Algrebra 2 polynomial function graph

Question

Find a polynomial function whose graph passes through (-1,-1),(0,-7),(1,-7),(2,11)

I get lost after trying to combine the equations, I’m stressing out and won’t be able to see my teacher in time to ask questions could someone walk me through the whole thing please?

Answer 1

Let's try to find a polynomial function of the form f(x) = ax³ + bx² + cx + d that passes through the given points.

Since (0, -7) is on the graph, d = -7.

So, f(x)= ax³ + bx² + cx - 7

Since (1, -7) is on the graph, a + b + c - 7 = -7

Since (-1, -1) is on the graph, -a + b - c - 7 = -1

Since (2, 11) is on the graph, 8a + 4b + 2c - 7 = 11

So, we have the system of equations:

a + b + c = 0

-a + b - c = 6

8a + 4b + 2c = 18

Add the first two equations to get 2b = 6. So, b = 3

Using the first and third equations, a + c = -3 and 8a + 2c = 6

So c = -a - 3 and 8a + 2(-a - 3) = 6. Therefore, 6a = 12. So, a = 2 and c = -5.

The values a = 2, b = 3, and c = -5 satisfy all three equations in the original system of equations.

So, f(x) = 2x³ + 3x² - 5x - 7 is a polynomial function that passes through all four of the given points.

Answer 2

f(x) = ax⁴ + bx³ + cx² + dx + e;

f(-1) = a - b + c - d + e = -1

f(0) = e = -7; f(1) = a + b + c + d - 7 = - 7

f(2) = 16a + 8b + 4c + 2d - 7 = 11

a - b + c - d = 6

a + b + c + d = 0

2a + 2c = 6; a + c = 3; c = 3 - a; - 2b - 2d = 6; b + d = - 3; d = - 3 - b

16a + 8b + 4(3 - a) + 2(-3 - b) = 18; 16a + 8b + 12 - 4a - 6 - 2b = 18; 12a + 6b = 12; 2a + b = 2; b = 2 - 2a

c = 3 - a; d = -3 - 2 + 2a = 2a - 5. We have infinitly many polinomials: f(x) = ax⁴ + (2 - 2a)x³ + (3 - a)x² +

(2a - 5)x - 7. If we take a = 1 we get: f(x) = x⁴ + 2x² - 3x - 7