**Sheri's work on finding the equation of a
parabola that is moved.**

Don went through the multiplication
of negative numbers with Sheri because she needed that idea to do the
graph. On a number line Don made sure Sheri understood that as one goes from ^{-}2
to ^{-}1 that is going **up** 1.

Sheri realized from the patterns that (^{-}3)^{2}
= ^{+}9 = 9. Sheri graphed the equation y = x^{2}^{
}(in mulberry, below). After she graphed y = x^{2
}, Don had her look at how the graph goes up from
(0,0)- go 1 to the right you go up 1, 1 to the right you go up 3, 1 to the right
you go up 5, 7, 9 etc. The parabola goes up odd numbers! Don asked her to find
the equation of the parabola if this original parabola is moved **up **2
units. Her answers were y = x^{3 }and
y = x^{2 }+ 2. (It turned
out that the second answer was arrived at talking with Mom's). Sheri proceeded
to graph y = x^{3 }next. In
this process Sheri and Don worked on (^{-}3)^{3}=
^{-}3 ^{x}^{-}3
^{x}^{-}3 = ^{-}27.
Sheri found that "a negative number raised to an odd power is negative and
a negative number raised to an even power is positive". (^{-}3)^{4}=
^{-}3 ^{x}^{-}3
^{x}^{-}3 ^{x}^{-}3 = ^{-}81. They also talked about (^{-}3)^{4
}is **not** the same as ^{-}(3^{4}).
T he graph of y = x^{3 }is
the blue graph and **not** a parabola. Then Sheri graphed y = x^{2
}+ 2 (in red), which moved the parabola **up** 2
units. Then Don gave Sheri the problem of moving the original parabola **to the
right** **3 units** (light green). The equation she gave for this was y = x^{2
}^{.} 3. (Now a very
important idea here is that even if Sheri's equation is not correct,** her
equation will give a graph that will be important, and do something different**).
Look at the graphs below- this one is in neon pink.

So Sheri's equation gave a graph that made the original one skinnier! She immediately said if you divide, the graph will be wider (the yellow one) and of course she was right. She didn't get the equation to move the graph to the right yet, but she did a fine job and learned lots about signed numbers and raising them to a power!

In trying to move the parabola to the right 3 units, Don and Sheri made the following table:

They talked about how the numbers in the third column are
different from those for the x^{2 }^{}values.
Sheri said they are the same, square numbers, but moved down. What do we have to
do to get 3 to go to 9? Subtract 3. So the new rule would be as Sheri wrote in
green, (x - 3)^{2 }= y. This is the hardest one to get, because you have
to subtract before squaring.

Mathman home