Fitting a parabola to data
Fitting a parabola to data is a Grade 7 math skill from Yoshiwara Intermediate Algebra where students find a quadratic equation y = ax^2 + bx + c that best models a given set of data points. Students either use three points to determine the exact equation or apply regression techniques.
Key Concepts
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 $C = 0.006v^2 0.5v + 16.2$.
Common Questions
How do you fit a parabola to data?
Choose three data points and substitute each into y = ax^2 + bx + c to create a 3x3 system of equations. Solve for a, b, and c.
When is a parabola a good model for data?
A parabola fits data that shows a curved, symmetric trend with a single turning point, such as projectile motion or profit-as-a-function-of-price.
What is quadratic regression?
Quadratic regression uses all data points (not just three) to find the best-fitting parabola by minimizing the sum of squared residuals.
How do you use a parabola model to make predictions?
Substitute the desired x-value into the fitted equation to predict the corresponding y-value.