



When
there are exponents in an equation other than 1,



the graph
of the equation will not be a straight line. 





Example: 


Y = X^{2}



The graph
of this puppy looks like this: 








It forms
a sort of U shape with the bottom point at X = 0, Y = 0. 


The left
side of the line is a mirror image of the right side. 


The line
that divides the two sides of the drawing is the Y axis. 


It is
called the line of symmetry of the equation. 





Something
funny happens with this equation. 


If we put
in some number for X and get a Y value, 


then put
in the same number but with a minus sign, 


you still
get the same Y value. 





Watch: 


X
= 3 
X
= 3 


Y= 
X^{2} 
Y= 
X^{2} 
Y= 
(3)^{2} 
Y= 
(3)^{2} 
Y= 
9 
Y= 
9 






When we
can put in a number for X with a plus sign or with a minus
sign 


and get
the same Y out both times, 


we call
the equation even. 





Neither
the "symmetrical" thing or the "even"
thing are that big a deal, 


but
knowing that kind of stuff impresses math people. 





OK, now
we'll talk about some more important stuff. 


We drew
the graph of Y = X ^{2}, 


and found
that the bottom point was at the point where X = 0 and Y = 0. 





We can
move or stretch the "U" this equation makes 


by
putting numbers into the equation at special places. 





If you
want to move the "U" up, 


just add
a number at the right end of the equation. 


To move
the "U" up 2 units, add a "+2" at the right end
of the equation. 


That
means we change Y = X ^{2} to Y = X
^{2} + 2: 











If you
want to move the "U" down, 


subtract
a number at the right side of the equation. 


If you
want to move the "U" down 1, 


stick a
1 at the right side of the equation. 


That
means we change Y = X ^{2} to Y = X
^{2}  1: 











If we
want to make the "U" skinnier, 


put a
number greater than 1 next to the "U" (called a
coefficient). 


The
larger the number, the skinnier the "U." 








If you
want to make the "U" wider, 


put a
fraction less than 1 next to the X. 


The
smaller the fraction, the wider the "U." 








If you
want to flip the "U" over and make it look like a conehead, 


put a
minus sign in front of the X. 


Even
though it's flipped over, all the other stuff still works the same
way. 








Now let's
move the "U" to the right. 


To do
that, SUBTRACT a number from the X like this. 


WATCH
CLOSELY ... 





OK, OK, 


I
know it looks backwards for a minus three to move the "U"
to the RIGHT, 


but it
does. 


The
reason is that X has to get up to 3 


before
the total inside the parentheses "( )" gets up to zero. 


If that
doesn't make sense to you, 


for now
you just have to remember that (X  (stuff)) works sort of
backwards. 





So
putting all this stuff together: 











So Y =
4(X + 1) ^{2} + 2 








copyright 2005 Bruce Kirkpatrick 
