curve fitting

Examples of curve fitting in the following topics:

• Fitting a Curve

  • Curve fitting with a line attempts to draw a line so that it "best fits" all of the data.
  • Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints.
  • Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function is constructed that approximately fits the data.
  • Fitted curves can be used as an aid for data visualization, to infer values of a function where no data are available, and to summarize the relationships among two or more variables.
  • Extrapolation refers to the use of a fitted curve beyond the range of the observed data, and is subject to a greater degree of uncertainty since it may reflect the method used to construct the curve as much as it reflects the observed data.

• Polynomial and Rational Functions as Models

  • Functions are commonly used in fitting data to a trend line.
  • We call this curve fitting.
  • Polynomial curves generated to fit points (black dots) of a sine function: The red line is a first degree polynomial; the green is a second degree; the orange is a third degree; and the blue is a fourth degree.

• Other Curves in Polar Coordinates

  • Some curves have a simple expression in polar coordinates, whereas they would be very complex to represent in Cartesian coordinates.
  • The following type of polar equation produces a petal-like shape called a rose curve.
  • The formulas that generate the graph of a rose curve are given by:
  • Complex graphs generated by the simple polar formulas that generate rose curves:$r=a\:\cos n\theta$ and $r=a\:\sin n\theta$ where $a≠0$.

• Parabolas As Conic Sections

  • The directrix is a straight line on the opposite side of the parabolic curve from the focus.
  • The parabolic curve itself is the set of all points that are equidistant from both the directrix line and the focus.
  • Any parabola can be repositioned and rescaled to fit exactly on any other parabola—that is, all parabolas are similar.

• The Vertical Line Test

  • The vertical line test is used to determine whether a curve on an $xy$-plane is a function
  • If any $x$-value in a curve is associated with more than one $y$-value, then the curve does not represent a function.
  • If a vertical line intersects a curve on an $xy$-plane more than once, then for one value of x the curve has more than one value of y, and the curve does not represent a function.
  • If all vertical lines intersect a curve at most once then the curve represents a function.
  • Explain why the vertical line test shows, graphically, whether or not a curve is a function

• Solving Problems with Logarithmic Graphs

  • $T$ would generate the expected curve, but the scale would be such that minute changes go unnoticed and the large scale effects of the relationship dominate the graph: It is so big that the "interesting areas" won't fit on the paper on a readable scale.

• Basics of Graphing Exponential Functions

  • That is, the curve approaches infinity as $x$ approaches infinity.
  • As $x$ takes on smaller and smaller values the curve gets closer and closer to the $x$-axis.
  • The curve approaches infinity zero as approaches infinity.
  • As $x$ takes on smaller and smaller values the curve gets closer and closer to the $x$ -axis.
  • That is, if the plane were folded over the $y$-axis, the two curves would lie on each other.

• Reflections

  • Reflections are a type of transformation that move an entire curve such that its mirror image lies on the other side of the $x$ or $y$-axis.
  • The result is that the curve becomes flipped over the $x$-axis.  
  • The result is that the curve becomes flipped over the $y$-axis.  

• Symmetry of Functions

  • To determine if a relation has symmetry, graph the relation or function and see if the original curve is a reflection of itself over a point, line, or axis.  
  • The axis splits the U-shaped curve into two parts of the curve which are reflected over the axis of symmetry.  
  • The curve is split into $2$ equivalent halves.  

• Asymptotes

  • In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity.
  • The coordinates of the points on the curve are of the form $(x, \frac {1}{x})$ where $x$ is a number other than 0.
  • The $x$-axis is a horizontal asymptote of the curve.
  • So the curve extends farther and farther upward as it comes closer and closer to the $y$-axis.
  • The $y$-axis is a vertical asymptote of the curve.