© carnegie learning - cusd 4 · 2014-07-30 · © carnegie learning 13.1 circles and polygons on...

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Circles and Parabolas 13

13.1 The Coordinate PlaneCircles and Polygons on the Coordinate Plane . . . . . . . 965

13.2 Bring On the AlgebraDeriving the Equation for a Circle . . . . . . . . . . . . . . . . . . 973

13.3 Is That Point on the Circle?Determining Points on a Circle . . . . . . . . . . . . . . . . . . . . 989

13.4 The ParabolaEquation of a Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . 997

13.5 Simply ParabolicMore with Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . 1019

Discus throwing is an ancient sport—at

least 3000 years old. This sport has been part of the

Olympics since the first Summer Olympics in 1896. Many ancient

Greek and Roman statues feature images of discus

throwers.

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13.1

LeArnIng gOALS

In this lesson, you will:

• Apply theorems to circles on the coordinate plane .• Classify polygons on the coordinate plane .• Use midpoints to determine characteristics of polygons .• Distinguish between showing something is true under certain conditions, and proving it is

always true .

The Coordinate PlaneCircles and Polygons on the Coordinate Plane

Have you ever seen a discus throw? It may seem like this track event does not require a lot of skill, but any experienced discus thrower will tell you otherwise.

Balance is an extremely important factor, as is rhythm. Orbit and delivery are key as well. A great discus thrower can release the discus at the right angle so that it catches the air underneath it for a longer throw.

Do you think any other sports use math as important factors?

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966 Chapter 13 Circles and Parabolas

13

Problem 1 Walk the Line

This lesson provides you an opportunity to review several familiar theorems while classifying polygons formed on a coordinate plane . Circles and polygons located on a coordinate plane enable you to calculate distances, slopes, and equations of lines .

1. Lauren and Jamie were practicing their discus throws . They each spun around in a circle to gain speed and then released their discus at a tangent to the circular spin . Jamie is left-handed, so she spun clockwise and released her discus at point T shown . Lauren is right-handed . She spun counterclockwise and released at point N . Both of the girls’ throws landed at point A .

a. Use the given information to algebraically show that if two tangents are drawn from the same point on the exterior of a circle, then the tangent segments are congruent .

b. What conclusions can you make about Lauren’s and Jamie’s discus throws?

x86

T (4, 3)

0C (0, 0)

2

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

4

N (0, –5)

A

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13.1 Circles and Polygons on the Coordinate Plane 967

13

2. Line AT is tangent to circle C at point T . Secant ___

AG intersects circle C at points N and G .

‹

___ › AT intersects

‹

___ › AG at point A . The coordinates of point A are (10, 0) . The center

of circle C is at the origin and the length of the radius is 6 units .

Use the given information to algebraically show that if a tangent and a secant intersect on the exterior of a circle, then the product of the lengths of the secant segment and its external secant segment is equal to the square of the length of the tangent segment .

x86

T

NG

2

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

4

A (10, 0)C (0, 0)0

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Problem 2 Work Your Quads

1. Four points on the circle were connected to form a square . Classify the polygon formed by connecting the midpoints of the sides of the square .

x

y

(0, y) (x, y)

(x, 0)(0, 0)

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2. Choose four points on the circle to form a quadrilateral that is not a square . Provide the coordinates and labels for each . Classify the polygon formed by connecting the midpoints of the sides of the quadrilateral .

x86

2

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

4

A (0, –7)

0

If you choose another set

of four points to form a quadrilateral would you come to the same conclusion about the

classification of the shape formed by connecting the

midpoints?

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3. Determine the shape formed by connecting the midpoints of the sides of an isosceles trapezoid .

a. Draw an isosceles trapezoid . Choose reasonable coordinates for each vertex .

b. Label the vertices of the trapezoid .

c. Determine the coordinates of the midpoint of each side .

x86

N

2

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

4

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4. Think back about the work you did in Questions 1 through 3 . Which were instances of showing something is true under certain conditions or proving something is always true? Explain your reasoning .

5. Describe the difference between showing something is true under certain conditions and proving something to be true under all conditions .

Be prepared to share your solutions and methods .

