Find a polynomial function whose graph passes through (5,13) (10,-10) and (0,4)
To find a polynomial function that passes through these three points, we can use the fact that a polynomial function of degree n has at most n+1 roots.
Let's assume the polynomial function has degree 2, so the function is in the form f(x) = ax^2 + bx + c.
Using the given points, we can form three equations to solve for the coefficients a, b, and c.
1) When x = 5, f(x) = 13:
13 = a(5^2) + b(5) + c
25a + 5b + c = 13 -> Equation (1)
2) When x = 10, f(x) = -10:
-10 = a(10^2) + b(10) + c
100a + 10b + c = -10 -> Equation (2)
3) When x = 0, f(x) = 4:
4 = a(0^2) + b(0) + c
c = 4 -> Equation (3)
Substituting equation (3) into equation (1):
25a + 5b + 4 = 13
25a + 5b = 9 -> Equation (4)
Substituting equation (3) into equation (2):
100a + 10b + 4 = -10
100a + 10b = -14 -> Equation (5)
Now we can solve equations (4) and (5) simultaneously to find the values of a and b.
Multiply equation (4) by 2 to eliminate the variable b:
50a + 10b = 18
Subtract equation (5) from this modified equation:
(50a + 10b) - (100a + 10b) = 18 - (-14)
-50a = 32
a = -32/50 = -16/25
Substitute the value of a into equation (4):
25*(-16/25) + 5b = 9
-16 + 5b = 9
5b = 9 + 16
5b = 25
b = 25/5 = 5
Now we have determined the values of a and b in the polynomial function f(x) = ax^2 + bx + c.
Substituting the values of a, b, and c = 4 into the equation:
f(x) = (-16/25)x^2 + 5x + 4
Therefore, the polynomial function that passes through the given points is f(x) = (-16/25)x^2 + 5x + 4.