Math algebra 2 polynomial

Question

Find a polynomial function whose graph passes through (-1,1), (0,-4), (1,-5), (2,4)

please help, I’ve been struggling for the past hour

Answer 1

1 = -a + b - c + d

4 = 0a + 0b + 0c + d, good starting point

5 = a + b + c + d

4 = 8a + 4b + 2c + d

Answer 2

Points in function are: (-1, 1), (0, -4), (1, -5) and (2, 4). Four points can be used to determine a cubic polynomial because the function can be expressed in the general form with four constants, namely:

y = ax³ + bx² + cx + d

Substituting the point values:

Point 2 (0, -4):

-4 = a(0)³ + b(0)² + c(0) + d

-4 = d

Point 3 (1, -5):

-5 = a(1)³ + b(1)² + c(1) - 4

-5 = a + b + c - 4

-1 = a + b + c (Eqn. 1)

Point 1 (-1, 1):

1 = a(-1)³ + b(-1)² + c(-1) - 4

5 = -a + b – c (Eqn. 2)

Point 4 (2, 4):

4 = a(2)³ + b(2)² + c(2) - 4

8 = 8a + 4b + 2c

4 = 4a + 2b + c (Eqn. 3)

Adding Eqn. 1 & 2 yields:

4 = 2b

2 = b

Subtracting Eqn. 1 from Eqn. 3 yields:

5 = 3a + b

5 = 3a + 2

3 = 3a

1 = a

Using Eqn. 1 we get:

-1 = a + b+ c

-1 = 1 + 2 + c

-4 = c

Therefore the polynomial equation is:

y = x³ + 2x² - 4x -4