5.1 graph and write equations of...

Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 3 197

5.1 Graph and Write Equations of ParabolasGoal p Graph and write equations of parabolas.Georgia

PerformanceStandard(s)

MM3G2b, MM3G2c

Your Notes

VOCABULARY

Parabola

Focus

Directrix

Axis of symmetry

Vertex

STANDARD EQUATION OF A PARABOLA WITH VERTEX AT THE ORIGIN

The standard form of the equation of a parabola with vertex at (0, 0) is as follows:

Equation Focus Directrix Axis of Symmetry

x2 5 4py (0, p) y 5 Vertical ( )

y2 5 4px (p, 0) x 5 Horizontal ( )

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198 Georgia Notetaking Guide, Mathematics 3 Copyright © McDougal Littell/Houghton Mifflin Company.

Your Notes

Graph x 51}3

y2. Identify the focus, directrix, and axis of

symmetry.

1. Rewrite the equation in standard form.

x 51}3y2 Write original equation.

5 Multiply each side by .

2. Identify the focus, directrix, and axis of symmetry. The

equation has the form y2 5 4px where p 5 . The

focus is (p, 0), or 1 , 0 2. The directrix is x 5 2p, or

x 5 . Because y is squared, the axis of symmetry

is the .

3. Draw the parabola by making

x

y

1

1a table of values and plotting points. Because p 0, the parabola opens to the .So, use only x-values.

x 1 2 3 4 5

y

Example 1 Graph an equation of a parabola

1. Graph y 5 21}4x2. Identify the focus, directrix, and

axis of symmetry.

x

y

1

1

Checkpoint Complete the following exercise.

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Your Notes

Write an equation of the parabola shown.

x

y

1

1(0, 0)

y 5 3

Solution

The graph shows that the vertex is and the directrix is y 5 2p 5 . Substitute for p in the standard form of the equation of a parabola.

Standard form, axis of symmetry

Substitute for p.

Simplify.

Example 2 Write an equation of a parabola

2. Write the standard form of the equation of the parabola with vertex at (0, 0) and the directrix

x 5 3}4.


Homework

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Match the equation with its graph.

1. x2 5 4y 2. x2 5 24y 3. y2 5 4x

A.

x

y

1

1

B.

x

y

1

1

C.

x

y1

2

Tell whether the parabola opens up, down, left, or right.

4. x2 5 6y 5. x2 5 27y 6. y2 5 24x

Graph the equation. Identify the focus and directrix of the parabola.

7. x2 5 4y 8. y2 5 2x 9. x2 5 212y

x

y

2

2

x

y

1

1

x

y

2

2

LESSON

5.1 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON

5.1 Practice continued

Write the standard form of the equation of the parabola with the given focus and vertex at (0, 0).

10. (1, 0) 11. (0, 3) 12. (24, 0)

13. (0, 24) 14. (7, 0) 15. (0, 7)

Write the standard form of the equation of the parabola with the given directrix and vertex at (0, 0).

16. x 5 1 17. y 5 21 18. x 5 23

19. y 5 24 20. x 5 6 21. y 5 8

22. Sailboat Race The course for a sailboat race includes a turnaround point marked by a stationary buoy. The sailboats must pass between the buoy and the straight shoreline. The boats follow a parabolic path past the buoy, which is 60 yards from the shoreline. Find an equation to represent the parabolic path, so that the boats remain equidistant from the buoy and the straight shoreline.



5.2 Graph and Write Equations of CirclesGoal p Graph and write equations of circles.Georgia


MM3G1a, MM3G1b, MM3G1c

Your Notes

VOCABULARY

Circle

Center

Radius

STANDARD EQUATION OF A CIRCLE WITH CENTER AT THE ORIGIN

The standard form of the equation of a circle with center at (0, 0) and radius r is as follows:

x2 1 y2 5

Graph y2 5 2x2 1 16. Identify the radius of the circle.

Solution1. Rewrite the equation y2 5 2x2 1 16 in standard form

as .

2. Identify the center and radius.

x

y

1

1

From the equation, the graph is a circle centered at the origin with radius r 5 5 .

