8.4 SAT Math - Transformations

Functions can move on the coordinate plane according to certain rules.  The basic graph of y = x, for example, or y = x² starts at the origin, or (0,0).  Once more numbers and mathematical operations get included, the graph moves. 

On the SAT, you don't need to know all the rules, but knowing how parabolas move is the most important. 

To make the graph:	Do this:
flip upside down	make it negative (y = -x2)
narrower	multiply by an integer (y = 4x2)
wider	multiply by a fraction (y = 1/5x2)
move up the graph	add to the quadratic (y = x2+ 1)
move to the left	add to the variable (y = (x+1)2)

Linear equations move up, down, and to the left and right based on the last two rules above as well.
-SAT Math Strategy: Graph It.  If you have a graphing calculator, don't waste time figuring out the transformations if you don't have to.  Graph it on your calculator and compare that graph to the original or the answer choices.  If you have a graphing calculator, it is very rare you will have a problem in which you can't just graph it.  So most of these transformations, while good to know, probably won't be necessary.

The other kind of transformation you'll see is something of an algebraic transformation.  This will be using the f(x) information we learned in Algebra, but will then require you to find the graph of the new function.  In this cases, you'll be given something like this:

If f(x) = 3x + 4, what is the graph of f(x + 2)?
You'll need to solve the algebraic function first, that is, put in (x + 2) for x in the original equation and clean it up:

From there, either make a quick graph of the new equation yourself to compare to the answer choices, or use your graphing calculator to compare this new graph to the ones given.

Rarely, you may find other types of transformation questions.  These deal with rotations, reflections, and points/lines of symmetry.

Rotation

A rotation rotates an object around a point.  Imagine you have stuck a pin in the center of rotation and turned the whole object.  It's often helpful to actually turn your test booklet while you hold your finger in the center of the object to really see how it rotated.

Reflection

A reflection is as if an object was held up to a mirror.  The new object will be a mirror image of the original.  Often, the line of reflection is an axis, but it doesn't have to be.  Again, it's often quite helpful to physically fold your test page along the line you believe to be the line of reflection to see where it is.

Objects that are symmetrical, that is, one half is the mirror image of the other, have a line of symmetry.  That's just the line where the reflection occurred.  Again, fold your SAT test paper in half if you're not sure.

An object that is symmetrical all the way around is said to have a point of symmetry.  With a point of symmetry, it's as if you're holding that pin in the point again.  You can rotate the object all the way around that point, and you'll end up with the same object.