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973

LeArnIng gOALS


• Use the Pythagorean Theorem to derive the equation of a circle given the center and radius .• Distinguish between the equation of a circle written in general form and the equation of a circle

written in standard form (center-radius form) .• Complete the square to determine the center and the radius of a circle .

Bring On the AlgebraDeriving the equation for a Circle

13.2

You stop a friend in the hallway to ask about his or her weekend. As you are talking, another friend shows up and joins the conversation. Soon, another and

then another and then another person joins in. Before long, your group will form a shape without even thinking about it. Your group will probably form a circle.

Try and notice this the next time it happens—either to you or in another group.

Why do you think people naturally form a circle when talking?

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Problem 1 Pythagorean Theorem Connections

Recall that a circle is the set of points on a plane equidistant from a given point .

If this circle is drawn on a coordinate plane, with its center point located at the origin and a point (x, y) on the circle, it is possible to write an algebraic equation for the circle .

1. Label the sides of the right triangle formed . Then, use the Pythagorean Theorem to solve for r2 .

(0, 0)

r

(x, y)

(x, 0)

Because your equation is true for

every point on the circle, it can be used as the equation

of a circle centered at the origin.

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13.2 Deriving the Equation for a Circle 975

2. Next, consider a circle with its center located at point (h, k) and a point (x, y) on the circle . Label the sides of the right triangle formed . Then, use the Pythagorean Theorem to solve for r2 .

(h, k)

r

(x, y)

3. How does the equation you wrote in Question 2 also describe a circle with its center point at the origin? Explain your reasoning .

The standard form of the equation of a circle centered at (h, k) with radius r can be expressed as

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

4. Write an equation for:

a. a circle with center at the origin and r 5 8 .

b. a circle with center (3, 25) and r 5 6 .

Pay attention to the form of the

equation. The coefficients of both

x and y are one.

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c. circle P .

x8 106

2

6

8

10

�2�2

42�4

�4

�6

�6

�8�10

�8

�10

y

4

P

0

d. circle Q .

x8 106

2

6

8

10

�2�2

42�4

�4

�6

�6

�8�10

�8

�10

y

4

Q0

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Problem 2 Transforming equations to Identify Key Characteristics

By expanding the binomial terms of the standard form of a circle, (x 2 h)2 1 (y 2 k)2 5 r2, the equation can be written as

x2 2 2hx 1 h2 1 y2 2 2ky 1 k2 5 r2, or

x2 1 y2 2 2hx 2 2ky 1 h2 1 k2 5 r2

The equation for a circle in general form is Ax2 1 Cy2 1 Dx 1 Ey 1 F 5 0, where A, C, D, and E are constants, A 5 C, and x fi y .

In order to identify the center and the radius of a circle written in general form, it is necessary to rewrite the equation in standard form . A mathematical process called completing the square is often necessary to rewrite the equation in standard form .

You can rewrite x2 1 y2 2 4x 2 6y 1 9 5 0 in standard form by completing the square .

First, use an algebraic transformation to remove the constant term from the variable expression . Write the resulting equation grouping the x-terms together and the y-terms together using sets of parentheses .

(x2 2 4x) 1 (y2 2 6y) 5 29

Next, complete the square within each parenthesis . To do this, examine the first two terms of each quadratic expression . Determine what the constant term would be if each expression were a perfect square trinomial . Add those constant terms to each side of the equation . Write the resulting equation .

(x2 2 4x 1 4) 1 (y2 2 6y 1 9) 5 29 1 4 1 9

Finally, factor the left side of the equation, which should be a perfect square trinomial . Write the resulting equation .

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

1. Identify the center point and radius of the circle described by the equation in the worked example .

Remember, h and k are constants;

they represent the center of the circle.

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2. Consider the general form of the equation 2x2 1 2y2 2 x 1 4y 1 2 5 0 .

a. How do the A and B values in this equation differ from the equation in the worked example? How can you tell this equation represents a circle?

b. How does this general form of a circle equation change your strategy to rewrite it in standard form?

c. Rewrite the equation in standard form .