3. Draw the circle. First plot several convenient points that are 4 units from the origin, such as (0, ), (4, ), (0, ), and (24, ). Then draw the circle that passes through the points.

Example 1 Graph the equation of a circle

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Your Notes

1. Graph x2 5 4 2 y2.

x

y

1

1

Identify the radius.


The point (23, 4) lies on a circle whose center is the origin. Write the standard form of the equation of the circle.

The circle’s radius r must be the distance between the center and (23, 4). Use the distance formula.

r 5 Ï}}}

( )2 1 ( )2

5 Ï}

5 Ï}

5

Use the standard form with r 5 to write an equation of the circle.

x2 1 y2 5 r2 Standard form

x2 1 y2 5 2 Substitute for r.

x2 1 y2 5 Simplify.

Example 2 Write an equation of a circle

Find an equation of the line tangent to the circle x2 1 y2 5 17 at (4, 21).

A line tangent to a circle is perpendicular to the radius at the point of tangency. The radius with endpoint

(4, 21) has slope m 5 5 , so the slope

of the tangent line at (4, 21) is the negative reciprocal

of , or . An equation of the tangent line is as

follows:

y 1 5 (x 2 ) Point-slope form

y 5 Solve for y.

Example 3 Find an equation of a line tangent to a circle

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Your Notes

2. Write the standard form of the equation of the circle with center at the origin that passes through the point (6, 23).

3. Find an equation of the line tangent to the circle x2 1 y2 5 34 at (23, 25).

4. In Example 4, suppose you are 16 miles east and 19 miles south of the lighthouse. Can you see the lighthouse beam?

Checkpoint Complete the following exercises.

Lighthouse The beam from Oak Island Lighthouse in North Carolina can be seen for up to 24 miles. You are 18 miles east and 9 miles south of the lighthouse. Can you see the lighthouse beam? Let the origin represent the lighthouse and the positive y-axis represent north.

Solution1. Write an inequality for the region lit by the beam.

This region is all the points that satisfy the following inequality: x2 1 y2 < 2

2. Substitute the coordinates (18, 29) into the inequality.

x2 1 y2 < 2 Inequality

<? 2 Substitute for x and y.

The inequality is .

You see the lighthouse beam.

Example 4 Write a circular model

Homework

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Name ——————————————————————— Date ————————————

LESSON

5.2 PracticeMatch the equation with its graph.

1. x2 1 y2 5 64 2. x2 1 y2 5 36 3. x2 1 y2 5 26

A.

x

y

2

2

B.

x

y

4

4

C.

x

y

3

3

Graph the equation. Identify the radius of the circle.

4. x2 1 y2 5 1 5. x2 1 y2 5 49 6. x2 1 y2 5 81

x

y

1

1

x

y

3

3

x

y

3

3

7. x2 1 y2 5 12 8. x2 1 y2 5 30 9. x2 1 y2 5 110

x

y

2

2

x

y

2

2

x

y

4

4



Write the standard form of the equation of the circle with the given radius and whose center is the origin.

10. 3 11. 6 12. Ï}

10

Write the standard form of the equation of the circle that passes through the given point and whose center is the origin.

13. (1, 0) 14. (2, 2) 15. (3, 22)

Write an equation of the line tangent to the given circle at the given point.

16. x2 1 y2 5 13; (2, 3) 17. x2 1 y2 5 20; (24, 2)

18. Radio A radio station has a broadcast radius of 40 miles. Your house is located 32 miles west and 21 miles south of the radio station. Is your house within the broadcast range?

LESSON


Name ——————————————————————— Date ————————————



5.3 Graph and Write Equations of EllipsesGoal p Graph and write equations of ellipses.Georgia


MM3G2b, MM3G2c

Your Notes

VOCABULARY

Ellipse

Foci

Vertices

Major axis

Center

Co-vertices

Minor axis

STANDARD EQUATION OF AN ELLIPSE WITH CENTER AT THE ORIGIN

Equation Major Axis Vertices Co-Vertices

x2}a2 1

y2}b2 5 1 Horizontal (6 , 0) (0, 6 )

x2}b2 1

y2}a2 5 1 Vertical (0, 6 ) (6 , 0)

The major and minor axes are of lengths 2a and 2b,respectively, where a > b > 0. The foci of the ellipse lie on the major axis at a distance of c units from the center, where c2 5 .

