Transcript Pop Quiz 8.1,8.2 - St. Monica Catholic Church

POP QUIZ 8.1,8.2
THE FOUR BASIC TYPES OF CONIC SECTIONS
8.2 PARABOLAS
ANALYZE THE EQUATION OF A PARABOLA
Write
in standard form. Identify the
vertex, axis of symmetry, and direction of opening of the parabola.
Original equation
Factor –1 from
the x-terms.
Complete the square
on the right side.
The 1 you added when
you completed the square
is multiplied by –1.
Answer: The vertex of this parabola is located at (–1, 4) and the
equation of the axis of symmetry is
The parabola opens downward.
.
Write
in standard form. Identify the
vertex, axis of symmetry, and direction of opening of the parabola.
Answer:
axis of symmetry:
GRAPH PARABOLAS
Graph
.
For this equation,
and
The vertex is at the origin.
Since the equation of the axis of symmetry is substitute some small
positive integers for x and find the corresponding y-values.
x
1
2
3
y
2
8
18
Since the graph is symmetric about the y-axis, the
points at (–1, 2), (–2, 8) and
(–3, 18) are also on the parabola. Use all of
these points to draw the graph.
Answer:
Graph
.
The equation is of the form
where
The graph of this equation is the graph of
in part a translated 1 unit right and 5 units down. The vertex is now at
(1, –5).
Answer:
Graph each equation.
a.
b.
Answer:
Answer:
GRAPH AN EQUATION NOT IN STANDARD FORM
Graph
First write the equation in the form
There is a y2 term, so
isolate the y and y2 terms.
Complete the square.
Add and subtract 4,
since
Then use the following information to draw the graph.
vertex: (3, 2)
axis of symmetry:
focus:
or
directrix:
direction of opening: left, since
length of the latus rectum:
or 1 unit
Answer:
Graph
Answer:
8.3 CIRCLES
WRITE AN EQUATION GIVEN THE CENTER AND
RADIUS
Landscaping The plan for a park puts the center of a circular pond, of
radius 0.6 miles, 2.5 miles east and 3.8 miles south of the park
headquarters. Write an equation to represent the border of the pond,
using the headquarters as the origin.
Since the headquarters is at (0, 0), the center of the pond is at (2.5, –
3.8) with radius 0.6 mile.
Equation of a circle
Simplify.
Answer: The equation is
Landscaping The plan for a park puts the center of a circular pond, of
radius 0.5 mile, 3.5 miles west and 2.6 miles north of the park
headquarters. Write an equation to represent the border of the pond,
using the headquarters as the origin.
Answer:
WRITE AN EQUATION GIVEN A DIAMETER
Write an equation for a circle if the endpoints of the diameter are at (2,
8) and (2, –2).
Explore
To write an equation for a circle, you must know the
center and the radius.
Plan
You can find the center of the circle by
finding the midpoint of the diameter.
Then you can find the radius of the circle
by finding the distance from the center to
one of the given points.
Solve
First, find the center of the circle.
Midpoint Formula
Add.
Simplify.
Now find the radius.
Distance Formula
Subtract.
Simplify.
The radius of the circle is 5 units, so
Substitute h, k, and r2 into the standard
form of the equation of a circle.
Answer: An equation of the circle is
Examine
Each of the given points satisfies the
equation, so the equation is reasonable.
8.3 CONTINUED…
POP QUIZ 8.2, 8.3
Write an equation for a circle if the endpoints of the diameter are at (3,
5) and (3, –7).
Answer:
WRITE AN EQUATION GIVEN A CENTER
AND A TANGENT
Write an equation for a circle with center at (3, 5) that is tangent to the
y-axis.
Sketch the circle. Since it
is tangent to the y-axis,
the radius is 3.
Answer: An equation of this circle is
.
Write an equation for a circle with center at (2, 3) that is tangent to the
x-axis.
Answer:
Find the center and radius of the circle with equation
Then graph the circle.
Answer: The center is at (0, 0) and the radius is 4.
The table lists some values for x and y that satisfy
the equation.
x
y
0
1
4
3.9
2
3.5
3
4
2.6
0
Since the circle is centered at
the origin, it is symmetric about
the y-axis. Therefore, the points
at (–1, 3.9), (–2, 3.5), (–3, 2.6)
and (–4, 0) lie on the graph.
The circle is also symmetric about the x-axis, so the points (–1,
(–2, –3.5), (–3, –2.6), (1, –3.9),
(2, –3.5), (3, –2.6), and (0, –4) lie on the graph.
Graph these points and draw the circle that
passes through them.
Answer:
–3.9),
Find the center and radius of the circle with equation
Then graph the circle.
