what is the equation for the graphs given the following information:
2. the graph of: y = asinbx has amplitude of 4 and a period of pi/6
3. a sinusoid graph with a maximum at A(0,3) and a minimum at (2pi, -3)
there are lots of online graphing sites: wolframalpha and desmos are two
#2. sin(kx) has a period of 2pi/k
so, a period of pi/6 means 2pi/k = pi/6 ==> k=12
y = 4sin(12x)
#3.
y varies from -3 to 3, so a = 3
max to min is 1/2 period, so you have a period of 4pi
cosine starts at a maximum, so
y = 3 cos(x/2)
To find the equation for the given graphs, we need to use the general form of a sinusoid equation: y = a*sin(bx + c) + d, where:
- a represents the amplitude of the graph.
- b determines the period of the graph.
- c is the phase shift of the graph.
- d is the vertical shift of the graph.
Let's solve each problem step by step:
Problem 2: The graph of y = asinbx has an amplitude of 4 and a period of π/6.
The amplitude (a) is equal to 4, and the period (T) is equal to π/6. The period is related to the value of "b" in the equation by T = 2π/|b|.
Since we know T = π/6, we can substitute it into the equation and solve for "b":
π/6 = 2π/|b|
Dividing both sides by 2π, we get:
1/6 = 1/|b|
Cross-multiplying and simplifying:
|b| = 6
To determine "b," we need to consider both positive and negative values since the absolute value was used. Therefore, b can be either 6 or -6.
So, the equation for the given graph can be written as:
y = 4*sin(6x) or y = 4*sin(-6x)
Problem 3: A sinusoid with a maximum at A(0,3) and a minimum at (2π, -3).
First, we note that the maximum value represents the amplitude (a), and the midpoint between the maximum and minimum values represents the vertical shift (d).
From the given information, we find that the amplitude (a) is 3 and the midpoint (d) is (3 + (-3)) / 2 = 0.
The maximum/minimum points help us determine the period (T) of the graph. The distance between two maximum or minimum points is equal to the period. In this case, the distance between A(0,3) and (2π, -3) is 2π.
Thus, the period (T) is equal to 2π.
From the equation T = 2π/|b|, we can solve for "b":
2π = 2π/|b|
Dividing both sides by 2π, we get:
1 = 1/|b|
Cross-multiplying and simplifying:
|b| = 1
Again, we consider both positive and negative values, so b can be either 1 or -1.
Now, since the graph has a maximum at A(0,3), there is no phase shift (c = 0).
Substituting all the values into the equation, we have:
y = 3*sin(x) + 0, or simply y = 3*sin(x)
Thus, the equation for the given sinusoid graph is y = 3*sin(x).