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Your Notes

Graph the equation 9x2 1 36y2 5 324. Identify the vertices, co-vertices, and foci of the ellipse.


9x2 1 36y2 5 324 Write original equation.

1 5 Divide each side by .

1 5 1 Simplify.

2. Identify the vertices, co-vertices,

x

y

2

2

and foci. Note that a2 5and b2 5 , so a 5 and b 5 . The denominator of the x2-term is that of the y2-term, so the major axis is . The vertices of the ellipse are at (6a, 0) 5 (6 , 0). The co-vertices are at (0, 6b) 5 (0, 6 ). Find the foci.

c2 5 a2 2 b2 5 5 , so c 5 Ï}

.

The foci are at (6Ï}

, 0), or about (6 , 0).

3. Draw the ellipse that passes through each vertex and co-vertex.

Example 1 Graph an equation of an ellipse

Write an equation of the ellipse that has a vertex at (0, 7), a co-vertex at (24, 0), and center at (0, 0).

Make a sketch of the ellipse.

x

y

2

2

By symmetry, the ellipse must also have a vertex at (0, ) and a co-vertex at ( , 0).

Because the vertex is on the and the co-vertex is on the ,the major axis is with a 5 , and the minor axis is with b 5 .

An equation is 5 1, or 5 1.

Example 2 Write an equation given a vertex and co-vertex

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Your Notes

Write an equation of the ellipse that has a vertex at (26, 0), a focus at (5, 0), and center at (0, 0).

SolutionMake a sketch of the ellipse.

x

y

2

2

Because the vertex and focus lie on the , the major axis is

, with a 5 and c 5 . To find b, use the equation c2 5 a2 2 b2.

5 2 b2

b2 5 2 5

b 5

An equation is 5 1, or 5 1.

Example 3 Write an equation given a vertex and a focus

1. Graph x2 1y2}25 5 1. Identify the

x

y

2

2

vertices, co-vertices, and foci of the ellipse.


2. Vertex: (29, 0) 3. Vertex: (0, 7)Co-vertex: (0, 4) Focus: (0, 23)

Checkpoint Write an equation of the ellipse with the given characteristics and center at (0, 0).

Homework

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Graph the equation. Identify the vertices, co-vertices, and foci of the ellipse.

1. x2

} 9 1

y2

} 4 5 1 2.

x2

} 25

1 y2

} 16

5 1 3. x2

} 36

1 y2

} 16

5 1

x

y

1

1

x

y

2

2

x

y

2

2

4. 2x2 1 50y2 5 50 5. 49x2 1 y2 5 49 6. 4x2 1 36y2 5 144

x

y

2

2

x

y

3

3

x

y

2

2

LESSON

5.3 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Write an equation of the ellipse with the given characteristics and center at (0, 0).

7. Vertex: (4, 0) 8. Vertex: (0, 23) 9. Vertex: (27, 0)

Co-vertex: (0, 2) Co-vertex: (22, 0) Co-vertex: (0, 21)

10. Vertex: (3, 0) 11. Vertex: (26, 0) 12. Co-vertex: (4, 0)

Focus: (1, 0) Focus: (22, 0) Focus: (0, 2 Ï}

5 )

The equations of a parabola, a circle, and an ellipse are given. Graph the equation.

13. x2 2 16y 5 0 14. 5x2 1 5y2 5 80 15. 4x2 1 16y2 5 64

x

y

4

4

x

y

2

2

x

y

2

2

16. Window An elliptical stained glass window is planned in the construction of a new offi ce building. The window will have a width of 8 feet and a height of 4 feet. Write an equation of the outline of the window that can be used on the blueprint. Assume the major axis of the window is horizontal.



