Workshop on Geometry Software

Week 3: Algebra and Geometry

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=a*x^2+b*x+c and the form y= a*(x-p)^2+q . In both cases a indicates the steepness of the parabola (larger a is steeper). As we did with lines, you should experiment with families like 2*x^2+b*x+5 as b varies, or y= a*(x-2)^2+3 as a 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.

1. Let f(x) = 2*x^2 + 4*x -3 and graph f(x). Use Geogebra to compute and to graph the derivative, f '(x). Create a slider and name it x0. Use Geogebra to graph the tangent line to the graph of f(x) at the point whose x-coordinate is x0. You should be able to move the slider for x0 and the tangent line should move along the parabola. Explain the relationship between the graphs of f(x), f '(x) and the tangent line.
2. Create sliders for a, p, q . Write a function whose graph has vertex (p,q) and steepness coefficient a. Construct the point D on this parabola with x-coordinate -1. 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.

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 did a similar exercise for the intersection of the cubic y= x^3+4*x+5*x+2 and the line y=x+a where a is a slider.

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.

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?

Due: Tuesday 9/28 at 5:00 pm.