Authors: Larry Berger & Michael Colton,Michael Colton,Manek Mistry,Paul Rossi,Workman Publishing
Up Your Score (45 page)
The following two questions relate to this graph.
1. What is the slope of the line that would connect point Q and point R?
Just plug the coordinates of the two points on the line into the formula:
2. What is the slope of the line that would connect point S and point T?
Formula of a Line
The formula
y
=
mx
+
b
describes a line with a slope of
m
and a
y
-intercept of
b
. (The
y
-
intercept
is the
y
-coordinate of the point where the line hits the
y
-axis.) This means that for any point on the line, you can figure out the
y
-coordinate by multiplying the slope by the point’s
x
-coordinate and then adding the
y
-intercept. Here’s an example:
3. The equation of a line is
y
= 3
x
+ 5. If the
x
-coordinate of a point on the line is 1, what is the
y
-coordinate of that point?
Answer:
y
= (3 × 1) 5 = 8
Note:
The SAT also has questions involving quadratic equations. It might give you two equations and ask you to compare their graphs. This is covered on
page 201
.
Another useful formula for a line is (y–y
1
) = m (x–x
1
), where m is the slope and (x
1
y
1
) is a point the line passes through.
Bar Graphs and Pie Charts
Since the SAT is big on practical reasoning ability, it usually includes lots of bar graphs and charts of real-world information that you have to interpret. The charts might indicate profits of a company on the
y
-axis and months on the
x
-axis, or amount of energy used on the
y
-axis and five different cities on the
x
-axis, or any two related things. The questions are usually pretty easy if you know the principles of coordinate geometry we have just reviewed. Start reading the graphs and pie charts in the bottom corner of the front page of
USA Today
. They’re incredibly inane, but if you get used to reading them, you’ll do fine when you get to questions about similar graphs on the SAT. Common sense. That’s all you need.
Geometry Basics
We start off with symbols:
| a line with A and B as points on the line |
| a ray with point A as an endpoint |
| a line with A and B as endpoints |
ΔABC ≅ ΔCDE | triangle ABC is congruent (equal) to triangle CDE |
Conic Sections
Circles and parabolas (and ellipses and hyperbolas—which you won’t see on the test) are called
conic sections
because they can all be formed by cutting a cone with a knife. (Only mathematicians would think of something like this.) Conic sections also go by another name: The Serpent’s Next Evil Invention. C’mon, he had to do s
omething
to replace those darling quantitative comparisons of his from the old SAT.
Circle
A circle looks pretty familiar:
A
circle
is defined as a series of points that are all equal in distance from the center. Its formula is:
(
x – h
)
2
+ (
y – k
)
2
=
r
2
(
h,k
) center,
r
= radius
Parabola
A parabola looks something like this:
A
parabola
is a set of points that all have the same distance from the focus (the dot) as from the directrix (the dotted line).
c
is the distance between the vertex (
h,k
) and the directrix, and is also the distance between the vertex and the focus.
The formula for a parabola is:
if it’s opening to the left or right
(
y – k
)
2
= 4
c
(
x – h
) (
h,k
) vertex,
c
= distance from vertex to directrix/focus
or, if it’s opening upward or downward
(
x – h
)
2
= 4
c
(
y – k
) (
h,k
) = vertex,
c
= distance from vertex to directrix/focus
Perpendicular and Tangent Lines
When two lines are
perpendicular
, they intersect at right angles. You can immediately tell that two lines are perpendicular if their slopes are opposite reciprocals of one another. For example,
are perpendicular to each other because the slope of the first line is –2 and the slope of the second is ½.
Tangent lines have to do with circles (or any other conic section, for that matter, but the SAT’s probably not going to ask about tangents to hyperbolas). Basically, a line is
tangent
to a circle if it makes a right angle with the circle’s radius and is outside the circle.
