Green Globs And Graphing Equations Green Globs and Graphing Equations has four parts to it which serve as a tool, a review or testing device, and two games. The menu choices are: Equation Plotter which is a graphing tool, Linear and Quadratic Graphs which is a review or testing device, Green Globs which challenges students graphing abilities in a game format, and Tracker which tests knowledge in a game format at a more advanced level. The graphs tested in L&QG and Tracker are lines, parabolas, circles, ellipses, and hyperbolas. The program graphs any type of equation, including trigonometry functions. One of the assets of this software is that it allows you to enter equations in a variety of formats. Equations do not have to be entered as a function; that is, you can enter X 2 + Y 2 = 4 to get a circle with center at the origin and radius of 2. To enter an exponent, type ^ then the number, or use the up and down arrows. The program will show it as a superscript. In other words, X^3 will appear as X 3. You do not have to type 4*X for 4X. You may omit multiplication signs and enter factors the way you would write them. The function f(x) = 4x 3 3x + 2 can be typed in as Y = 4X^3 3X+2, which will be displayed as Y = 4X 3 3X+2. Division is entered as /. An equation with two or more terms in the numerator or denominator, or with two or more factors in the denominator, must have these enclosed in parentheses. For example, 2 + X 3-2X should be entered as (2+X)/(3 2X); 5 should be entered as (X +1)(2X - 3) 5/((X+1)(2X 3)) or 5/(X+1)/(2X 3). Use ABS(X) for x, SQR(X) for x, LOG(X) for ln x (which is log e x), EXP(X) for e x, ATN(X) for arctangent x (which is tan 1 (x)), SIN(X) for sine x, COS(X) for cosine x, TAN(X) for tangent x, and P for π. All of the functions must have parentheses enclosing their arguments. An ellipse can be entered as (X 5)^2/4 + (Y+2)^2/9 = 1. Lines can be entered as 3X+2Y=4. The program will complain if you use too many parentheses or if you enter functions as 3X+2=Y instead of Y=3X+2. Use the pull down menus at the top to change utilities or options. Problem 1: Equation Plotter This program draws a graph of the equation entered. You can print out the graph or copy and paste it into another document. From the Programs menu, choose Equation Plotter. From the Equation Plotter menu choose the type of grid on which you want to graph. Notice that one of the rectangular grids is for graphing trigonometric functions ( 2π to 2π). Now just type in your equation and press return or click on Graph Equation to graph. Schoaff, 2002 Green Globs, p. 147
figure 1 Objective: To graph the following equations to test how this program works. a. y = 4x 2 3x + 2 describe the graph: vertex: b. y = c. 5 (x +1)(2x - 3) describe the graph: asymptotes: x 2 4 + ( y - 3)2 = 1 9 describe the graph: center: axes: d. Write two equations of your own, that are different than those given, and graph them. equation: equation: description: description: Schoaff, 2002 Green Globs, p. 148
Problem 2: Linear and Quadratic Graphs This option tests your ability to write equations for the graphs that are displayed. You get to decide what kind of graph is to be drawn. The program displays the graph and you type in your guess for the equation. Your equation is graphed in a different color. If you are wrong, you can correct your guess and try again or press A for the answer. When you get an equation correct on the first try, you move to the next higher level. From the Programs menu choose Linear and Quadratic Graphs. Then from the Linear and Quadratic Graphs menu choose the types of equations you want to be challenged with. Your menu choices are Lines, Parabolas, Circles, Ellipses, Hyperbolas, and Mixed graphs. Try several of each type. The following general equations should help you with this section. line y = mx + b slope = m, y-intercept = b circle (x h) 2 + (y k) 2 = r 2 center = (h, k), radius = r parabola y = a(x h) 2 + k vertex = (h, k), vertical stretch = a ellipse (x h) 2 a 2 + (y k)2 b 2 = 1 center = (h, k), horizontal axis = 2a, vertical axis = 2b hyperbola (x h) 2 a 2 (y k)2 b 2 = 1 center = (h, k), horizontal graph, distance between vertices = 2a, slope of asymptotes = b/a (y k) 2 b 2 (x h)2 a 2 = 1 center = (h, k), vertical graph, distance between vertices = 2b, slope of asymptotes = b/a (x h)(y k) = a (see sample screen in Figure 2.) center = (h, k), use test point to determine a. Objective: From the Linear & Quadratic Graphs exercise, to test your ability at writing equations for each graph displayed. Indicate how many of each type you tried: type Lines Parabolas Circles Ellipses Hyperbolas Mixed Graphs # tried Schoaff, 2002 Green Globs, p. 149
figure 2 Problem 3: Green Globs game This game plots thirteen globs on the screen. You are to type in a sequence of equations that will go through the globs. When a glob is hit, it explodes and disappears. (Sometimes if you are off by just a little, you will still manage to hit a corner of the glob.) For each shot (equation), you get 1 point for the first glob, 2 points for the second glob, 4 points for the third glob, etc. Therefore, it is in your best interest to hit as many globs as possible with a single equation. There are buttons that will allow you to clean up the screen or start the game over. The game ends when all the globs are gone. From the Programs menu choose Green Globs. From the Green Globs menu choose the game level and whether or not you want to read the instructions. Novice game: There are thirteen globs. You can use polynomials, conics, square roots (SQR), absolute value (ABS), natural logarithms (LOG) and exponential (EXP) functions. You may not use trigonometric functions. (see Figure 3) Expert game: There are thirteen globs and five shot absorbers (white circles). Trigonometric functions SIN, COS, TAN, and ATN are allowed. If you hit a white shot absorber, the graph stops. This prevents someone from wiping out all globs with an equation like y = 8 cos(100x). Pencil and paper is very useful. For polynomial functions, you can start with the roots and then work with the points to determine the coefficients. For example, x = (y 8)(y+1)(y+11) will cross Schoaff, 2002 Green Globs, p. 150
the y-axis at 8, 1, and 11. If a glob is located at ( 6,4), then plug 4 into the equation to get x = (4 8)(4+1)(4+11) = 300. Since x must be 6, not 300, multiply by 6 / 300. Guess will be x = (y 8)(y+1)(y+11) * 6 / 300. If there are vertical columns of points at x = 3 and x = 4, then try adding the term 1 / (x 3.5) 2. This creates a vertical asymptote at x = 3.5, which sends ends of the graph going to + close to x = 3 and x = 4. The term 1/(x 3.5) also creates an asymptote, but one end goes to and one end to +. figure 3 Objective: To play at least one Green Globs game, either level. Record your score. Don t be afraid to use lines, they can be high scoring. Level: Score: Partner: How many equations did you use? What type of equation did you use the most often? Print out your winning screen. Schoaff, 2002 Green Globs, p. 151
Problem 4: Tracker This option asks you to find equations for several hidden graphs. There are two types of graphs for each game, such as parabolas and lines. You have two types of actions at your disposal. You can shoot a Probe across the screen to get clues about the hidden graphs. A probe is a vertical or horizontal line. Every time a probe crosses a type #1 graph, you will see an X. Every time a probe crosses a type #2 graph, you will see a circle. The first probe costs you 1 point, the second one 2 points, the third one 3 points, and so on. (see Figure 4) Once you feel you know an equation for one of the graphs, you can switch to Tracker to type in your guess. Each wrong tracker costs you ten points. Scoring is 30 points for a correct line, 35 points for a parabola, 40 points for a circle, 45 points for an ellipse, and 50 points for a hyperbola. figure 4 Objective: To play at least one Tracker game, either level or shapes. Record your score. Level: Score: Partner: Solution Equations: Print out your winning game. Schoaff, 2002 Green Globs, p. 152