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Thread: Advanced Techniques for Algebra

  1. #1
    Sketchpad Developer
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    Advanced Techniques for Algebra

    We had a pretty interesting Tech Support question that came in recently: "I'd like to create a graph of the relation |y|=x^3-x. How do I do it?"

    The problem is that Sketchpad knows how to graph functions but not relations. Fortunately, in this case we can consider two related functions, based on the definition of absolute value: |y| = y when y ≥ 0, and |y| = -y when y < 0. So we can graph y = x^3 - x when y ≥ 0, and also graph -y = x^3 - x when y < 0. Simplifying and rewriting to put our conditions in term of x, to show the relation we need to graph these two functions:
    y = x^3 - x when x^3 - x ≥ 0
    y = -x^3 + x when -x^3 + x < 0.
    [11/28/10: I corrected the condition above to show "<". The sketch was, and remains, correct. -stek]
    Once we've figured this out, the remaining challenge is to properly restrict the domain of these functions. I show how to do this in the enclosed sketch, by using a boolean function that returns 1 or 0 depending on whether the condition (x^3 - x) is true or false.
    Graph_absolute_value_relation.gsp
    Last edited by Stek; 11-27-2010 at 10:55 PM. Reason: Minor correction

  2. #2

    Easy way?

    Just a thought. What about the intersection of the graphs (y=x^3-x) and (x=constant,when y>=0)
    Attached Files

  3. #3

    An Idea: Combine this with the "Define function from Drawing" feature

    This was fun. As a teaching tool, I also thought of having students draw a function by hand (make it cross the x-axis a few times) and then define a function from their drawing --let's call it t(x). Hide the drawing, then display the graph of abs(y) = t(x) to generalize what is happenining when the relation abs(y) = f(x) is graphed. This is my first post, so I have no idea if my attachment will work.
    Attached Files

  4. #4
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    Nice use of the Marker tool to draw an interesting function for this technique!

  5. #5
    I like this idea with the marker tool. I would instead, not hide the drawing, but once I have graphed the absolute value of the drawing function, drag the drawing around and see how the absolute value function changes.

  6. #6
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    Graph an Equation in Two Unknowns

    Another interesting tech support question asked about graphing an equation in two unknowns, so I created a sketch and put it on the Sketch Exchange, here:
    http://sketchexchange.keypress.com/s...n-two-unknowns

    You can graph any equation you like. (You must rewrite the equation so that the left side is set equal to zero.) You can also change the region of the plane within which you want the graph to be shown.

    Graphing relations is challenging: because y is not a function of x, the sketch has to scan (x, y) values on the plane looking for values that satisfy the equation. Because it would take too long (forever!) to scan every possible point, my sketch uses an action button to scan hundreds of thousands of points looking for values that are close to solutions, based on a tolerance that you control. The resulting graph is generated as a trace.

    [Technical summary: there's a parametrically-colored point on the sweep line that is white when the result is outside the value of tolerance. When the result is closer, the point is colored according to how close it is. This colored point is used to create a traced locus that moves with the sweep line. The traces of the locus show the graph.]

  7. #7
    I have a question maybe someone may answer this question, I am teaching how to graph inequalities, what I can't figure out is how can I shade below or above the line.

  8. #8
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    Graphing Inequalities

    I use a set of custom tools to graph inequalities. I've just posted these tools here: http://sketchexchange.keypress.com/s...equality-tools. To use them, create the corresponding function, and then choose the inequality tool you want to use (y>x, y≥x, y<x, y≤x, x>y, x≥y, x<y, or x≤y).


    Quote Originally Posted by Mr. Nolasco View Post
    I have a question maybe someone may answer this question, I am teaching how to graph inequalities, what I can't figure out is how can I shade below or above the line.

  9. #9
    Thank you! I played around with the file and it works, I know my students and I would greatly appreciate it.

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