Find a trigonometric equation in the form y = a sin (b (x-c)) which has a minimum of (3, -5) and a zero value of (6, -1).
There is a graph with the minimum and the zero value labeled. It is wavy with suggests that it is a sin curve.
What do you mean by a "zero" value? You do not include a vertical shift term. The minimum of -5 indicates to me that
y = 5sin(b(x-c))
I assume your zero value is supposed to help define the period...
What I don't understand about this problem is that what the letters mean?
I am sorry I forgot the + d at the end.
ah! that makes all the difference!
Since the center line is at y = -1, the distance to the extremes is the amplitude. So, the amplitude is 4 and the vertical shift is -1. That means
y = 4sin(b(x-c))-1
since sin(0) = 0, that means y+1=0 at x=6, so the horizontal shift is 6
the distance from the zero value to the minimum is 3, so the period is 12.
Now the period of sin(kx) is 2π/k. That means k=π/6
y = 4sin(π/6 (x-6)) - 1
see the graph at
http://www.wolframalpha.com/input/?i=plot+y%3D4sin(%CF%80%2F6+(x-6))+-+1,+y%3D-1+for+0%3C%3Dx%3C%3D12
Thank you so much Steve!
To find the equation of a trigonometric function in the form y = a sin(b(x - c)), we need to determine the values of a, b, and c based on the given information.
First, let's analyze the information provided. We are given a minimum point at (3, -5) and a zero value at (6, -1).
The minimum point (3, -5) is the lowest point on the graph, which means that the amplitude of the sine function is the absolute value of the y-coordinate of the minimum point. In this case, the amplitude is 5, so a = 5.
The zero value (6, -1) means that the sine function crosses the x-axis at x = 6. Since the general equation for a sine function crossing the x-axis is x = c + nπ (where n is an integer), we can determine c. In this case, x = 6 when c + nπ = 6. By solving for c, we find c = 6 - nπ.
Now, we need to determine the period of the sine function, which is the distance between two consecutive zero values. In this case, we have one zero value at (6, -1), so we need to find the next zero value. Since the sine function has a periodicity of 2π, the period is 2π/b. Therefore, we can find b using the formula b = 2π/(next zero value - current zero value). The next zero value can be obtained by adding the period to the x-coordinate of the zero value we have. In this case, the next zero value is at x = 6 + 2π/b.
Finally, we can substitute the values of a, b, and c into the general form equation y = a sin(b(x - c)) to obtain the specific equation. In this case, the equation would be y = 5 sin(b(x - (6 - nπ))) or y = 5 sin(b(x - 6 + nπ)).
Please note that there could be multiple solutions for b and n that satisfy the given conditions.