5.4 Graph and Write Equations of HyperbolasGoal p Graph and write equations of hyperbolasGeorgia


MM3G2b, MM3G2c

Your Notes

VOCABULARY

Hyperbola

Foci

Vertices

Transverse Axis

Center

STANDARD EQUATION OF A HYPERBOLA WITH CENTER AT THE ORIGIN

Equation Transverse Axis Asymptotes Vertices

x2}a2 2

y2}b2 5 1 Horizontal y 5 6 x (6 , 0)

y2}a2 2

x2}b2 5 1 Vertical y 5 6 x (0, 6 )

The foci lie on the transverse axis, c units from the center, where c2 5 .

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Your Notes

Graph 36y2 2 9x2 5 324. Identify the vertices, foci, and asymptotes of the hyperbola.

Solution

x

y

2

2


5 1

2. Identify the vertices, foci, and asymptotes. Note that a2 5and b2 5 , so a 5 and b 5 . The y2-term is , so the transverse axis is and the vertices are (0, 6 ). Find the foci.

c2 5 a2 1 b2 5 5 , so c 5 Ï}

The foci are at (0, 6Ï}

) ø (0, 66.7).

The asymptotes are y 5 6a}bx, or y 5 .

3. Draw the hyperbola. Draw a rectangle centered at the origin that is 2a 5 units high and 2b 5 units wide. The asymptotes pass through opposite corners of the rectangle. Then, draw the hyperbola passing through the vertices and approaching the asymptotes.

Example 1 Graph an equation of a hyperbola

1. Graph x2

}49 2

y2}9 5 1. Identify

x

y

2

2

the vertices, foci, and asymptotes of the hyperbola.


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Your Notes

Write an equation of the

x

y

2

6

(4, 0)

(5, 0)

(−4, 0)

(−5, 0)

hyperbola with foci at (25, 0) and (5, 0) and vertices at (24, 0) and (4, 0).

The foci and vertices lie on the -axis equidistant from the

origin, so the transverse axis is and the center is the origin. The foci are

each units from the center, so c 5 . The vertices are each units from the center, so a 5 .

Because c2 5 a2 1 b2, you have b2 5 c2 2 a2. Find b2.

b2 5 c2 2 a2 5 5

Because the transverse axis is horizontal, the standard form of the equation is as follows:

5 1 Substitute for a and for b2.

5 1 Simplify.

Example 2 Write an equation of a hyperbola

2. Write an equation of the hyperbola with foci at (0, 27) and (0, 7) and vertices at (0, 25), (0, 5).


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Your Notes

3. Write an equation for the hyperbolic cross section of the lamp in Example 3 if the vertices are at (63, 0) and the foci are at (65, 0). If the lamp is 15 inches high, how wide is the base?


Lamp The diagram shows the

x

y(0, 4)

(2, 0)

(0, −4)

(−2, 0)

hyperbolic cross section of a lamp. Write an equation for the cross section of the lamp. The lamp is 10 inches high. How wide is the base?

Solution

1. From the diagram, a 5and b 5 .

Because the transverse axis is ,an equation for the cross section of the lamp is as follows:

5 1, or 5 1

2. Find the x-coordinate at the lamp’s bottom edge. Because the lamp is 10 inches tall, substitute y 5 into the equation and solve.

5 1 Substitute for y.

x2 5 Solve for x2.

x 5 Take positive square root of each side.

So, the lamp has a width of 2x or 2( ) ø inches.

Example 3 Solve a multi-step problem

Homework

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Match the equation with its graph.

1. x2

} 4 2

y2

} 4 5 1 2.

x2

} 4 2

y2

} 9 5 1 3.

y2

} 9 2

x2

} 16

5 1

A.

x

y

2

4

B.

x

y

2

2

C.

x

y

2

4

Graph the equation. Identify the vertices, foci, and asymptotes of the hyperbola.

4. x2

} 1 2

y2

} 4 5 1 5.

x2

} 9 2

y2

} 9 5 1 6.

y2

} 25

2 x2

} 36

5 1

x

y

1

1

x

y

2

2

x

y

3

3

LESSON

5.4 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


7. y2

} 25

2 x2

} 9 5 1 8.

x2

} 49

2 y2

} 64

5 1 9. y2

} 16

2 x2

} 25

5 1

x

y

3

3

x

y

4

4

x

y

4

4

Write an equation of the hyperbola with the given foci and vertices.

