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.
Find a polynomial function whose graph passes through (-1,-4) (0,3) (1,4) (2,11)
The polynomial function is y=
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