


Coordinate Geometry
The new SAT has stepped up the emphasis on coordinate
geometry, which is the study of geometric shapes on the coordinate
plane. If you think the coordinate plane is a ferocious type of
new jet fighter, don’t worry. We’re about to clear this all up for you.
The Coordinate Plane
The coordinate plane is where all the magic happens. It’s
the space in which coordinate geometry exists. Pretty snazzy.
Every point on a coordinate plane can be mapped by using
two perpendicular number lines. The xaxis
defines the space from left to right. The yaxis
defines the space up and down. And the two meet at a point called
the origin.
Every point on the plane has two coordinates. Because
it’s the center of the plane, the origin gets the coordinates (0,0).
The coordinates of all other points indicate how far they are from
the origin. These coordinates are written in the form (x, y).
The xcoordinate is the point’s location along
the xaxis (its distance either to the left or right
of the origin). If the point is to the right of the origin, the xcoordinate
is positive. If the point is to the left of the yaxis,
the xcoordinate is negative.
The ycoordinate of a point
is its location along the yaxis (either
up or down from the origin). If the point is above the xaxis,
its ycoordinate is positive, and
if the point is below the xaxis,
its ycoordinate is negative. So the
point labeled (2,2) is 2 to the right
and 2 above the origin. The point labeled (–7,–5) is 7 to
the left and 5 below the origin.
Are We There Yet? Distance on the
Coordinate Plane
The SAT tests to see if you can find the distance between
any two points on the coordinate plane. It also tests to see if
you can find the midpoint between two points on the test. This news
should make you happy. Why? Because these are easy questions that
you can answer easily as long as you know the two necessary formulas.
Now we’re going to make sure you know those two formulas. Memorize
them.
The Distance Between Two Points
If you know the coordinates of any two points—we’ll call
them (x_{1},y_{1}) and (x_{2},y_{2})—you
can find their distance from each other with the aptly named distance
formula:
Let’s say you were suddenly overcome by the desire to
calculate the distance between the points (4,–3) and (–3,8). Just
plug the coordinates into the formula:
Finding Midpoints
As for the midpoint between the two points (x_{1}, y_{1}) and (x_{2}, y_{2}),
the formula to use is
In other words, the x and ycoordinates
of the midpoint are the averages of the x and ycoordinates
of the endpoints. To find the midpoint of the points (6,0) and (3,7),
Lines in the Coordinate Plane
You already know that a line is just an infinite set of
points arrayed in a straight formation. But once you stick one of
those “infinite set of points” into a coordinate plane, it has all
sorts of properties you can analyze. And the SAT will make sure
you know how to analyze ’em.
The Slope of a Line
A line’s slope is a measurement of how steeply that line
climbs or falls as it moves from left to right. If you want the
technical jargon, slope is a line’s vertical change divided by its
horizontal change. Or, if you prefer the poetic version,
Slope is “the rise over run.”
If you’ve got two points on a line, once again (x_{1}, y_{1}) and (x_{2}, y_{2}),
the slope of that line can be calculated using the following formula:
The variable most often used to represent slope is m.
So, for example, the slope of a line that contains the
points (–2, –4) and (6, 1) is
Positive and Negative Slopes
The slopes of some lines are positive, the slopes of others
are negative. Whether a line has a positive or negative slope is
easy to tell just by looking at a graph of the line. If the line
slopes uphill as you trace it from left to right, the slope is positive.
If a line slopes downhill as you trace it from left to right, the
slope is negative. Uphill = positive. Downhill = negative.
You can get a sense of the magnitude of the slope of a
line by looking at the line’s steepness. The steeper the line, the
greater the slope; the flatter the line, the smaller the slope.
Note that an extremely positive slope is larger then
a moderately positive slope, while an extremely negative slope is smaller then
a moderately negative slope.
Check out the lines below and try to determine whether
the slope of each line is negative or positive and which has the
greatest slope:
Lines a and b have
positive slopes, and lines c and d have
negative slopes. In terms of slope magnitude, line a > b > c > d.
Slopes You Should Know by Sight
There are certain easytorecognize slopes that it pays
to recognize by sight. Knowing how to pick them out instantly will
save you precious time.
 A horizontal line has a slope of zero. Since there is no “rise,” y_{2} – y_{1} = 0, m =
 A vertical line has an undefined slope. In this case, there is no “run,” and x_{2} – x_{1} = 0. So, m = and any fraction with zero in its denominator is, by definition, undefined.
 A line that makes a 45º angle with a horizontal line has a slope of either 1 or –1, depending on whether it’s going up or down from left to right. In this case, the rise equals the run: y_{2} – y_{1} = x_{2} – x_{1}, or y_{2} – y_{1} = –(x_{2} – x_{1}).
Of the four lines pictured below, which has a slope of 0,
which has a slope of 1, which has a slope of –1,
which has an undefined slope?
Line a has slope 0 because
it’s horizontal. Line b has slope
–1 because it slopes downward at 45º as you move from left to right.
Line c has slope 1 because it slopes upward
at 45º as you move from left to right. Line d has
undefined slope because it is vertical.
The Slopes of Parallel and Perpendicular Lines
The slopes of parallel and perpendicular lines always
have the same relationships.