10. Foci: (2, 0), (22, 0) 11. Foci: (0, 5), (0, 25)

Vertices: (1, 0), (21, 0) Vertices: (0, 3), (0, 23)

12. Photography A hyperbolic mirror can be used to take panoramic photographs. A camera is pointed toward the vertex of the mirror and is positioned so that the lens is at the focus that is farther from the vertex of the mirror. An equation for the cross

section of the mirror is x2 }

16 2

y2

} 4 5 1 where x and y are measured in inches. How far

from the mirror is the lens? Round your answer to two decimal places.



5.5 Translate and Classify Conic SectionsGoal p Translate conic sections.Georgia



Your Notes

VOCABULARY

Conic sections

General second-degree equation

Discriminant

STANDARD FORM OF EQUATIONS OF TRANSLATED CONICS

In the following equations, the point (h, k) is the vertex of the parabola and the center of the other conics.

Circle (x 2 h)2 1 (y 2 k)2 5 r2

Parabola (y 2 k)2 5 4p(x 2 h) Horizontal axis(x 2 h)2 5 4p(y 2 k) Vertical axis

Ellipse(x 2 h)2}

a2 1(y 2 k)2}

b2 5 1 Horizontal axis

(x 2 h)2}

b2 1(y 2 k)2}

a2 5 1 Vertical axis

Hyperbola(x 2 h)2}

a2 2(y 2 k)2}

b2 5 1 Horizontal axis

(y 2 k)2}

a2 2(x 2 h)2}

b2 5 1 Vertical axis

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Your Notes

Graph (x 1 3)2 1 (y 2 2)2 5 4.

1. Compare the given equation to the standard form of an equation of a circle. The graph is a circle with center at (h, k) 5 ( , ) and radius r 5 Ï

}

5 .

2. Plot the center. Then plot several points that are each units from the center:

5

5

5

5

x

y

1

13. Draw a circle through the points.

Example 1 Graph the equation of a translated circle

1. Graph (x 2 2)2 1 (y 1 3)2 5 9.

x

y1

1Identify the center and radius.

2. Graph (x 1 3)2}

9 2(y 2 1)2}

25 5 1.

x

y

2

2


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Your Notes

Write an equation of the parabola whose vertex is at (2, 1) and whose focus is at (5, 1).

1. Determine the form of the

x

y

2

2

equation. Sketch the parabola. The parabola opens to the

and has the form (y 2 k)2 5 4p(x 2 h) where

p 0.

2. Identify h and k. The vertex is at (2, 1), so h 5 and k 5 .

3. Find p. The vertex (2, 1) and focus (5, 1) both lie on the line , so the distance between them is

⏐p⏐ 5 ⏐ ⏐ 5 , and p 5 or p 5 .Because p 0, it follows that p 5 , so 4p 5 .

The equation is .

Example 2 Write an equation of a translated parabola

3. Write an equation of the parabola whose vertex is at (23, 21) and whose focus is at (23, 1).


CLASSIFYING CONICS USING THEIR EQUATIONS

Discriminant Type of Conic

B2 2 4AC 0, B 5 0, and A 5 C Circle

B2 2 4AC 0 and either B Þ 0 or A Þ C Ellipse

B2 2 4AC 0 Parabola

B2 2 4AC 0 Hyperbola

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Your Notes

Classify the conic given by 4x2 2 y2 1 8x 2 12 5 0. Then graph the equation.

Solution

Note that A 5 , B 5 , and C 5 , so the value of the discriminant is: B2 2 4AC 5 5 .

Because B2 2 4AC 0, the conic is a .

To graph the conic, first complete

x

y

1

1

the square in x.

4x2 2 y2 1 8x 2 12 5 0

4x2 1 8x 2 y2 5 12

2 y2 5 12

2 y2 5 12 1

2 y2 5

5 1

From the equation, the axis is ,(h, k) 5 , a 5 , and b 5 . Use these facts to draw the conic.

