**Shaleen moves a parabola and something else happens!**

**Shaleen graphed the function
x(x+2)=y which was derived from the Shuttle Puzzle or Peg Game (see chapter 6,
#3 in Don's sample problems):**

**In the process Don had him get points
where the x-coordinates were negative and he was able to do that. He also noted
that the curve (a parabola) was symmetrical, and the equation for the axis of
symmetry was x= ^{-}1. Then Don asked Shaleen to move this graph 2 units
to the right and write the equation. His graph with the equations he figured out
is shown below:**

**Shaleen ended up with 2 straight
lines, instead of the curve. The equation for right side he found was x-
3 = y, and Don suggested that x > or = to 2 had to be added; the equation he found for the
left side of his graph was x + y = 1, again Don suggested x < or = 2,
otherwise he would not have one piece like the parabola. Don also
showed Shaleen the absolute value function |x|=y, and his graph could be written
in terms of the absolute value as | x - 2 | - 1 = y. They looked at what
Shaleen did here, and the parabola, and saw the differences between the 2
functions.**

**Fine job
Shaleen!**

**This is an example of the student getting an answer
that doesn't work, and instead of just telling the student he is wrong, the
teacher has the student see that his result gives some new mathematics, which is
important!**

**Now we can look at the equations that
Julia and Kathryn found to move the Peg Game parabola:**

**Below Don wrote the equation or
function for the the Peg Game parabola in the vertex form (in
orange) and shows, for himself, that
this is equivalent to the what we all get x(x+2)
= y; then he wrote the equation of
parabola that's moved 2 units to the right in vertex form and shows that is
equivalent to what Julia and Kathryn obtained.**

**The vertex form of the parabola is
a(x - h) ^{2} + k = y, where a is the coefficient of the x^{2}
term, h is the x-coordinate of the vertex (the lowest or highest point of the
parabola) and k is the y-coordinate of the vertex. For the Peg Game equation
a=1, h=^{-}1, and k=^{-}1, shown above. **

**For the Peg Game function, the input or x
is the number of pairs of pegs, and the output or y, is the number of moves to
interchange the colored pegs. One gets 1->3 (one pair of pegs takes 3 moves
to interchange the colored pegs), 2->8=2*4, 3->15=3*5, and 4->24=4*6.
From these pairs of numbers and the pattern, it's relatively easy to see that
1->1*(1+2) =3 and 2->2*(2+2)=8, and 3->3*(3+2)=15 and x-> x*(x+2).
Julia showed Don that for the shifted parabola, 5->15=5*(5-2)
and therefore found the function to be x*(x-2). **

**Fine job Julia
and Kathryn!**

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