The zeros of a quartic function are -3, -1, and 2 (order 2). Determine
a). an equation for the function that satisfies this condition.
b). an equation for a function that satisfies this condition and passes through the point (1, 4)
a) Since the zeros are -3, -1, and 2 (order 2), the factors of the quartic function are (x + 3), (x + 1), and (x - 2)^2. Therefore, the equation for the function is:
f(x) = k(x + 3)(x + 1)(x - 2)^2
where k is a constant. To find the value of k, we can use the fact that the function passes through the point (1, 4). Substituting x = 1 and y = 4 into the equation, we get:
4 = k(1 + 3)(1 + 1)(1 - 2)^2
4 = k(4)(2)(1)
4 = 8k
k = 0.5
Therefore, the equation for the quartic function is:
f(x) = 0.5(x + 3)(x + 1)(x - 2)^2
b) To find an equation for a function that passes through the point (1, 4) and satisfies the given zeros, we need to determine the value of the constant k in the general equation for the quartic function.
Using the coordinates of the point (1, 4), we have:
4 = 0.5(1 + 3)(1 + 1)(1 - 2)^2
4 = 0.5(4)(2)(1)
4 = 4
Therefore, the constant k does not change from the previous part and the equation for the function that passes through the point (1, 4) is:
f(x) = 0.5(x + 3)(x + 1)(x - 2)^2