Find a polynomial function whose graph passes through (-1,-4) (0,3) (1,4) (2,11)
The polynomial function is y=
PLEASE DO THE RIGHT ANSWER
To find the polynomial function that passes through the given points, we can use the concept of interpolation. Since we have four points, we can use a degree 3 polynomial function (a cubic) to obtain a unique solution.
Step 1: Set up the general form of a cubic polynomial function:
y = ax^3 + bx^2 + cx + d
Step 2: Substitute the x and y values of each given point into the equation:
For the point (-1, -4):
-4 = a(-1)^3 + b(-1)^2 + c(-1) + d
For the point (0, 3):
3 = a(0)^3 + b(0)^2 + c(0) + d
For the point (1, 4):
4 = a(1)^3 + b(1)^2 + c(1) + d
For the point (2, 11):
11 = a(2)^3 + b(2)^2 + c(2) + d
Step 3: Solve the system of equations to find the coefficients a, b, c, and d.
-4 = -a + b - c + d (Equation 1)
3 = d (Equation 2)
4 = a + b + c + d (Equation 3)
11 = 8a + 4b + 2c + d (Equation 4)
Using Equation 2, we can determine that d = 3.
Substitute d = 3 in Equations 1, 3, and 4:
-4 = -a + b - c + 3
4 = a + b + c + 3
11 = 8a + 4b + 2c + 3
Simplifying the equations further:
-a + b - c = -7 (Equation 5)
a + b + c = 1 (Equation 6)
8a + 4b + 2c = 8 (Equation 7)
Multiplying Equation 5 by 4 and adding it to Equation 6 eliminates b:
-4a + 4b - 4c = -28
a - 3c = -27 (Equation 8)
Multiplying Equation 5 by 2 and subtracting it from Equation 7 eliminates c:
8a + 4b + 2c - 2a + 2b - 2c = 16 - 14
6a + 6b = 2
a + b = 1/3 (Equation 9)
Using Equation 9, we can solve for a:
a = 1/3 - b
Plugging the value of a in Equation 8:
(1/3 - b) - 3c = -27
1/3 - b - 3c = -27
1 - 3b - 9c = -81
-3b - 9c = -82 (Equation 10)
Using Equation 9 to solve for b:
b = 1/3 - a
Plugging the value of b in Equation 10:
-3(1/3 - a) - 9c = -82
-1 + 3a - 9c = -82
3a - 9c = -81
a - 3c = -27 (Equation 11)
Equation 11 is the same as Equation 8. Hence, the values of a and c remain the same, regardless of introducing b.
Therefore, a = 1/3 and c = 9.
Step 4: Return to the general form of the cubic polynomial function and substitute the values of a, b, c, and d:
y = ax^3 + bx^2 + cx + d
y = (1/3)(x^3) + b(x^2) + 9x + 3
Hence, the polynomial function that passes through the given points is:
y = (1/3)(x^3) + bx^2 + 9x + 3
Since the value of b cannot be determined with the given information, this is the most accurate polynomial function that can be determined using the given points.