All SAT II Math II Resources
Example Questions
Example Question #1 : Parabolas
Which of the following equations represent a parabola?
The parabola is represented in the form . If there is a variable in the denominator or as an exponent, it is not a parabola.
The only equation that has an order of two is:
Example Question #1 : Coordinate Geometry
Refer to the above figure. The circle has its center at the origin. What is the equation of the circle?
The equation of a circle with center and radius is
The center is at the origin, or , so . To find , use the distance formula as follows:
Note that we do not actually need to find .
We can now write the equation of the circle:
Example Question #2 : Coordinate Geometry
Refer to the above diagram. The circle has its center at the origin; is the point . What is the length of the arc , to the nearest tenth?
First, it is necessary to determine the radius of the circle. This is the distance between and , so we apply the distance formula:
The circumference of the circle is
Now we need to find the degree measure of the arc. We can do this best by examining this diagram:
The degree measure of is also the measure of the central angle formed by the green radii. This is found using the relationship
Using a calculator, we find that . We can adjust for the location of :
which is the degree measure of the arc.
Now we can calculate the length of the arc:
Example Question #1 : Circles, Ellipses, And Hyperbolas
On the coordinate plane, the vertices of a square are at the points with coordinates . Give the equation of the circle.
The figure in question is below.
The center of the circle can be seen to be the origin, so, if the radius is , the equation will be
.
The circle passes through the midpoints of the sides, so we will find one of these midpoints. The midpoint of the segment with endpoints and can be found by using the midpoint equations, setting :
The circle passed though this midpoint . The segment from this point to the origin is a radius, and its length is equal to . Using the following form of the distance formula, since we only need the square of the radius:
,
set :
Substituting in the circle equation for , we get the correct response,
Example Question #1 : Circles, Ellipses, And Hyperbolas
Find the diameter of the circle with the equation .
Start by putting the equation in the standard form of the equation of a circle by completing the square. Recall the standard form of the equation of a circle:
, where the center of the circle is at and the radius is .
From the equation, we know that .
Since the radius is , double its length to find the length of the diameter. The length of the diameter is .
Example Question #1 : Circles, Ellipses, And Hyperbolas
A triangle has its vertices at the points with coordinates , , and . Give the equation of the circle that circumscribes it.
None of these
The circumscribed circle of a triangle is the circle which passes through all three vertices of the triangle.
In general form, the equation of a circle is
.
Since the circle passes through the origin, substitute ; the equation becomes
Therefore, we know the equation of any circle passing through the origin takes the form
for some .
Since the circle passes through , substitute ; the equation becomes
Solving for :
Now we know that the equation takes the form
for some .
Since the circle passes through , substitute ; the equation becomes
Solving for :
The general form of the equation of the circle is therefore
Example Question #2 : Coordinate Geometry
Which of the following symmetries applies to the graph of the relation
?
I) Symmetry with respect to the origin
II) Symmetry with respect to the -axis
III) Symmetry with respect to the -axis
I, II, and III
None of these
I only
II only
III only
II only
The relation
is a circle with center and radius .
In other words, it is a circle with center at the origin, translated right units and up units.
or
is a circle translated right 4 units and up zero units. The upshot is that the circle moves along the -axis only, and therefore is symmetric with respect to the -axis, but not the -axis. Also, as a consequence, it is not symmetric with respect to the origin.
Example Question #1 : Coordinate Geometry
Refer to the above figure.
Which of the following functions is graphed?
The correct answer is not given among the other responses.
Below is the graph of :
The given graph is the graph of shifted 6 units left (that is, unit right) and 3 units up.
The function graphed is therefore
where . That is,
Example Question #131 : Geometry
Refer to the above figure.
Which of the following functions is graphed?
Below is the graph of :
If the graph of is translated by shifting each point to the point , the graph of
is formed. If the graph is then shifted upward by three units, the new graph is
Since the starting graph was , the final graph is
, or,
Example Question #1 : Coordinate Geometry
Refer to the above figure.
Which of the following functions is graphed?
Below is the graph of :
If the graph of is translated by shifting each point to the point , the graph of
is formed. If the graph is then shifted right by four units, the new graph is
Since the starting graph was , the final graph is
, or
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