Example 3 Classify a conic

4. Classify the conic given by 2x2 1 2y2 2 5x 1 3y 2 1 5 0.


Homework

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Graph the equation. Identify the important characteristics of the graph.

1. (x 2 2)2 1 y2 5 4 2. (x 2 1)2

} 9 1

(y 2 3)2

} 4 5 1

x

y

1

1

x

y

1

1

3. (x 2 4)2 5 4(y 2 2) 4. (x 1 3)2

} 16

2 (y 2 1)2

} 25

5 1

x

y

2

2

x

y

4

4

5. (x 1 3)2

} 16

1 (y 2 1)2

} 25

5 1 6. (x 1 4)2 1 (y 1 4)2 5 36

x

y

2

2

x

y2

2

LESSON

5.5 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Write an equation of the conic section.

7. Circle with a center at (3, 4) and a radius of 3

8. Parabola with vertex (2, 1) and focus at (2, 3)

9. Ellipse with vertices at (1, 1) and (7, 1) and co-vertices at (4, 21) and (4, 3)

10. Hyperbola with vertices at (2, 22) and (2, 2) and foci at (2, 24) and (2, 4)

Use the discriminant to classify the conic section.

11. x2 1 8x 1 2y 1 24 5 0 12. 2x2 1 2y2 2 6x 2 4y 1 16 5 0

13. x2 2 y2 2 2x 1 7y 5 0 14. x2 1 9y2 1 2x 2 18y 2 20 5 0



Classify the conic section and write its equation in standard form. Then graph the equation.

15. x2 1 y2 2 6x 2 2y 1 1 5 0 16. y2 2 12x 2 72 5 0

x

y

1

1

x

y

4

4

17. 9x2 1 4y2 1 54x 2 16y 1 61 5 0 18. x2 1 y2 1 8x 1 2y 1 1 5 0

x

y

1

1

x

y

2

2

19. 225x2 1 16y2 2 150x 2 96y 2 481 5 0 20. x2 1 9y2 2 8x 2 90y 1 232 5 0

x

y

4

4

x

y

2

2

21. Sprinkler System A sprinkler system shoots a stream of water that follows a parabolic path. The nozzle is fastened at ground level. The water reaches a maximum height of 16 feet and a horizontal distance of 32 feet from the nozzle. Find the equation that represents the path of the water.

LESSON


Name ——————————————————————— Date ————————————



5.6 Solve Quadratic SystemsGoal p Solve quadratic systems.Georgia


MM3G1d, MM3G1e

Your Notes

VOCABULARY

Quadratic system

Solve the system using a graphing calculator.

y2 2 3x 2 1 5 0 Equation 12x 2 y 5 6 Equation 2

1. Solve each equation for y.

Equation 1 Equation 2y2 2 3x 2 1 5 0 2x 2 y 5 6

y2 5 2y 5

y 5 y 5

2. Graph the equations. Use the

IntersectionX=5 Y=4

calculator’s intersect feature to find the coordinates of the intersection points. The graphs of y 5 and y 5 intersect at

( , ). The graphs of y 5 and y 5 intersect at ( , ).

The solutions are ( , ) and ( , ). Check the solutions by substituting the coordinates of the points into each of the original equations.

Example 1 Solve a linear-quadratic system by graphing

1. x2 1 y2 5 18

y 5 2x 1 6

Checkpoint Use a graphing calculator to solve system.

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Your Notes

Solve the system using substitution.

x2 1 y2 5 13 Equation 1

y 5 x 1 5 Equation 2

SolutionSubstitute x 1 5 for y in Equation 1 and solve for x.

x2 1 y2 5 13 Equation 1

x2 1 ( )2 5 13 Substitute for y.

x2 1 5 13 Expand the power.

5 0 Combine like terms.

5 0 Divide each side by .

5 0 Factor.

x 5 or x 5 Zero product property

The corresponding y-values are y 5 5 and y 5 5 . The solutions are ( , ) and ( , ).

