[LON-CAPA-users] New Problem type for graphing
Rex Abert
ABERTR at tcc.fl.edu
Thu Nov 14 12:28:18 EST 2019
I would like to announce to those using Lon-Capa for mathematics (Pitt,
Purdue, Simon Fraser, et al), especially high school and first and
second-year college math, that I have developed a suite of graphing
questions for LC. These are not meant to replace Geogebra or
FunctionPlot response questions. Geogebra is good at what it does.
Given a function, it will graph it nicely. FunctionPlot response is
great for plotting, say, experimental results and fitting a rough curve
to data. I wanted LC questions that would turn it around: give the
students a function to graph, and have the student plot critical
features to create a graph, and have LC evaluate the correctness of the
graph.
These questions call on libraries written using Javascript, and I have
written libraries supporting the graphing of polynomials, absolute-value
functions, explonential and logarithmic functions, rational functions,
and trigonometric functions. Yet to come are graphers for conic
sections and piecewise-defined functions. (Maybe others?)
There is probably room for improvement of these questions and
libraries. For example, I paid little attention to accessibility concerns.
There might be browser issues with older browsers, Microsoft browswers,
or mobile devices.
The interface might benefit from a little tweaking. There might yet be
some unexpected bugs that pop up with certain randomizations. I welcome
any feedback, constructive criticism, and/or suggestions.
It is my hope that one day these might be added to LC as new question
types. Perhaps some enthusiastic graduate student could be employed to
do this. All the required libraries and supporting files are in
/res/tccfl/abertr/Graphing/
While these files are open source, I hope that no one will grab the code
and port it to other LMS platforms, such as Lumen/MyOpenMath.
1. Polynomials
In its general form, given n points selected by the user, with at least
one point that is not an x-intercept, a degree n-1 polynomial is fit to
the points using Newton's interpolating polynomial. The instructor can
specify that more points than necessary be input, for example, demand
that the student select 5 points for a quadratic function (if the points
are on the graph correctly, then the coefficients of higher-degree terms
will be 0). This same library is used for absolute value functions, but
if and when I develop a library for piecewise functions, absolute value
functions might better be shifted there. For samples, see
/res/tccfl/abertr/CollegeAlgebra/QuadraticFunctions/QuadraticGraph01.problem
through
/res/tccfl/abertr/CollegeAlgebra/QuadraticFunctions/QuadraticGraph09.problem
/res/tccfl/abertr/CollegeAlgebra/LinearEquations/AbsoluteValueGraph01.problem
/res/tccfl/abertr/CollegeAlgebra/LinearEquations/AbsoluteValueGraph02.problem
/res/tccfl/abertr/CollegeAlgebra/LinearEquations/GraphOfALine.problem
/res/tccfl/abertr/CollegeAlgebra/LinearEquations/GraphOfALine02.problem
/res/tccfl/abertr/CollegeAlgebra/LinearEquations/GraphOfALine03.problem
For higher-degree polynomials:
/res/tccfl/abertr/Precalculus/Polynomials/DrawPolyGraph01.problem
/res/tccfl/abertr/Precalculus/Polynomials/DrawPolyGraph02.problem
2. Rational Functions
Given a rational function P(x)/Q(x), the student will graph the function
by plotting x- and y-intercepts, vertical asymptotes, and the end
behavior (horozontal asymptote or other). The numerator and denominator
must be polynomials, although if they are not strictly polynomials, it
might still work. The zeros of the numerator and denominator must be
real. P(x) and Q(x) with imaginary roots, such as x^2 + 1, is not
supported (yet?). For examples, see
/res/tccfl/abertr/Precalculus/PolynomialsAndRationalFunctions/graphRationalFunction01.problem
through
/res/tccfl/abertr/Precalculus/PolynomialsAndRationalFunctions/graphRationalFunction07.problem
3. Exponential and Logarithmic Functions
Given one of these functions, the student creates the graph by plotting
at least 3 points AND the asymptote, even if the asymptote is one of the
coordinate axes. Incorrect graphs made by including both a horizontal
and vertical asymptote, which I've seen many students do on paper, is
rather crudely supported. See
/res/tccfl/abertr/Precalculus/LogarithmExponential/GraphExponential01.problem
through
/res/tccfl/abertr/Precalculus/LogarithmExponential/GraphExponential06.problem
and
/res/tccfl/abertr/Precalculus/LogarithmExponential/GraphLogarithmic01.problem
through
/res/tccfl/abertr/Precalculus/LogarithmExponential/GraphLogarithmic05.problem
4. Trigonometric Functions
The student will select the type of graph to be drawn: sine & cosine, or
tangent & cotangent, or secant & cosecant. The student will then enter 5
features for each: 5 points for a wave, two asymptotes and three points
for tangent and cotangent, or 3 asymptotes and two points for secant and
cosecant. The student may also select the type of scale to use for the
horizontal axis, either a scale labeled with rational numbers or a scale
labeled with irrational numbers (multiples of pi). The selection will be
driven by the period and phase shift of the given function. For examples,
/res/tccfl/abertr/Trigonometry/Graphing/DrawTrigGraph01.problem
through
/res/tccfl/abertr/Trigonometry/Graphing/DrawTrigGraph12.problem
Best regards,
Rex Abert
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