Answer:
center (0, 0);
GRAPH AN EQUATION NOT IN STANDARD FORM
Find the center and radius of the circle with equation
Then graph the circle.
Complete the square.
(–3, 0)
Answer: The center is
at (–3, 0) and the
radius is 4.
Find the center and radius of the circle with equation
Then graph the circle.
Answer:
center (–4, 2);
(–4, 2)
8.4 ELLIPSES
HELPFUL HINTS:
1)
2)
3) If the
term has the greater denominator, the
foci are on the x-axis
4) If the
term has the greater denominator, the
foci are on the y-axis.
WRITE AN EQUATION FOR A GRAPH
Write an equation for the ellipse shown.
In order to write an equation for
the ellipse, we need to find the
values of a and b for the
ellipse. We know that the length
of the major axis of any ellipse
is 2a units. In this ellipse, the
length of the major axis is the
distance between (0, 5) and
(0, –5). This distance is 10
units.
Divide each side by 2.
The foci are located at (0, 4) and (0, –4), so c = 4. We can use the
relationship between a, b, and c to determine the value of b.
Equation relating a, b, and c
and
Solve for b2.
Since the major axis is vertical, substitute 25 for a2 and
9 for b2 in the form
Answer: An equation of the ellipse is
Write an equation for the ellipse shown.
Answer:
http://www.youtube.com/watch?v=sPhQWWdFe1M
WRITE AN EQUATION GIVEN THE LENGTH OF THE AXES
Sound A listener is standing in an elliptical room 150 feet wide and 320 feet
long. When a speaker stands at one focus and whispers, the best place for the
listener to stand is at the other focus.
Write an equation to model this ellipse, assuming the major axis is horizontal
and the center is at the origin.
The length of the major axis is 320 feet.
Divide each side by 2.
The length of the minor axis is 150 feet.
Divide each side by 2.
Substitute
and
Answer: An equation for the ellipse is
into the form
How far apart should the speaker and the listener
be in this room?
The two people should stand at the two foci of the ellipse. The distance
between the foci is 2c units.
Equation relating a, b, and c
Take the square root
of each side.
Multiply each side by 2.
Substitute
and
Use a calculator.
Answer: The two people should be about 282.7 feet apart.
Sound A listener is standing in an elliptical room 60 feet wide and 120 feet
long. When a speaker stands at one focus and whispers, the best place for the
listener to stand is at the other focus.
a. Write an equation to model this ellipse, assuming the major axis is
horizontal and the center is at the origin.
Answer:
b.
How far apart should the speaker and the listener be in this room?
Answer:
103.9 feet apart
GRAPH AN EQUATION IN STANDARD FORM
Find the coordinates of the center and foci and the
lengths of the major and minor axes of the ellipse with
equation
Then graph the equation.
The center of this ellipse is at (0, 0).
Since
and since
The length of the major axis is 2(6) or 12 units, and the length of the minor
axis is 2(3) or 6. Since the x2 term has the greatest denominator, the major
axis is horizontal.
Equation relating a, b, and c
Take the square root of each side.
The foci are at
and
You can use a calculator
to find some approximate nonnegative values for
x
y
x
0
3
1
2
3
4
2.96
2.83
2.60
2.24
5
6
1.66
0
and y that satisfy the equation.
Since the ellipse is centered at the origin, it
is symmetric about the y-axis. So, the
points at
(1, 2.96) and (–1, 2.96) lie on the
graph.
The ellipse is also symmetric about
the x-axis, so the points at (1, –2.96) and (–1,
–2.96) also lie on the graph.
Graph the intercepts (–6, 0) (6, 0) (0, 3) and (0, –3)
and draw the ellipse that passes through them and
the other points.
Answer:
center: (0, 0);
foci:
major axis: 12;
minor axis: 6
Find the coordinates of the center and foci and the
lengths of the major and minor axes of the ellipse with
equation
Then graph the equation.
Answer:
center: (0, 0);
foci:
major axis: 10;
minor axis: 4
GRAPH AN EQUATION NOT IN STANDARD FORM
Find the coordinates of the center and foci and the
lengths of the major and minor axes of the ellipse
with equation
Then graph the ellipse.
Complete the square to write in standard form.
Original equation
Complete the squares.
Write the trinomials
as perfect squares.
Divide each side
by 36.
Answer:
The center is (3, 2) and the foci are located at
and
The length of the major
axis is 12 units and the length of the minor axis is 6.
Find the coordinates of the center and foci and the
lengths of the major and minor axes of the ellipse
with equation
Then graph the ellipse.
Answer:
center: (–2, 3);
foci:
major axis: 10;
minor axis: 4
HOMEWORK:
P. 845, 8.3 #4,5,8,9,10-20 EVENS
P.846, 8.4 #2-16 EVENS