MDE generates descriptions via templates. Description templates contain phrases and sentences into which specific mathematical values and descriptive terms (adjectives, adverbs) are inserted depending on the mathematical features present in a solution/graph and their values, and the mode of description desired. Some examples are listed in the table below.
MDE supports addition of custom templates and description modes. The modes listed below are included in Version 1.0 of the MDE SDK.
MDE Text Description Examples
Equation Type 
Describer mode = "visual" 
Describer mode = "math" 
NULL SET 
Your input equation is y 2 = y. The graph of the equation is a null set. The equation has no solution. 
Your input equation is y 2 = y. The graph of the equation is a null set. The equation has no solution. 
SINGLE POINT 
Your input equation is x^2 +y^2 = 0.0. The graph of the equation is a single point. The single point solution is (0, 0). 

ALL POINTS 
Your input equation is x = x. The solution is the set of all points. The solution will not be graphed. 

VERTICAL LINE 
Your input equation is x = 0.0. The graph of the equation is a vertical line. The slope is undefined. 
Your input equation is x = 0.0. The graph of the equation is a vertical line. The slope is undefined. The graph has an inclination of 90 degrees or approximately 1.571 radians. The xintercept is 0. The equation is a linear equation. The domain of the equation is {x such that 0 <= x <= 0}. The range of the equation is {y such that infinity < y < infinity}. 
HORIZONTAL LINE 
Your input equation is y = 0.0. The graph of the equation is a horizontal line. It is flat with a slope of 0. 
Your input equation is y = 0.0. The graph of the equation is a horizontal line. It is flat with a slope of 0. The graph has an inclination of 0 degrees or 0 radians. The yintercept is 0. The equation is a linear equation. The domain of the equation is {x such that infinity < x < infinity}. The range of the equation is {y such that 0 <= y <= 0}. 
TWO PARALLEL LINES 
Your input equation is x^2 = 1.0. The graph of the equation is two parallel lines. The lines are a distance of 2 units apart. They have an inclination of 90 degrees. 
Your input equation is (y 1.0*x)^2 = 1.0. The graph of the equation is two parallel lines. The lines are a distance of approximately 1.414 units apart. They have an inclination of 45 degrees. The x intercepts are 1, 1. The y intercepts are 1, 1. The equation is a degenerate parabola. 
TWO INTERSECTING LINES 
Your input equation is x^2 y^2 = 0. The graph of the equation is two intersecting lines. The lines cross at the point (0, 0) and have inclinations of 45 degrees and 45 degrees. 
Your input equation is (x 2.0*y)^2 y^2 = 0. The graph of the equation is two intersecting lines. The lines cross at the point (0, 0) and have inclinations of approximately 18.435 degrees and 45 degrees. The xintercept is 0. The yintercept is 0. 
SLOPING LINE 
Your input equation is y = 3*x. The graph of the equation is a line. It rises steeply from left to right with a slope of 3.
Note that MDE has the ability to change qualitative words like "steeply" depending on the line's characteristics. 
Your input equation is y = 3*x. The graph of the equation is a line. It rises steeply from left to right with a slope of 3. The graph has an inclination of approximately 71.565 degrees or approximately 1.249 radians. The xintercept is 0. The yintercept is 0. The ascending region is {x such that infinity < x < infinity}. The equation is a linear equation. The domain of the equation is {x such that infinity < x < infinity}. The range of the equation is {y such that infinity < y < infinity}. 
PARABOLA 
Your input equation is y = 1.0*x^2 +0.0. The graph of the equation is a parabola. It opens to the North. Focal length can be a measure of a parabola's width . The focal length of this parabola is 0.25. This is a good 'reference parabola' to compare other parabolas to. What happens to the focal length and parabola width when you change the coefficient of x^2? Enter y=c*x^2, with c=1. Then change c to see what happens to the parabola. 
Your input equation is y = 2.0*x^2 +0.0. The graph of the equation is a parabola. The vertex is located at the point (0, 0). The curve has an axis of symmetry which is the line given by 1*x = 0. Its axis of symmetry is oriented at an angle of 90 degrees from the positive x axis. In other words, the curve opens to the North. The focus is located at the point (0, 0.125). The focal length is 1/8. The directrix is the line given by 8*y+1 = 0. The angle of inclination of the directrix is 0. The xintercept is 0. The yintercept is 0. The ascending region is {x such that 0 <= x < infinity}. The descending region is {x such that infinity < x <= 0}. The equation is a conic section. The domain of the equation is {x such that infinity < x < infinity}. The range of the equation is {y such that 0 <= y < infinity}. 
