5. In Chapter 5, you studied the
parabola. The parabola is one
of a family of curves called
conic sections. Conic
sections are formed by the
intersection of a double right
cone and a plane. There are
four types of conic sections:
circles, ellipses, hyperbolas,
and parabolas.
Although the parabolas you studied in Chapter 5 are
functions, most conic sections are not. This means
that you often must use two functions to graph a
conic section on a calculator.
6. When you take the square root of both sides of
an equation, remember that you must include
the positive and negative roots.
Remember!
A circle is defined by its center and its radius. An
ellipse, an elongated shape similar to a circle, has
two perpendicular axes of different lengths.
7. Graph each equation on a graphing calculator.
Identify each conic section. Then describe the
center and intercepts.
Example 1A: Graphing Circles and Ellipses on a
Calculator
(x – 1)
2
+ (y – 1)
2
= 1
Step 1 Solve for y so that the expression can be
used in a graphing calculator.
Subtract (x – 1)
2
from both sides.(y – 1)
2
= 1 – (x – 1)
2
Take square root of both sides.
Then add 1 to both sides.
8. Example 1A Continued
Step 2 Use two equations to see the complete graph.
Use a square window on your
graphing calculator for an accurate
graph. The graphs meet and form a
complete circle, even though it might
not appear that way on the calculator.
The graph is a circle with center (1, 1)
and intercepts (1,0) and (0, 1).
Check Use a table to
confirm the intercepts.
9. 4x
2
+ 25y
2
= 100
Step 1 Solve for y so that the expression can be
used in a graphing calculator.
Subtract 4x
2
from both sides.25y
2
= 100 – 4x
2
Take the square root of both
sides.
y
2
= 100 – 4x
2
25
Divide both sides by 25.
Example 1B: Graphing Circles and Ellipses on a
Calculator
10. Step 2 Use two equations to see the complete graph.
Use a square window on your graphing
calculator for an accurate graph. The
graphs meet and form a complete ellipse,
even though it might not appear that way
on the calculator.
The graph is an ellipse with center (0, 0)
and intercepts (±5, 0) and (0, ±2).
Check Use a table to
confirm the intercepts.
Example 1B Continued
11. Graph each equation on a graphing calculator.
Identify each conic section. Then describe the
center and intercepts.
x
2
+ y
2
= 49
Step 1 Solve for y so that the expression can be
used in a graphing calculator.
Check It Out! Example 1a
12. Step 2 Use two equations to see the complete graph.
Check Use a table to
confirm the intercepts.
Check It Out! Example 1a Continued
13. 9x
2
+ 25y
2
= 225
Step 1 Solve for y so that the expression can be
used in a graphing calculator.
Check It Out! Example 1b
14. Step 2 Use two equations to see the complete graph.
Check Use a table to
confirm the intercepts.
Check It Out! Example 1b Continued
15. A parabola is a single curve, whereas a hyperbola
has two congruent branches. The equation of a
parabola usually contains either an x
2
term or a y
2
term, but not both. The equations of the other
conics will usually contain both x
2
and y
2
terms.
Because hyperbolas contain two curves that open
in opposite directions, classify them as opening
horizontally, vertically, or neither.
Helpful Hint
16. Graph each equation on a graphing calculator.
Identify each conic section. Then describe the
vertices and the direction that the graph opens.
Example 2A: Graphing Parabolas and Hyperbolas on a
Calculator
Step 1 Solve for y so that the expression can be
used in a graphing calculator.
y = – x
21
2
y = – x
21
2
17. Example 2A Continued
Step 2 Use the equation to see the complete graph.
The graph is a parabola
with vertex (0, 0) that
opens downward.
y = – x
21
2
18. Example 2B
Step 1 Solve for y so that the expression can be
used in a graphing calculator.
y
2
– x
2
= 9
y
2
= 9 + x
2
Add x
2
to both sides.
Take the square root of both
sides.
19. Example 2B Continued
Step 2 Use two equations to see the complete graph.
The graph is a hyperbola
that opens vertically
with vertices at (0, ±3).
and
20. Graph each equation on a graphing calculator.
Identify each conic section. Then describe the
vertices and the direction that the graph opens.
Step 1 Solve for y so that the expression can be
used in a graphing calculator.
2y
2
= x
Check It Out! Example 2a
21. Step 2 Use two equations to see the complete graph.
Check It Out! Example 2a Continued
22. Step 1 Solve for y so that the expression can be
used in a graphing calculator.
x
2
– y
2
= 16
Check It Out! Example 2b
23. Step 2 Use two equations to see the complete graph.
Check It Out! Example 2b Continued
24. Every conic section can be defined in terms of
distances. You can use the Midpoint and Distance
Formulas to find the center and radius of a circle.
25. Because a diameter must
pass through the center of
a circle, the midpoint of a
diameter is the center of
the circle. The radius of a
circle is the distance from
the center to any point on
the circle and equal to half
the diameter.
The midpoint formula uses averages. You can
think of xM as the average of the x-values and yM
as the average of the y-values.
Helpful Hint
26. Find the center and radius of a circle that has
a diameter with endpoints (5, 4) and (0, –8).
Example 3: Finding the Center and Radius of a Circle
Step 1 Find the center of the circle.
Use the Midpoint Formula with the endpoints
(5, 4) and (0, –8).
( , ) = (2.5, –2)5 + 0
2
4 – 8
2
27. Example 3 Continued
Step 2 Find the radius.
Use the Distance Formula with (2.5, –2) and
(0, –8)
The radius of the circle is 6.5
Check Use the other endpoint (5, 4) and the
center (2.5, –2). The radius should equal 6.5 for
any point on the circle.
The radius is the same using (5, 4).
r = ( 5 – 2.5)2
+ (4 – (–2))2
28. Find the center and radius of a circle that has
a diameter with endpoints (2, 6) and (14, 22).
Step 1 Find the center of the circle.
Check It Out! Example 3
29. Step 2 Find the radius.
Check Use the other endpoint (2, 6) and the
center (8, 14). The radius should equal 10 for any
point on the circle.
Check It Out! Example 3 Continued
30. Lesson Quiz: Part I
Graph each equation on a graphing calculator.
Identify each conic section. Then describe the
center and intercepts for circles and ellipses, or
the vertices and direction that the graph opens for
parabolas and hyperbolas.
1. x
2
– 16y
2
= 16
2. 4x
2
+ 49y
2
= 196
3. x = 6y
2
4. x
2
+ y
2
= 0.25
31. Lesson Quiz: Part II
5. Find the center and radius of a circle that
has a diameter with endpoints (3, 7) and
(–2, –5).
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