Write a function to represent the set of ordered pairs. {(2, -8), (3, -18), (4, -32), (5, -50)}

y = -2x²

Step-by-step explanation:

The set of ordered pairs is:

(x₁, y₁) = (2, -8)

(x₂, y₂) = (3, -18)

(x₃, y₃) = (4, -32)

(x₄, y₄) = (5, -50)

First let's check if this is linear.  For even increments of x, Δy is:

Δy₂₁ = y₂ − y₁ = -18 − -8 = -10

Δy₃₂ = y₃ − y₂ = -32 − -18 = -14

Δy₄₃ = y₄ − y₃ = -50 − -32 = -18

Δy isn't constant, so this isn't linear.  However, the difference of the differences is constant:

Δy₃₂ − Δy₂₁ = -14 − -10 = -4

Δy₄₃ − Δy₃₂ = -18 − -14 = -4

So this is a quadratic.

y = ax² + bx + c

To find the coefficients of a, b, and c, we can either plug in three points from the set and solve the system of equations:

-8 = a(2)² + b(2) + c

-18 = a(3)² + b(3) + c

-32 = a(4)² + b(4) + c

Or, if it's simple, we can use a little trial and error.

-8 = -2 (2)²

-18 = -2 (3)²

-32 = -2 (4)²

So the function is:

y = -2x²