Example 2 Solve a linear-quadratic system by substitution

2. Solve the system using substitution.

y2 1 6x 2 3 5 0

y 5 2x 2 1


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Your Notes

3. Solve the system by elimination.

2x2 1 2y2 2 26 5 0

2x2 1 y 1 7 5 0


Solve the system by elimination.

x2 1 y2 2 9 5 0 Equation 1x2 2 6x 1 y2 2 6y 1 9 5 0 Equation 2

1. Multiply Equation 1 by and add the equations to eliminate the x2 and y2 terms.

Multiply Equation 1 by .

x2 1 y2 2 6x 2 6y 1 9 5 0 5 Add.

y 5 Solve for y.

2. Substitute for y in Equation 1 and solve for x.

x2 1 y2 2 9 5 0 Equation 1

x2 1 2 9 5 0 Substitute for y.

5 0 Simplify.

5 0 Factor binomial.

x 5 or x 5 Zero product property

3. Substitute for x in .

IntersectionX=3 Y=0

y 5 5 Substitute.

y 5 5 Substitute.

The solutions are and . Check the solutions by

graphing the equations of the system.

Example 3 Solve a quadratic system involving two circles

Homework

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Solve the system using a graphing calculator.

1. x2 1 y2 2 16 5 0 2. x2 2 y2 2 8 5 0 3. x2 2 y 1 5 5 0 2x 1 y 2 2 5 0 x 1 y 1 2 5 0 x 2 y 1 5 5 0

4. x2 1 2y2 5 12 5. (x 1 3)2 2 y 5 4 6. 2x2 2 2(y 2 2) 5 4 y 5 3 x 5 1 x2 1 y2 5 25

7. Multiple Choice Which ordered pair is a solution of the linear-quadratic system below?

x2 2 x 1 y2 5 21 x 1 3y 5 2

A. (1, 5) B. (5, 21) C. (24, 2) D. (23, 2)

Solve the system using substitution.

8. x2 1 y2 5 32 9. x2 1 y 2 3 5 0 10. x2 2 2x 1 4y 5 3 y 5 x x 1 y 5 1 x 2 y 5 3

11. 2x2 1 2y2 5 14 12. 3x 1 y2 1 2 5 9 13. x2 2 3y2 1 3 5 0 x 2 y 5 1 x 5 y 1 1 x 2 2y 1 2 5 0

LESSON

5.6 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


14. x2 1 y 5 10 15. 2x2 2 x 2 y2 1 55 5 0 16. y2 2 y 1 4x 1 10 5 0 3x 1 y 5 6 2x 1 y 5 5 22x 1 4y 5 2

Solve the system.

17. x2 2 y2 1 6x 2 8 5 0 18. 2x2 1 y2 2 3x 5 0 2x2 1 y2 2 4x 1 2 5 0 x2 1 y2 1 x 1 21 5 0

19. x2 2 y2 2 7y 2 12 5 0 20. 2x2 1 3y2 5 9 2x2 1 2y2 1 4y 2 6 5 0 2x2 1 y2 5 1

21. Farming A farmer has 30 feet of fence to enclose a rectangular area that borders a barn. No fence is needed along the barn. Is it possible for the farmer to enclose 112 square feet? If possible, fi nd the dimensions of the enclosure.



5.7 Use Figures in Three-Dimensional SpaceGoal p Understand the three-dimensional rectangular

coordinate system.

GeorgiaPerformanceStandard(s)


Your Notes

Draw a rectangular prism having a diagonal with endpoints (3, 22, 3) and (0, 2, 21). Find the length of the diagonal.

1. Draw a rectangular prism. Plot and connect (3, 22, 3) and (0, 2, 21). The difference of the x-coordinates is

⏐3 2 0⏐5 , so the prism has a length of units. The difference of the y-coordinates is ⏐22 2 2⏐5 ,

so the prism has a width of units. The difference of the z-coordinates is ⏐3 2 (21)⏐5 , so the prism has a height of units. Draw the rectangular prism.

