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Figure 5. Figure

Figure4 Figure .

( u + 2, −4( v − 1)) lies on the parent parabola the second coordinate −4( v − 1) is the square of first coordinate u + 2. More simply put, 4( v − 1) = ( u + 2) 2 . To emphasize that we are describing the shape of ordered pair ( u + 2, −4( v − 1)) on the parent parabola in the statement −4( v − 1)= ( u + 2) 2 we enclose first coordinate u + 2 and second coor- dinate −4( v − 1) in the statement, −4( v − 1) = ( u + 2 ) 2 . Why is this statement valuable? Recall that we are looking for the shape of ordered pair ( u , v ) on the solid parabola. If we interpret the statement, −4( v − 1) = ( u + 2) 2 , from the perspective of “standing” on the solid parabola, we see a relation- ship between coordinates u and v of ordered pair ( u , v ) on the solid parabola. To emphasize this relationship we enclose u and v in statement −4( v − 1) = ( u + 2) 2 . We describe this shape of ( u , v ) on the solid parab- ola in words: take second coordinate v and add −1, then multiply this quantity by −4 to get the square of the quantity take first coordinate u and add 2 . Using X to stand for “take the first coordinate” and Y to stand for “take the second coordinate” we get equation −4( Y − 1) = ( X + 2) 2 for the solid parabola. Upon reflection, we took points with or- dered pair ( u , v ) on the solid parabola. Then we moved those points to the parent parabola where we knew the shape of ordered pairs, wrote a state- ment, and then re-interpreted the statement in terms of the chosen ordered pair ( u , v ) on the solid parabola.

Program (2) : Again we compare the given solid parabola with the dotted parent parabola in Fig- ure 3. For this second sequence of motions we begin by shifting the solid parabola vertically −1 unit followed with a vertical flip (see dashed parab- ola in Figure 5). Rather than scaling the second coordinates by a factor of 4, we scale the first coor- dinates by a factor of (1/2) (see dashed/dotted pa- rabola in Figure 5). Finally we apply a horizontal shift 1 unit (see dotted parabola in Figure 5). This completes the move from the solid parabola to the parent parabola. We write the corresponding program, Yshift −1| Yflip | Xscale (1/2)| Xshift 1 to move ( u , v ) to ((1/2) u + 1, −( v − 1)). Since ((1/2) u + 1, −( v − 1)) lies on the parent parabola, the second coordinate −( v − 1) is the square of first coordinate (1/2) u + 1 . More simply put, −( v − 1) = ((1/2) u + 1) 2 . We change perspectives from the parent parabola back to the mystery parabola by changing shapes from The statement −( v − 1) = ((1/2) u + 1) 2 , with u and v enclosed for emphasis, describes the shape of ordered pair ( u , v ) for points on the solid parabola: take second coordinate v and add −1, multiply this quantity by −1 to get the square of the quantity, −( v − 1) = ((1/2) u + 1 ) 2 to −( v − 1) = ((1/2) u + 1) 2 .

take first coordinate u multi- ply by (1/2) then add 1 . The equation for the solid parabola is −( Y − 1) = ((1/2) X + 1) 2 . We show that the two equa- tions derived from the two programs are equivalent:

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