Determine the equation of the member of the family of functions with zeros at 2 (order 2), 3 (order 2) and 4 (order 1) that passes through the point 1, 36. Show step-by-step calculations.
y = a(x-2)^2 (x-3)^2 (x-4)
plug in the point (1,36) to find a
36 = a(-1)^2 (-2)^2 (-3)
continue
To determine the equation of the member of the family of functions with the given zeros and passing through the point (1, 36), we can start by noting that if p is a zero of multiplicity n, then the function can be factored as (x - p)^n.
Given that the zeros are 2 (order 2), 3 (order 2), and 4 (order 1), we can write the factors of the function as follows:
(x - 2)^2 * (x - 3)^2 * (x - 4)
To find the equation of the specific function, we need to determine the constant coefficient. For this, we can substitute the given point (1, 36) into the equation:
(1 - 2)^2 * (1 - 3)^2 * (1 - 4) = k * (1 - 2)^2 * (1 - 3)^2 * (1 - 4)
Simplifying, we have:
(-1)^2 * (-2)^2 * (-3) = k * (-1)^2 * (-2)^2 * (-3)
36 = k * 36
Dividing both sides by 36, we find:
k = 1
Therefore, the equation of the member of the family of functions with the given zeros and passing through the point (1, 36) is:
f(x) = (x - 2)^2 * (x - 3)^2 * (x - 4)
To determine the equation of the member of the family of functions, we need to consider the given information: zeros at 2 (order 2), 3 (order 2), and 4 (order 1), and the point it passes through 1, 36.
Step 1: Start with the general form of a polynomial function:
f(x) = a(x - r₁)^m₁(x - r₂)^m₂...(x - rₙ)^mₙ
where 'a' is a constant, 'r' represents the zeros of the function, and 'm' represents their respective multiplicities.
Step 2: We are given the zeros and their multiplicities. We have (2, 2), (3, 2), and (4, 1).
So far, our equation looks like:
f(x) = a(x - 2)^2(x - 3)^2(x - 4)
Step 3: Next, we substitute the point (1, 36) into our equation to solve for 'a'. Since the point lies on the function, when we substitute 'x' with 1, it should give us 36.
So, plugging in the values for 'x' and 'f(x)', we have:
36 = a(1 - 2)^2(1 - 3)^2(1 - 4)
Step 4: Simplify the equation:
36 = a(-1)^2(-2)^2(-3)
36 = a(1)(4)(-3)
36 = -12a
Step 5: Solve for 'a':
36/(-12) = a
a = -3
Step 6: Now that we have the value of 'a', substitute it back into our function:
f(x) = -3(x - 2)^2(x - 3)^2(x - 4)
Therefore, the equation of the member of the family of functions with zeros at 2 (order 2), 3 (order 2), and 4 (order 1) that passes through the point 1, 36 is:
f(x) = -3(x - 2)^2(x - 3)^2(x - 4)
These step-by-step calculations helped determine the equation by considering the given zeros, their multiplicities, and the given point.