Determine the equation of a cubic function that goes through the point (1,7) and has a zero at x=-1 and another zero at x=3 that is also a turning point
Done by Anonymous in the post below when you were David
To determine the equation of the cubic function, let's start by using the given zeros and the turning point.
1. Zero at x = -1: This means that the function will have a factor of (x + 1).
2. Zero at x = 3: This means that the function will have a factor of (x - 3).
3. Turning point: Since the turning point is not specified, we'll assume it is at (h, k). This means that the function will have the form of (x - h)^2 + k.
Now we can write the equation in factored form:
f(x) = A(x + 1)(x - 3)(x - h)^2 + k
To find A, we'll use the given point (1, 7):
7 = A(1 + 1)(1 - 3)(1 - h)^2 + k
Simplifying the equation:
7 = A(-2)(1 - h)^2 + k
Now let's substitute the x-coordinate of the turning point (-1) for h to eliminate h:
7 = A(-2)(1 - (-1))^2 + k
7 = A(-2)(2)^2 + k
7 = A(-2)(4) + k
7 = -8A + k
At this point, we have a system of two equations:
1. 7 = -8A + k
2. 7 = A(-2)(1 - h)^2 + k
To solve the system and find the values of A and k, we need more information about the turning point.
To determine the equation of a cubic function that satisfies the given conditions, we can start by considering the general form of a cubic function:
f(x) = ax^3 + bx^2 + cx + d
where a, b, c, and d are the coefficients we need to find.
Given that the cubic function has a zero at x = -1, we can deduce that (x + 1) is a factor of the function. Similarly, since the cubic function has a zero at x = 3, we can conclude that (x - 3) is also a factor of the function.
To find the equation, we can multiply these two factors together, which gives us:
(x + 1)(x - 3) = x^2 - 2x - 3
Now, to find the value of the coefficient a, we can use the fact that the cubic function has a turning point at some coordinate (h, k). The x-coordinate of the turning point, h, is the average of the zeros, which is given by:
h = (-1 + 3)/2 = 2/2 = 1
Since the turning point also lies on the graph of the cubic function, we can substitute the x-coordinate, h = 1, and the y-coordinate, k = 7, into the function:
f(1) = a(1)^3 + b(1)^2 + c(1) + d = 7
Simplifying the equation:
a + b + c + d = 7
Now, we have three equations with four unknowns (a, b, c, and d):
1) a(-1)^3 + b(-1)^2 + c(-1) + d = 0 (for x = -1)
2) a(3)^3 + b(3)^2 + c(3) + d = 0 (for x = 3)
3) a + b + c + d = 7 (for x = 1)
We can solve this system of equations to find the values of a, b, c, and d.
Substituting x = -1 into the first equation:
-a + b - c + d = 0
Substituting x = 3 into the second equation:
27a + 9b + 3c + d = 0
Now, we have a system of three linear equations:
-a + b - c + d = 0
a + b + c + d = 7
27a + 9b + 3c + d = 0
Solving this system will give us the values of a, b, c, and d, which we can then substitute into the general form of the cubic function to obtain the equation.