State an equation of a sinusoidal graph with the following properties.
The central axis is at y = 5
Two consecutive minimums are found at (3, −1) and (13, −1
find values for a,b,c and d
State an equation of a sinusoidal graph with the following properties.
The central axis is at y = 5
Two consecutive minimums are found at (3, −1) and (13, −1)
find values for a,b,c and d
To find the values for a, b, c, and d in the equation of the sinusoidal graph, we need to use the general form of a sinusoidal equation:
y = a*sin(b(x - c)) + d
Given that the central axis is at y = 5, we have d = 5.
Now let's find the values for a, b, and c using the consecutive minimums as reference points.
1. Find the amplitude (a): The amplitude is the distance from the central axis to the maximum or minimum point. In this case, the distance from the central axis (y = 5) to the minimum point (y = -1) is 6 units. So, a = 6.
2. Find the period (P) or the distance between two consecutive minimums: In this case, the two consecutive minimums are at (3, -1) and (13, -1). The x-coordinates of these points correspond to one complete cycle of the sinusoidal graph. So, P = 13 - 3 = 10.
3. Calculate the angular frequency (b): The angular frequency is calculated using the formula b = 2π/P. In this case, b = 2π/10 = π/5.
4. Find the horizontal shift (c): The horizontal shift represents how many units the graph is shifted to the right or left. Since the first minimum occurs at (3, -1), we can use this point to find c. The equation x - c = 0 should hold true at this point. Therefore, c = 3.
Now we can write the equation of the sinusoidal graph:
y = 6*sin((π/5)(x - 3)) + 5
So, the values for a, b, c, and d are as follows:
a = 6
b = π/5 (approximately 0.628)
c = 3
d = 5
To find the equation of a sinusoidal graph with the given properties, we can start by noting that the central axis corresponds to the midline of the sinusoidal graph. In this case, the central axis is located at y = 5.
The equation of a sinusoidal graph can be written in the form:
f(x) = a sin(bx + c) + d
Here's how we can determine the values of a, b, c, and d:
1. Amplitude (a):
The amplitude of a sinusoidal graph represents the vertical distance between the maximum or minimum value and the midline (central axis). In this case, since the midline is at y = 5 and the minimum value is -1, the amplitude can be calculated as:
Amplitude = |minimum value - midline| = |-1 - 5| = 6
Thus, the value of a is 6.
2. Period (b):
The period of a sinusoidal graph refers to the length of one complete cycle. Since we are given two consecutive minimums at (3, -1) and (13, -1), the distance between these points represents one complete cycle.
Distance between two points = 13 - 3 = 10
Therefore, the period can be calculated as the distance between the points, which means b is equal to:
b = 2π / period = 2π / 10
3. Phase Shift (c):
The phase shift of a sinusoidal graph represents the horizontal shift of the graph. In this case, the graph is not shifted horizontally, as the minimum point at (3, -1) is the starting point of the graph. That means there is no phase shift, so c is equal to zero.
4. Vertical Shift (d):
The vertical shift refers to the vertical displacement of the entire graph. In this case, the graph is shifted upwards by 5 units (since the central axis is at y = 5), so the value of d is 5.
Therefore, the equation of the sinusoidal graph with the given properties is:
f(x) = 6 sin(2πx / 10) + 5
The central axis is at y = 5
y = sin(x)+5
Two consecutive minima are found at (3, −1) and (13, −1)
So, the period is 10, the amplitude is 5+1=6, and the shift is 3. cos has max at x=0, but we have a minimum, so
y = -6cos(π/5 (x-3))+5
see the graph at
http://www.wolframalpha.com/input/?i=plot+-6cos%28%CF%80%2F5+%28x-3%29%29%2B5,+x%3D0..15