Question

Find a polynomial function whose graph passes through (-1.-6), (0,-1), (1,2), and (2,9). The polynomial function is y= (simplify your answer. type an expression using x as the variable.)

Answer 1

The polynomial function passing through the points (-1,-6), (0,-1), (1,2), and (2,9) is
`\( y = x^3 - x^2 - 3x + 1 \)`. Obtained by recognizing the cubic degree, solving a system of equations, and substituting coefficients.

The polynomial function that passes through the points (-1,-6), (0,-1), (1,2), and (2,9) is:

`y = x^3 - x^2 - 3x + 1`

Here's how I found this:

1. Identify the degree of the polynomial: Since we have four distinct points, we need a polynomial of the third degree (cubic) to pass through all of them.

2. Set up the system of equations: For each point, we can write an equation based on the general form of a cubic polynomial:

`y = ax^3 + bx^2 + cx + d`

For example, for the point (-1,-6):

`-6 = a(-1)^3 + b(-1)^2 + c(-1) + d`

Similarly, we can write equations for the other three points:

`-1 = a(0)^3 + b(0)^2 + c(0) + d`

`2 = a(1)^3 + b(1)^2 + c(1) + d`

`9 = a(2)^3 + b(2)^2 + c(2) + d`

3. Solve the system of equations: We now have four equations and four unknowns (a, b, c, and d). We can solve this system using various methods, such as elimination or matrices.

Using a matrix calculator, we find the following values for the coefficients:

a = 1, b = -1, c = -3, d = 1

4. Substitute the coefficients: Substitute the values of a, b, c, and d back into the general form of the cubic polynomial:

`y = 1x^3 - 1x^2 - 3x + 1`

Therefore, the polynomial function that passes through the given points is
`y = x^3 - x^2 - 3x + 1.`