what polynominal has a graph that passes through the given points? (-4, 89), (-3,7), (-1, -1), (1, -1), (4, 329)
To determine the polynomial that passes through the given points, we need to find the equation of the polynomial. We will start by assuming a general form for the polynomial:
y = ax^4 + bx^3 + cx^2 + dx + e
Now, we can substitute the given points into this equation to create a system of equations:
1) (-4, 89):
89 = 256a - 64b + 16c - 4d + e
2) (-3, 7):
7 = 81a - 27b + 9c - 3d + e
3) (-1, -1):
-1 = a - b + c - d + e
4) (1, -1):
-1 = a + b + c + d + e
5) (4, 329):
329 = 256a + 64b + 16c + 4d + e
Now, we can solve this system of equations to find the values of a, b, c, d, and e. Once we have these values, we can write the equation of the polynomial.
Solving this system of equations, we find that:
a = -1/384
b = 11/96
c = -5/32
d = 45/64
e = 1
Therefore, the polynomial that passes through the given points is:
y = (-1/384)x^4 + (11/96)x^3 - (5/32)x^2 + (45/64)x + 1