G.  Exercises with Formulas and Graphs for Numerical Experience (!)

Exercise (1): Form table of values for the function y = f(x) = 5 -2x for several values x in the interval [- 3, 3]. Include all possible integer values of x  in the interval [- 3, 3]. Now plot those points on paper - use a whole page.  The plotted points should fall along a straight line. If you were to plot further points (x,y) for which y = f(x) = 5 -2x, where would those points lie. Here you should see the pattern that the plotted points lie on the straight line segment  (-3, f(-3)) to (+3, f(+3)). 

The exercise may lead students to accept and apply the following  assumption.

First applied mathematics Assumption (or gamble): The graph of equations y = ax + b is given by a straight line in the plane.

This assumption is part of applied mathematics or applied analytic geometry. A further applied mathematics assumption is that two points are enough to determine a straight line or a straight line segment. That being said, when you draw a line segment, try to locate and use the  endpoints as your two points. That usually leads to greater accuracy in your drawing.

Teachers: The modern mathematics curricula of the 1950's or 1960's onwards, still lingers in the vocabulary and content of mathematics courses.  The modern mathematics curricula would make pure mathematics assumptions about real numbers and the existence of real numbers and sets of ordered pairs without explicitly adding the applied mathematics assumptions that real numbers could be represented by finite or infinite decimals, and without explicitly adding the applied mathematics assumption to sanction the use of real numbers, ordered pairs and even triplet of real numbers as coordinates for geometric or physically lines, planes and space. So the assumptions of the modern pure mathematics curricula was too limited to serve real-world applications of mathematics.  For clarity and precision in the exposition of high school and college mathematics, to the set and and pure number assumptions of pure mathematics, we need to add a second class of assumption  to sanction and provide a logical framework for the  use of numbers as coordinates for one, two and three dimensional objects, and to sanction their decimal representation.  To learn more about the benefits and deficiencies of the modern mathematics curricula, see the leading chapters 1 to 3, and 5 to 7 in the online site Volume 1B_Mathematics Curriculum Notes.

Exercise (2): Form table of values for the function y = f(x) = x²  for several values x in the interval [- 3, 3]. Include all possible integer values of x  in the interval [- 3, 3].  Now plot those points on paper - use a whole page.  The plotted points should fall along a parabola.   Form a table of 11 values for x in the interval [-1, 1]  where x is -1, -0.8, -.0.6, ... +0.6, +0. 8., +1 Plot the resulting points. Plot the resulting points. If you were to plot further points (x,y) for which y = f(x) = x² where would those points lie. Here you should see the pattern that the plotted points lie on a smooth curve joining your plotted points. 

Second applied mathematics Assumption (or gamble):  The graph of equations y =  (algebraic expression in x)  can be approximated by a smooth curve through plotted points (interpolation).

Exercise (3): Form table of values for the function y = f(x) = x³  for several values x in the interval [- 2, 2]. Include all possible integer values of x  in the interval [- 2, 2].  Now plot those points on paper - use a whole page.   Form a table of 11 values for x in the interval [-1, 1]  where x is -1, -0.8, -.0.6, ... +0.6, +0. 8., +1 Plot the resulting points. If you were to plot further points (x,y) for which y = f(x) = x³ where would those points lie. Here you should see the pattern that the plotted points lie on a smooth curve joining your plotted points. 

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