Graphs, Functions, and Transformations
This topic seems impossibly complicated, but
you can get through it if you do one simple thing: ignore all the
fancy terms and just act like a robot.
Here is an example of a function
For every value of x
you put in the right side
of the equation, the function spits out one (and only one) corresponding
value. Pretty robotic. Let’s jam x
= 0 into the function
and see what comes out.
So if x is zero, the corresponding value
of f(x) is 5.
Now, on a graph, f(x)
acts as the y-value, so you’ve just discovered the
point (0, 5). Seem familiar? It should, since that’s point B,
your y-intercept on page . The function is nothing more
than the equation for line AC in the form y = mx + b,
only f(x) is masquerading as y.
Think about it. If you understand linear equations in
y = mx + b, then
you understand how a linear function appears on a graph. Granted,
not every function is going to be a straight line: you can get some
funky functions if you start squaring terms. In general, though, whatever
function you’re given, put in values for x and
see what values for y are spit out. Then graph
Transformations can be tricky. Again, acting
robotic can make them easier. You’ll be given an initial function,
and then this function will be “transformed” in some manner. Here’s
an example using a linear function:
||Which line shows the function f(x)
= 3x +1 after f(x)
has been transformed to f(x –
All right, robot, here are your instructions:
Take (x –1) and place it into the
this new equation.
the new slope and y-intercept of this “transformed
function,” locate the new line.
You can replace f(x)
with y anytime you want.
The y-intercept (b in y = mx + b)
is –2, so the new, transformed line has a point at (0, –2). Only
line p crosses that point, so D is
That’s a transformation. Good work. Power down, robot.
Imagine if the world’s strongest person took a line, grabbed
it in both hands, and then somehow bent it. The resulting
shape would be a parabola, a U-shaped curve
that can open either upward or downward on a coordinate graph.
Parabolas occur when the x
value in a
function is squared. A common equation for a parabola is:
The equation for our parabola is y
You can see this is similar to the equation for a line,
only we’ve added a squared term to the front of it.
Parabolas have many characteristics. The two key ideas
you have to learn for the SAT are how to:
Determine whether the parabola opens upward or downward
the vertex—the bottom point of the U—of
To determine whether the parabola opens upward or downward,
look at the a
value (the number in front of the
squared term). The parabola shown in the diagram conforms to the
. The a
is positive, which is why the parabola is heading upward.
The vertex is
a bit trickier. This point is found by the very complicated formula:
It would be nice if this formula was at the front of every
Math section, but no such luck. For the parabola
the vertex ends up at
. If you don’t
believe us, block out about five minutes of time and work it out
Thankfully, that’s the last geometry concept we’re going
to cover. Now that your mind is crammed with the right facts, it’s
time to show you the best way to put this knowledge to use.