2. Find the length of the diagonal.

x

z

y

Let (x1, y1, z1) 5 (3, 22, 3) and

(x2, y2, z2) 5 (0, 2, 21).

d 5 Ï}}}}

(x2 2 x1)2 1 (y2 2 y1)2 1 (z2 2 z1)2

5

5 5

Example 1 Apply distance formula in 3-space

VOCABULARY

Three-dimensional coordinate system

DISTANCE BETWEEN TWO POINTS IN 3–SPACE

The distance d between (x1, y1, z1) and (x2, y2, z2) is

d 5 Ï}}}}

(x2 2 )2 1 (y2 2 )2 1 (z2 2 )2.

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Your Notes

1. Draw a rectangular prism having a diagonal with endpoints (1, 1, 0) and (2, 22, 4). Find the length of the diagonal.

x

z

y


Graph the equation x 1 3y 1 9z 5 9.

Solution

1. Find three points at which the plane intersects the axes.

Intersection of x-axis:x 1 3( ) 1 9( ) 5 9

x 5

The graph intersects the x-axis at .

Intersection of y-axis:1 3y 1 9( ) 5 9

y 5

The graph intersects the y-axis at .

Intersection of z-axis: z

y

x

1 3( ) 1 9z 5 9z 5

The graph intersects the z-axis at .

2. Plot and connect the points. Then shade the resulting region.

Example 2 Recognize and graph the equation of a plane

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Your Notes

Write an equation of a sphere in standard form with center (2, 21, 6) and radius r 5 3.

Solution

1. Write the standard form of an equation for a sphere. (x 2 x0)2 1 (y 2 y0)2 1 (z 2 z0)2 5 r2

2. Identify x0, y0, and z0. The center is at (2, 21, 6), so x0 5 , y0 5 , and z0 5 .

3. Find r2. Because r 5 3, r2 5 .

The standard form of the equation is.

Example 3 Write an equation for a sphere

2. Graph the equation 4x 1 y 2 2z 5 8.

x

z

y

3. Write an equation of a sphere in standard form with center (1, 3, 24) and radius r 5 6.


Homework

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Name ——————————————————————— Date ————————————

LESSON

5.7 PracticeDetermine the coordinates of points A and B.

1. z

y

x

AB

2. z

y

x A

B

3. z

y

x A

B

Plot the points A and B in 3-space.

4. A(2, 1, 2), B(0, 4, 1) 5. A(21, 5, 3), B(0, 0, 4) 6. A(3, 23, 21), B(2, 4, 25)

z

y

x

z

y

x

z

y

x

Find the distance between the points.

7. (0, 1, 4) and (2, 0, 2) 8. (5, 21, 0) and (1, 3, 22)

9. (22, 4, 8) and (0, 3, 6) 10. (21, 21, 7) and (4, 5, 2)



Draw a rectangular prism having a diagonal with the given endpoints. Find the length of the diagonal.

11. (1, 0, 3) and (3, 22, 1) 12. (2, 1, 2) and (0, 24, 4)

x

z

y

x

z

y

Graph the equation of the plane.

13. x 1 2y 1 2z 5 4 14. 3x 1 2y 2 z 5 6

x

z

y

x

z

y

Write an equation of the sphere in standard form with the given center and radius.

15. (3, 0, 21); r 5 4 16. (22, 5, 8); r 5 7

17. Parade Balloon Suppose a spherical parade balloon is sketched in a three-dimensional coordinate system with the origin at the center of the balloon. Each axis is given in feet. The diameter of the balloon is 20 feet. Write an equation that represents the spherical balloon.

LESSON


Name ——————————————————————— Date ————————————



Words to ReviewGive an example of the vocabulary word.

Parabola

x

y

1

1

Directrix

Vertex of a parabola

Center of a circle

Focus of a parabola

Axis of symmetry

Circle

x

y

1

1

Radius

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Ellipse

x

y

1

1

Major axis

Co-vertices

Hyperbola

x

y

2

2

Foci, vertices of an ellipse

Center of an ellipse

Minor axis

Foci, vertices of a hyperbola

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Transverse axis

Conic sections

x

y

1

1

Discriminant

Center of a hyperbola

General second-degree equation

Quadratic system

Three-dimensionalcoordinate system

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5.1 graph and write equations of...

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