Lesson 3: Curve Fitting

New Concept

Curve fitting finds a quadratic model, $y = ax^2 + bx + c$, for a set of data. Just as two points define a line, three non-collinear points define a unique parabola by creating a system of equations we can solve.

What’s next

Next, you’ll master this concept through practice cards on solving these systems, then use interactive examples to perform quadratic regression on real-world data.

Property

A quadratic equation can be written in the form

Examples

• To find the parabola through $(1, 3)$, $(3, 5)$, and $(4, 9)$, we solve the system: $a + b + c = 3$, $9a + 3b + c = 5$, and $16a + 4b + c = 9$. The solution is $a=1, b=-3, c=5$, so the equation is $y = x^2 - 3x + 5$.

• A parabola passes through $(0, 8)$, $(1, 5)$, and $(2, 6)$. The system is $c = 8$, $a+b+c=5$, and $4a+2b+c=6$. Substituting $c=8$ gives $a+b=-3$ and $4a+2b=-2$. The solution is $a=2, b=-5, c=8$, so $y = 2x^2 - 5x + 8$.

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 $C = 0.006v^2 - 0.5v + 16.2$.

Property

A graphing calculator can find the quadratic regression equation, $y = ax^2 + bx + c$, that provides the best fit for a set of data. This is done by entering the data points into lists and using the quadratic regression command (often QuadReg). The calculator computes the parameters $a$, $b$, and $c$ that minimize the overall error for all data points.

Examples

• Given data for speed versus driving cost, enter speeds in list L1 and costs in L2. Using the QuadReg function yields the regression equation $C = 0.0057v^2 - 0.47v + 15.56$, which models the overall trend.

• For data on radiation dosage versus egg survival, a calculator's QuadReg provides the best-fit model, such as $y = (3.65 \times 10^{-7})x^2 - 0.001x + 1.02$, without needing to solve equations manually.

Property

Using the wrong type of function to fit the data is a common error in making predictions. Even though a calculator can always compute a regression equation, that equation is not necessarily appropriate for your data. The chosen model must be reasonable for the situation it describes.

Examples

• Modeling the height of a clock's minute hand with a quadratic function works for a short time but fails over an hour. The motion is periodic, so a different type of function would be a more appropriate model.

• A quadratic regression might fit a company's stock price for one week, but it cannot be trusted for long-term prediction because market behavior is far too complex and not inherently parabolic.