HYPERBOLA 
Your input equation is x^2/(1.0^2) y^2/(1.0^2) = 1. The graph of the equation is a hyperbola. The graph consists of two separate pieces called branches that approach each other as if they would cross, but then bend back away from each other. The points on each piece where the branches are closest together are called vertices. The vertices are located at the points (1, 0), (1, 0). The midpoint of the line segment between the vertices is called the center of the hyperbola. The center is at (0, 0). Way out on each branch, a hyperbola is nearly straight and actually approaches a straight line called an asymptote. The equations of the asymptotes are: 1*x1*y = 0 and 1*x+1*y = 0. 
Your input equation is x^2/(2.0^2) y^2/(1.0^2) = 1. The graph of the equation is a hyperbola. The center is at (0, 0). The vertices are located at the points (2, 0), (2, 0). The eccentricity is approximately 1.118 . The focal length is approximately 2.236. The equation of the transverse axis is 1*y = 0. The length of the semitransverse axis is 2. The equation of the conjugate axis is 1*x = 0. The length of the semiconjugate axis is 1. The foci are located at the points (2.236, 0), (2.236, 0). The equations of the asymptotes are: 1*x2*y = 0 and 1*x+2*y = 0.The x intercepts are 2, 2. The equation is a conic section. 
ELLIPSE 
Your input equation is x^2/(1.0^2) +y^2/(3.0^2) = 1. The graph of the equation is an ellipse. Ellipses are oval shaped curves. How 'flat' or how rounded the oval is depends on the length of the major axis compared to the length of the minor axis. The longer the major axis compared to the minor axis, the 'flatter' the ellipse. Another term for flatness is eccentricity. The major axis of this ellipse with length 6 is approximately 3 times the length of the minor axis with length 2. This ellipse is pretty 'flat'. It's a nice long oval. 
Your input equation is x^2/(2.0^2) +y^2/(1.0^2) = 1. The graph of the equation is an ellipse. The center is at (0, 0). The eccentricity is approximately 0.866 . The semimajor axis is half the distance across the ellipse along the longest of its axes. The length of the semimajor axis is 2. The major axis is given by the line 1*y = 0. The major axis inclination is 0 degrees. The semiminor axis is half the distance across the ellipse along its shortest principal axis. The length of the semiminor axis is 1. The minor axis is given by the line 1*x = 0. The minor axis inclination is 90 degrees. The foci are located at the points (1.732, 0), (1.732, 0). The focal length is approximately 1.732. The x intercepts are 2, 2. The y intercepts are 1, 1. It is a closed curve. The equation is a conic section. The domain of the equation is {x such that 2 <= x <= 2}. The range of the equation is {y such that 1 <= y <= 1}. 
CIRCLE 
Your input equation is x^2 +y^2 = 9.0. The graph of the equation is a circle. The center is at (0, 0). The width of the circle is 6. 
Your input equation is x^2 +y^2 = 9.0. The graph of the equation is a circle. The center is at (0, 0). The radius is 3. The x intercepts are 3, 3. The y intercepts are 3, 3. It is a closed curve. The equation is a conic section. The domain of the equation is {x such that 3 <= x <= 3}. The range of the equation is {y such that 3 <= y <= 3}. 
POLYNOMIAL 
Your input equation is y = x^3. This is the graph of a cubic polynomial . The curve rises from the far lower left to an inflection point at the point (0, 0) and rises to the far upper right 

POLAR ROSE 
Your input equation is r = 1.0*sin(3.0*theta). The graph of the equation is a polar rose. The graph looks like a 3bladed propeller with its blades symmetric about the origin. 

RATIONAL FUNCTION 
Your input equation is y = x/(1 +x^2). This is the graph of a function. The curve is nearly flat from a horizontal asymptote at the line y = 0 at the far left to a local minimum at the point (1, 0.5), rises to a local maximum at the point (1, 0.5) and is nearly flat to a horizontal asymptote at the line y = 0 at the far right . 

DATA DESCRIPTION EXAMPLE: ALTITUDE (ALT) VS TIME 
The ALT(M) curve has the following characteristics. The portion of the graph in the visible window consists of a single continuous graph. The curve rises from a boundary point at the point (0.05, 0) to a local maximum at the point (7.05, 216.615) and falls to a boundary point at the point (14.2, 0.974). 
