What polynomial has a graph that passes through the given points? (-2,2)(-1,-1)(1,5)(3,67)
looks like a second degree, so we can test that.
y=ax^2+bx + c
now do the data points...
2=4a-2b + c (-2,2)
-2=a-b+c (-1,-2)
5=a+b+c (1,5)
now you have three equations, three unknwns
solve by any method,then check the last data pont.
To find the polynomial that passes through the given points, we can use the concept of interpolation. In this case, since we have four points, we can find a polynomial of degree three (a cubic polynomial) that satisfies these points.
To simplify the process, let's represent the coordinates of the given points as (x₁, y₁), (x₂, y₂), (x₃, y₃), and (x₄, y₄).
For the given points:
Point 1: (-2,2) => (x₁, y₁) = (-2, 2)
Point 2: (-1,-1) => (x₂, y₂) = (-1, -1)
Point 3: (1,5) => (x₃, y₃) = (1, 5)
Point 4: (3,67) => (x₄, y₄) = (3, 67)
Now, let's proceed to find the polynomial.
Step 1: Set up the general form of a cubic polynomial: f(x) = ax³ + bx² + cx + d.
Step 2: Plug in the given points into the polynomial expression:
Using the point (-2, 2),
y₁ = a(-2)³ + b(-2)² + c(-2) + d
2 = -8a + 4b - 2c + d
Using the point (-1, -1),
y₂ = a(-1)³ + b(-1)² + c(-1) + d
-1 = -a + b - c + d
Using the point (1, 5),
y₃ = a(1)³ + b(1)² + c(1) + d
5 = a + b + c + d
Using the point (3, 67),
y₄ = a(3)³ + b(3)² + c(3) + d
67 = 27a + 9b + 3c + d
Step 3: Set up a system of equations using the equations from step 2:
-8a + 4b - 2c + d = 2
-a + b - c + d = -1
a + b + c + d = 5
27a + 9b + 3c + d = 67
Step 4: Solve the system of equations to find the values of a, b, c, and d.
Solving these equations can be done manually or by using various methods like substitution or elimination. Alternatively, you can use numerical solver tools or software to quickly find the solution.
After obtaining the values for a, b, c, and d, substitute them back into the general form of the polynomial function f(x) = ax³ + bx² + cx + d.
Thus, you can find the polynomial function that passes through the given points (-2,2), (-1,-1), (1,5), and (3,67).