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By the Application of Areas The x-coordinate (h) of the vertex equals the bisection of the length B or B/2, and each quadrant of B" defined by the intersecting yellow lines includes the area (h") . The y-coordinate (K) of the vertex equals the sum of the areas with shared corners at the intersection of the green lines, if one of the areas equals h" a quadrant of B" with a corner at the intersection of the yellow lines. Or the y-coordinate (K) of the vertex equals the area of the square with a shared corner at the intersection of green lines and the its opposite corner at the intersection of the yellow lines. For example select the YAxis, go to the third frame and see a large square (K=-201/4) squares equal the sum of areas with a shared corner at the intersection of green lines, including h" a quadrant of B". Then click NEXT AREA and see a square (K =-1/4) squares with a shared corner at the intersection of green lines and an opposite corner at the intersection of yellow lines, included in h" a quadrant of B". The y-coordinate (K) of the vertex will always be negative in these examples associated with the pythabacus. The y-intercept (C2=K+h") for K=-201/4 (third frame of YAxis) will equal the area of (K) that elbows around an area equal a quadrant of B". In this case the elbow C2 equals (-201/4 + 61/4) = -14 squares. Or (C2=K+h) for K =1/4 (fourth frame of YAxis) will equal an area of a quadrant of B" that elbows around the area equal (K). In this case the elbow C2 equals (-1/4 + 61/4) = 6 squares. Therefore the y-coordinate (K) will also equal the area (C2-h"). Now by the application of areas, all the information necessary to graph a parabola in the cartesian plane, the roots, y-intercept and coordinates of the vertex can be visually derived. Use the Pythabacus above to explore more equations and areas. |
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