Lesson 9: Modeling with Polynomial Functions

Property

A cubic function with three x-intercepts $r_1$, $r_2$, and $r_3$ can be written in factored form as:

where $a$ is a nonzero constant that determines the vertical stretch and orientation.

Examples

Property

For equally spaced $x$-values, if the $n$th finite differences are constant and nonzero, then the data can be modeled by a polynomial of degree $n$. The finite differences are calculated by repeatedly finding differences of consecutive $y$-values: $\Delta y = y_{i+1} - y_i$.

Examples

Property

The simplest way to fit a parabola to a set of data points is to pick three of the points and find the equation of the parabola that passes through those three points. This creates a quadratic model, $y = ax^2 + bx + c$, for the relationship shown in the data.

Examples

• A ball's height is measured at three times: $(1, 21)$, $(2, 24)$, and $(3, 21)$. Fitting a parabola gives the model $h = -3t^2 + 12t + 12$, which describes the ball's trajectory.

• Data for driving cost at different speeds are $(50, 6.20)$, $(60, 7.80)$, and $(70, 10.60)$. We can fit a parabola $C = av^2 + bv + c$ to model how cost changes with speed, finding $C = 0.006v^2 - 0.5v + 16.2$.