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the mystery parabola to first coordinate u + 2 on parent parabola by adding 2 which corresponds to applying Xshift 2. We move second coordinate v on the mystery parabola to second coordinate −4( v − 1) on the parent parabola by adding −1, multiplying by −1 and multiplying by 4 which corresponds to applying Y shift −1| Yflip | Yscale 4. Joining these programs together we move ( u , v ) to ( u + 2, −4( v − 1)) by applying Xshift 2| Yshift −1| Yflip | Yscale 4. By uncovering moving directions in an equation we learn to see in a new way. All this information — the two parabolas, two shapes of ordered pairs and directions for moving from one parabola to the other — is embedded in the equa- tion. We just have to see it. Problem 2: Let Y = 4( X − 3) 2 − 5 be an equation for the graph of a mystery parabola. Write a program that moves this mystery parabola to the parent parabola. First we sweep all the Y stuff to the left side of the equation: (1/4)( Y + 5) = ( X − 3) 2 . This puts the original equation in parent form: Ystuff =( Xstuff ) 2 . Next take ( u , v ) on (1/4)( Y + 5)= ( X − 3) 2 so that (1/4)( v + 5) = ( u − 3) 2 . Using our “eye-training” and “hopping” skills we see two shapes of ordered pairs: ( u , v ) on the mystery parabola and ( u − 3, (1/4)( v + 5)) on the parent parabola. To move ( u , v ) to ( u − 3, (1/4)( v + 5)) we move u to u − 3 by adding −3 which corresponds to Xshift −3, and we move v to (1/4)( v + 5) by adding 5 and multiplying by (1/4) which corresponds to Y shift 5| Yscale (1/4). Joining these two programs together we move the mystery parabola to the parent parabola under Xshift −3| Yshift 5| Yscale (1/4). Wrapping Up. We call the relationship between coordi- nates of ordered pairs on a parabola the shape of ordered pairs ; it’s the algebraic shape of parabolas. Given a parabola we have shown how to build up its equation by moving the given parabola to the parent parabola where we know the shape of or-

Sometimes we lump together Y expressions and just call them Y stuff and we call X expressions Xstuff . With this language we write equations in

parent form as Ystuff = ( Xstuff ) 2 . From Equations to Motions.

We have shown that equations for parabolas can be built up and described in parent form cY + d = ( aX + b ) 2 . Now we pose the converse situation: can we begin with an equation with its mystery parabola, uncover a sequence of motions embedded in the equation, and from this sequence write a program that moves the mystery parabola to the parent parabola? Response. We take another look at the fir st program’s equation −4( Y − 1) = ( X + 2) 2 that has a parabola for its graph. We know the parabola (see Figure 2) but we pretend this is a new example where we are given an equation and seek to uncov- er the motions that lead to its formation. We call this “unknown” parabola the mystery parabola. Take points ( u , v ) on −4( Y − 1) = ( X + 2) 2 so that −4( v − 1) = ( u + 2) 2 . This statement shows the shape of ordered pair ( u , v ) on the mystery parabo- la. We emphasize this fact by enclosing u and v in −4( v − 1) = ( u + 2) 2 . The next step involves a change of per- spectives only this time the change goes the other way. After looking at statement −4( v − 1)= ( u + 2) 2 from the perspective of “standing” on the mystery parabola, we “hop” to the parent parabola and see the shape of ordered pairs ( u + 2, −4( v − 1)) where −4( v − 1) is the square of u + 2. We emphasize this change of perspectives from the mystery parabola to the parent parabola by changing shapes from the statement −4( v − 1) = ( u + 2) 2 to −4( v − 1) = ( u + 2 ) 2 . In other words we see two shapes of or- dered pairs simultaneously in statement −4( v − 1) = ( u + 2) 2 : one shape for the mystery parabola and the other shape for the parent parabo- la (changes we see as we “hop” back and forth between parabolas). Knowing these two shapes, we uncover moving directions to move from the mys- tery parabola to the parent parabola by interpreting the order of arithmetic operations that move ( u , v ) to ( u + 2, −4( v − 1)). We move first coordinate u on

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