Answer: A cubic polynomial function has the form f(x) = ax^3 + bx^2 + cx + d.
A horizontal tangent line is a line that is parallel to the x-axis and it occurs at a critical point of the function, where the derivative of the function is 0.
We know that a cubic function has 3 critical points and thus 3 derivative that could be 0, so to find the equation of the cubic function we will use these two points to find the value of a, b, c, and d.
We know that the critical points of a cubic function are found by solving f'(x) = 0, where f'(x) is the derivative of the function, so we have:
f'(x) = 3ax^2 + 2bx + c = 0
Now we can use the two points to find the value of a, b, c, and d.
We know that the point (1,-2) is on the graph of the function, so we can substitute it into the equation to find d:
f(1) = a + b + c + d = -2
And we know that the point (-1,2) is on the graph of the function, so we can substitute it into the equation to find d:
f(-1) = -a + b - c + d = 2
Now we can use these two equations to find the value of a, b, and c:
a + b + c + d = -2
-a + b - c + d = 2
2a + 2d = 0
a = -d
-a + b - c + d = 2
b - c = 4
3ax^2 + 2bx + c = 0
3(-d)x^2 + 2bx - c = 0
Now we can use these equations and the critical points to find the value of a, b, c, and d
3(-d)x^2 + 2bx - c = 0, (1,-2) => 3d+2b-c = 0
3(-d)x^2 + 2bx - c = 0, (-1,2) => 3d-2b+c = 0
b-c = 4
Solving these equations, we have:
a = -1/3
b = 1
c = -5/3
d = -2
And the equation for the function is:
f(x) = -1/3x^3 + x^2 - 5/3x - 2
We can now sketch the graph of the function by using the information that we have. The cubic function will have an vertex of symmetry at (0,-2) and two horizontal tangent lines at (1,-2) and (-1,2) .
Since a < 0, the function is concave down, so the graph will be opened downwards and will have a local minimum at (0,-2) .
It is important to note that this is one possible equation for a cubic function that has horizontal tangent lines at (1, −2) and (−1, 2), but there may be other possibilities.
Explanation: