fall-2017-final

Programs.

Programs play a central role in our ap- proach. Underlying these programs are the order of operations associated with the order of individual motions. It is the application of operations that create the algebraic expressions for coordinates. Suppose we apply a horizontal shift 2 units to the solid parabola in Figure 2 so that the image parabola has its vertex at (0, 1) on the Y -axis (see dashed parabola in Figure 6). Let this be our new starting location; in other words, our task is to write an equation for the dashed parabola shown in Fig- ure 6. To this end we write three different programs to move this dashed parabola to the dotted parent parabola. (Note that we are dealing only with sec- ond coordinates to make the moves.) We number these programs (3), (4) and (5). Program (3) : we apply Yshift −1| Yflip | Yscale4 . The order of arithmetic operations is: add −1, multiply by −1, and multiply by 4. The corresponding alge- braic expression is 4(−1)( v + (−1)) = −4( v − 1). Program (4) : we apply Yflip | Yshift 1| Yscale 4. The order of arithmetic operations is: multiply by −1, add 1, and multiply by 4. The corresponding alge- braic expression is 4((−1) v + 1) = 4(− v + 1). Program (5) : we apply Yscale 4| Yshift −4| Yflip . The order of arithmetic operations is: multiply by 4, add −4, and multiply by −1. The corresponding alge- braic expression is (−1)(4 v + (−4)) = −(4 v − 4). In our discussion of programs (3), (4) and (5) we went from programs to expressions. Now we go from expressions to programs. Consider expression (3): −4( v − 1). We begin with second coordinate v . The first operation applied to v is add −1. The result is v − 1. Next we multiply by 4. The result is 4( v − 1). Finally we multiply by −1. The result is −4( v − 1). The order of operations is: add −1, multiply by 4, multiply by −1. The correspond- ing order of motions is shift vertically −1, scale vertically by factor 4, flip vertically. The corre- sponding program is Y shift −1| Yscale 4| Yflip . We read expressions 4(− v + 1) and −(4 v − 4) similarly for programs (4) and (5). These three expressions are mathematically equivalent to expression (6) −4 v + 4. We interpret the order of operations for expression (6): take second coordinate v , multiply by 4, multiply by −1

Figure6 Figure .

(we could multiply by −1 first and multiply by 4 second), add 4. The corresponding motions are: scale vertically by factor 4, flip vertically, shift vertically 4 units. We move the dashed parabola to the parent parabola by applying program Yscale 4| Yflip | Yshift 4. We observe that this sixth program is yet another program distinct from the others but it still moves the dashed parabola to the parent parabola. In short, each of these four algebraic expressions expresses a different program that we can uncover by interpreting the corresponding expressions’ order of operations. Parent Form. In general, suppose we are given a parabola and suppose we take ( u , v ) on this mystery parabo- la and move its points to the parent parabola by applying successive basic motions. We may apply any number of motions, we may shift, flip and scale all over the coordinate plane but eventually we arrive at the parent parabola. The associated operations of addition and multiplication simplify to algebraic expressions of the form au + b for first coordinates and cv + d for second coordinates, where a ≠ 0, b , c ≠ 0, d are real numbers (flips occur when a or c are negative). These individual motions make up programs that move ( u , v ) to ( au + b , cv + d ) that lie on the parent. The resulting shape of ordered pairs ( au + b , cv + d ) on the par- ent parabola is cv + d = ( au + b ) 2 . At this point in our approach we change perspectives from the parent parabola to the mystery parabola and con- sider the shape of ( u , v ) on the mystery parabola. The equation is cY + d = ( aX + b ) 2 .

Virginia Mathematics Teacher vol. 44, no. 1

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