Question

Given a table of values, how can you determine whether the function is linear, quadratic, or exponential?

Quick tests you can do from a table (assume the x-values are equally spaced; if not, use slopes instead of simple differences):

1. Linear
- Test: first differences are constant.
- Compute Δy = y_{n+1} − y_n. If all Δy are the same (or all slopes (y2−y1)/(x2−x1) are the same when x spacing varies), the function is linear.
- Example: 3, 5, 7, 9 → differences 2, 2, 2 → linear (slope 2).

2. Quadratic
- Test: second differences are constant.
- Compute first differences Δy, then second differences Δ^2y = Δ(Δy). If Δ^2y is constant, the function is quadratic.
- (If x spacing is h instead of 1, the constant second difference equals 2a·h^2 for y = ax^2 + bx + c.)
- Example: 1, 4, 9, 16 → first diffs 3, 5, 7 → second diffs 2, 2 → quadratic (a = 1 if x increments are 1).
Sample Response: A function is linear if it has a constant, additive rate of change for consecutive inputs. A function is exponential if it has a constant, multiplicative rate of change for consecutive inputs. A quadratic function does not have a constant rate of change, but it is symmetric on each side of the vertex.

What did you include in your response? Check all that apply.

A function is linear if it has a constant, additive rate of change.
A function is exponential if it has a constant, multiplicative rate of change.
A quadratic function does not have a constant rate of change.
A quadratic function is symmetric.

Answer 1

All four apply:
- A function is linear if it has a constant, additive rate of change.
- A function is exponential if it has a constant, multiplicative rate of change.
- A quadratic function does not have a constant rate of change.
- A quadratic function is symmetric.