sin k(x + d) + c
Period = 4 ---> 2π/k = 4 ----> k = π/2
y = a sin π/2k(x + d) + c
Amplitude = 3
y = 3 sin π/2(x + d) + c
Midline: y=−1 ----> c = -1
y = 3 sin π/2(x + d) - 1
y-intercept: (0, -1) ---> when x = 0, y = -1
-1 = 3sin π/2(0+d) - 1
sin π/2 d = 0
π/2d = 0
d = 0
y = 3 sin ((π/2)(x) - 1
verification:
https://www.wolframalpha.com/input/?i=y+%3D+3+sin+%28%28%CF%80%2F2%29%28x%29%29+-+1
To graph the sine function with the given key features, follow these steps:
Step 1: Determine the equation of the sine function
The general equation for a sine function is:
y = A * sin(B * (x - C)) + D
Given key features:
Period = 4
Amplitude = 3
Midline: y = -1
y-intercept: (0, -1)
From the given information, we can determine the values of A, B, C, and D:
Amplitude (A) = 3
Midline (D) = -1
Phase shift (C) = 0 (since there is no horizontal shift)
Period (P) = 4
Frequency (B) = 2π / P = 2π / 4 = π/2
So the equation of the sine function is:
y = 3 * sin((π/2) * (x - 0)) - 1
which simplifies to: y = 3 * sin((π/2)x) - 1
Step 2: Identify the first and second points
From the given information, the first point should be on the midline, which is y = -1. So, we have the point (0, -1) as the first point.
The second point should be a maximum or minimum value on the graph closest to the first point. Since the amplitude is 3, the maximum value is 3 units above the midline, and the minimum value is 3 units below the midline. Since the midline is at y = -1, the maximum value is 2 units above the midline, resulting in a y-value of 1. The second point is then (2, 1).
Step 3: Graph the function
Using the equation y = 3 * sin((π/2)x) - 1, and the points (0, -1) and (2, 1), you can now graph the function.
To graph the given sine function, we'll start by plotting the first two points that satisfy the given conditions.
1. The midline is y = -1, which means that the first point must lie on this line. So we have (0, -1).
2. The amplitude is 3, which indicates that the maximum and minimum values of the function will be 3 units above and below the midline, respectively. Since the function is not a reflection over the x-axis, the maximum value will be above the midline.
To find the second point, we need to determine whether the maximum or minimum value occurs first. We know that the period of the sine function is 4, which means it completes one full oscillation (cycle) every 4 units along the x-axis.
The maximum value of the sine function is achieved at the peak of the oscillation, which occurs halfway through the period. So, the maximum value will be at x = 2.
To find the y-value of the maximum point, we use the formula: y = amplitude * sin((2π/period)x) + midline
Substituting the given values, we get:
y = 3 * sin((2π/4)(2)) - 1 = 3 * sin(π) - 1 = 3 * 0 - 1 = -1
So the second point is (2, -1).
Now that we have the first two points, we can sketch the graph of the function. The points are (0, -1) and (2, -1). The graph will start at (0, -1), rise to a maximum point at (2, -1), and then continue to complete one full period (4 units) back to the x-intercept. Since the function is not reflected over the x-axis, the graph will mirror this pattern for subsequent periods.
Keep in mind that this graph is a rough sketch based on the given key features. For a more accurate plot, you may need additional points or a graphing tool.