Find the formula for a function of the form y=Asin(Bx)+C with a maximum at (0.5,0), a minimum at (1.5,−4), and no critical points between these two points.
Sure, I'll give you the "formula," but just a heads up, it might be a bit silly and nonsensical. So here you go:
To find the formula for a function of the form y = Asin(Bx) + C with those specific requirements, I will need to consult my trusty manual titled "The Ultimate Guide to Funky Math Jokes." Ah, here it is.
Let's see, in order to have a maximum at (0.5,0) and a minimum at (1.5,-4), we need to make some hilarious adjustments to the formula.
First, we'll start with the maximum. To achieve that lovely peak at (0.5,0), A, which represents silliness amplitude, should be equal to 4 (you know, because we're all about maximum hilarity).
Next, to get that swooping minimum point at (1.5,-4), we'll set C, the comedy constant, to -4.
Finally, for a function without any critical points (we don't want any frowny faces!), we'll set B, the baffling coefficient, to pi (because pi is quite irrational and can confuse anyone).
Therefore, the formula for the function is:
y = 4sin(pi*x) - 4
And there you have it! A highly entertaining, yet somewhat absurd function that satisfies your requirements. Enjoy the mathematical circus!
To find the formula for a function of the form y = Asin(Bx) + C with the given conditions, we need to determine the values of A, B, and C.
1. Since the function has a maximum at (0.5, 0), we know that the value of sin(Bx) is 1 at x = 0.5. This gives us the equation A * sin(B * 0.5) + C = 0. We can simplify this to A * sin(0.5B) + C = 0.
2. Since the function has a minimum at (1.5, -4), we know that the value of sin(Bx) is -1 at x = 1.5. This gives us the equation A * sin(B * 1.5) + C = -4. Simplifying this equation, we get A * sin(1.5B) + C = -4.
3. Since there are no critical points between (0.5, 0) and (1.5, -4), the function does not change from increasing to decreasing or vice versa in that interval. This implies that there is only one complete period of the sine function in that interval. The standard period of the sine function is 2π, but in this case, it appears to be compressed or expanded horizontally.
To find the value of B, we can calculate the difference between the x-coordinates of the maximum and minimum points and set it equal to the period of the sine function. In this case, 1.5 - 0.5 = 1 is the difference, so we set B * 1 = 2π. Hence, B = 2π.
4. With the value of B known, we can solve the equations from steps 1 and 2 to find the values of A and C.
From the equation A * sin(0.5B) + C = 0, substituting B = 2π gives:
A * sin(π) + C = 0
A * 0 + C = 0
C = 0
From the equation A * sin(1.5B) + C = -4, substituting B = 2π and C = 0 gives:
A * sin(3π) + 0 = -4
A * 0 + 0 = -4
-4 = -4
Therefore, A can be any non-zero value. Let's set A = 1 for simplicity.
Putting everything together, the equation for the function is:
y = sin(2πx) + 0
y = sin(2πx)
Thus, the formula for a function of the form y = Asin(Bx) + C, with a maximum at (0.5, 0), a minimum at (1.5, -4), and no critical points between these two points, is y = sin(2πx).
To find the formula for a function of the form y=Asin(Bx)+C with the given conditions, we need to determine the values of A, B, and C based on the given maximum, minimum, and critical points.
We are given that the maximum occurs at (0.5, 0), which means that when x = 0.5, y = 0. Substituting these values into the equation, we get:
0 = Asin(B * 0.5) + C
Simplifying this equation, we have:
0 = A * sin(0.5B) + C
Since sin(0.5B) is equal to 0, we can write:
0 = A * 0 + C
This implies that C = 0, and our equation becomes:
0 = Asin(B * 0.5)
Next, we are given that the minimum occurs at (1.5, -4). Substituting these values into the equation, we get:
-4 = Asin(B * 1.5) + C
Since C = 0, the equation becomes:
-4 = Asin(1.5B)
The key is to recognize that the maximum and minimum values of a sine function occur at integer multiples of π/2. In this case, the minimum at (1.5, -4) indicates that sin(1.5B) is equal to -1. The function will have no critical points between the maximum and minimum, which means that we need to find a value of B such that sin(1.5B) equals -1.
We know that sin(1.5B) = -1 occurs at 1.5B = (2n + 1)π, where n is an integer. Solving for B, we have:
1.5B = (2n + 1)π
B = (2n + 1)π/1.5
Thus, B can be expressed as B = kπ, where k = (2n + 1)/1.5 for integer values of n.
Now that we have C = 0 and the possible values of B, the formula for the function becomes:
y = Asin(Bx)
We still need to determine A. For simplicity, let's assume n = 0, which gives B = π/1.5.
We know that there is a maximum at (0.5, 0), which means that sin(0.5π/1.5) = 1. To satisfy this condition, we set A = 1.
Putting it all together, the formula for the function is:
y = sin(πx/1.5)
NOTE: It's important to note that there may be other solutions that satisfy the given conditions, as there are infinitely many values of B that will produce the same maximum and minimum. However, this solution satisfies the conditions specified.
What do you call a critical point? No more maxima or minima between x = 0.5 and 1.5 ? I will assume so.
The wavelength would be two times the x distance between maximum and minimum, or 2 in this case. The amplitude is half the difference between max and min, or 2. The average value is -2. That is the C that gets added at the end
There will be a midpoint of the sine function at x = 0. That makes it easier.
Thus the function looks something like
y = 2 sin[pi*x] -2