# Workshop on Geometry Software

Math 241
Spring 2015
Professor: Mike O'Sullivan

### Week 3: Algebra and Geometry

First Day:

We started by graphing lines. We reviewed slope-intercept, and point-slope, equations for a line. We also looked at the standard form, a*x+b*y=c, and parametric equations for a line. Here is an example of a parameteric equation X= (2,3) + t*(-1,5). This line goes through the point (2,3) and has direction vector (-1,5) so its slope is -5. We also explored what happens when you look at a family of lines, say y=m*x+5 as m varies.

We graphed parabolas using the standard form y=m*x^2+b*x+c and the form y= m*(x-p)^2+q . In both cases m indicates the steepness of the parabola (larger m is steeper). As with lines, we experimented with families like 2*x^2+b*x+5 as b varies, or y= m*(x-2)^2+3 as m varies.

We also worked with linear and quadratic functions. If f and g are functions you can multiply them, add them, compose them, take derivatives, in a very natural way.

Assignment:
1. Create sliders for m, p, q . Write a function whose graph has vertex (p,q) and steepness coefficient m. Construct the point D on this parabola with x-coordinate 2. Using the derivative of your function, find the equation of the tangent line to the parabola at D. Vary each of the parameters p, q, m by itself and describe the behavior of the intersection point and tangent line in each case.

Second Day:

We talked about the intersection of a line with a parabola and observed that there may be 0 1 or 2 points in the intersection. There is 1 point in the intersection if the line is tangent. Algebraically, this corresponds to getting a root of multiplicity two when you substitute the equation for the line into the equation for the parabola. Having 0 points in the intersection corresponds to the substitution giving complex roots. We constructed the intersection points of the line and parabola and animated the slope of the line. This shows how the intersection points move as the slope changes.

We then looked at the intersection of a fixed line with a family of curves y^2=x^3+a*x. Animating a we see how the intersection changes as the cubic curve changes.

Finally, we worked on illustrating a standard precalculus problem: Inscribe a rectangle in a semicircle and investigate the area as the rectangle varies.

Assignment:
1. Create a worksheet for the intersection of y=x+1 with the family of curves y^2=x^3+a*x. Animate a and describe what happens to the intersection points as a varies. What is happening algebraically when you try to solve the equations to find the intersection points?
2. Finish the worksheet on the rectangle in semicircle. Animate the rectangle and show the area of the rectangle as you vary it. Describe the construction and the lessons learned from it for your students.