 The slopes of parallel lines are always the same. If one line has a slope of m, any line parallel to it will also have a slope of m.
 The slopes of perpendicular lines are always the opposite
reciprocals of each other. A line with slope m is
perpendicular to a line with a slope of –
^{1} /_{m. }
In the figure below, lines q and r both
have a slope of 2, so they are parallel. Line s is perpendicular
to both lines q and r,
so it has a slope of –^{1}/
_{2. } .
Equation of a Line
Coordinate geometry is actually where algebra and geometry
meet. Coordinate geometry allows you to “graph” algebraic equations.
For the new SAT, you need to know the equation of a line and how
to graph that equation. The equation of a line is
where m is the slope of
the line, and b is the yintercept
of the line (the ycoordinate of the
point where the line intersects the yaxis).
As long as you know the slope of the line and the yintercept,
you can write the equation of the line.
To sketch a line whose equation you know, first plot the yintercept,
and then use the slope of the line to plot another point. Connect
the two points to form your line. The figure below graphs the line y =
–2x + 3.
Since the yintercept is
at 3, the line crosses the yaxis
at (0, 3). And since the slope is equal to –2,
the line descends two units for every one unit it moves in the positive x direction.
In other words, from (0,3), the line moves one unit
to the right and two units down, to point (1,1). You
could graph the line using those two points.
Finding the x and yIntercepts
of a Line
The yintercept of a line
is the ycoordinate of the point where
the line intersects the yaxis. The xintercept
of a line is the xcoordinate of the
point where the line intersects the xaxis.
You can find either the x or yintercept
of a line by using the slopeintercept form of the line equation.
Finding the intercepts is very straightforward. To find
the yintercept, set x =
0 and solve for y. To solve
for the xintercept, set y =
0 and solve for x. For example,
if you’ve got the line equation y = –3x +
2, the yintercept equals 2,
since y = –3(0) + 2 = 2. To find the xintercept,
set y = 0 and solve:
Parabolas Attack! Quadratic Equations Invade the
Coordinate Plane
When a quadratic equation is graphed on the coordinate
plane, the result is a parabola, which is a giant maneating insect.
Actually, it’s a harmless, Ushaped curve that can open either upward
or downward.
If the SAT covers parabolas at all, it’ll most likely
do one of these things:
 Give you an equation and ask you to choose which graph matches the equation.
 Give you a graph and ask you to choose which equation matches the graph.
You can answer either of these questions as long as you
can read the quadratic equation to identify the location of a parabola’s
vertex and determine whether the parabola opens upward or downward.
Here’s how to do it.
The equation for a parabola looks like this:
where a, b,
and c are constants. By plugging a, b,
or c into the correct formulas, you can
figure out where the vertex is and whether the parabola opens upward
or downward:

The vertex of the parabola is located at point (–
^{b}/_{2a} , c –^{b2}/_{4a} ).  The parabola opens upward if a > 0, and downward if a < 0.
So, if you’re given the quadratic equation , you know that the parabola opens
upward, since a > 0. And you could
figure out the vertex by plugging in. The xcoordinate
would be
And the ycoordinate would
be
Put it all together, and you’ve got a parabola that looks
something like this:
Of course, if you had a graphing calculator, you wouldn’t
have to go through any of this calculation at all. This is one of
the many reasons we steadfastly demand that you have a graphing
calculator and know how to use it for the new SAT.
Transformations
There’s just one more bit of coordinate geometry you have
to know for the new SAT: how slight changes to a function change
the way that the graph of that function looks in the coordinate
plane. There are two different kinds of transformations you have
to know how to deal with: shifts and stretches.
Shifts
Imagine a graph. No, better yet, look at this graph:
It’s a pretty simple graph: a parabola that has a vertex
at the origin. To put it into math, f(x)
= x^{2}. A shift
of this graph would occur when the parabola remains exactly the
same shape but is shifted over either vertically or horizontally
so that the vertex no longer rests on the origin.
To get a vertical or horizontal shift, you have to do
slightly different things, but each type of shift has one thing
in common: addition or subtraction.
Horizontal Shifts
To get a horizontal shift, in which the graph
moves either to the left or right, you use addition. But this time,
you need to add within the parentheses. So, if you want to move the
graph of the function f(x) two
spaces to the right, you make the function f(x +
2). If you want to move it four spaces to the left, you make
the function f(x – 4).
Vertical Shifts
Vertical shifts are extremely easy. If you want the image
to shift up two spots, just add the number 2 to it. If you want
it to shift down four spots, subtract the number 4. So, an equation
of a parabola that is two spaces above the origin would look like
this: f(x) + 2 = x^{2} +
2. And an equation that’s four spaces below would look like
this f(x) – 4 = x^{2} –
4.
Stretches
Imagine a graph. No better yet, look at that same example
we showed you before.
Stretching a graph makes it either fat or thin.
A graph stretches when a function is multiplied, whether
that multiplication is 3f(x) or f(3x).
If a function is multiplied by a number greater than 1, it gets
taller and thinner, while if it is multiplied by a number less than
1, it gets stubbier and wider.
That’s all the SAT covers on geometry. Just one more SAT
Math chapter to go. It’s got the longest name—Data, Statistics,
and Probability—but it’s the shortest of the bunch. Almost there.
. . .