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3. Determine if each equation represents a circle . If so, describe the location of the center and the radius .

a. x2 1 y2 2 2x 1 4y 1 4 5 0

b. x2 1 2y2 2 x 1 10y 1 25 5 0

c. 2x2 1 2y2 2 5x 1 8y 1 10 5 0

d. x2 1 y2 2 10x 1 12y 1 51 5 0

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Problem 3 Match game

Cut out each circle equation . Then tape each equation into the table on the following page with its corresponding center and radius .

A . (x)2 1 (y 2 2)2 1 5 5 10 B . (x 1 1)2 1 (y 2 2)2 5 25

C . (x 2 1)2 1 (y 1 2)2 5 10 D . x2 1 y2 2 2y 1 4 5 100

E . (x 2 1)2 1 (y 2 2)2 5 25 F . x2 1 y2 2 2x 1 4y 5 0

G . x2 1 y2 2 2x 2 4y 2 95 5 0 H . x2 1 y2 1 2x 2 4y 5 5

I . (x 2 1)2 1 (y 1 2)2 5 25 J . (x)2 1 (y 2 2)2 5 10

K . (x 1 1)2 1 (y 2 2)2 5 100 L . (x 2 1)2 1 (y 2 2)2 5 5

M . x2 1 y2 2 x 2 4y 2 5 5 0 N . (x 2 1)2 1 (y 1 2)2 5 100

O . (x 1 1)2 1 (y 2 2)2 5 5 P . x2 1 y2 2 4y 2 21 5 0

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Cir

cle

Eq

uati

ons

Cen

ter

at (0

, 2)

Cen

ter

at (1

, 2)

Cen

ter

at (1

, 22)

Cen

ter

at (2

1, 2

2)

Rad

ius

at 5

Rad

ius

at 1

0

Rad

ius

at √

__

5

Rad

ius

at √

___

10

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Problem 4 Twice (or Thrice) As nice

1. Circle P is represented by the equation (x 2 4)2 1 (y 1 1)2 5 36 .

a. Determine the equation of a circle that has the same center as circle P but whose circumference is twice that of circle P .

b. Determine the equation of a circle that has the same center as circle P but whose circumference is three times that of circle P .

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c. Determine the equation of a circle that has the same center as circle P but whose area is twice that of circle P .

d. Determine the equation of a circle that has the same center as circle P but whose area is three times that of circle P .

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2. Complete the table . Then, identify any pattern(s) you notice .

Equation of Circle P r 5 6 (x 2 4)2 1 (y 1 1)2 5 36

2 Times the Circumference of Circle P




n Times the Circumference of Circle P

3. Complete the table . Then, identify any pattern(s) you notice .

Equation of Circle P r 56 (x 2 4)2 1 (y 1 1)2 5 36

2 Times the Area of Circle P




n Times the Area of Circle P

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4. Circle F is represented by the equation x2 1 (y 2 3)2 5 25 .

a. Determine the equation of a circle that has the same center as circle F but whose circumference is 10 times that of circle F .

b. Determine the equation of a circle that has the same center as circle F but whose area is 10 times that of circle F .


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LeArnIng gOALS


• Use the Pythagorean Theorem to determine if a point lies on a circle on the coordinate plane given the circle’s center at the origin, the radius of the circle, and the coordinates of the point .

• Use the Pythagorean Theorem to determine if a point lies on a circle on the coordinate plane given the circle’s center not at the origin, the radius of the circle, and the coordinates of the point .

• Use rigid motion to transform a circle about the coordinate plane to determine if a point lies on a circle’s image given the pre-image’s center, radius, and the coordinates of the point .

• Determine the coordinates of a point that lies on a circle given the location of the center point and the radius of the circle .

• Use the Pythagorean Theorem to determine the coordinates of a point that lies on a circle .

Is That Point on the Circle?Determining Points on a Circle

Beginning in about the 1970s, people in many different countries began reporting formations formed in fields, created by flattening down crops in certain ways.

These came to be known as crop circles.

At first, people thought that weather or even aliens were creating these formations, but it turned out that groups of people would go into fields at night and create the crop circles themselves. Many of these formations are extremely complex and beautiful.

13.3

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Problem 1 Identifying Points on a Circle

In this problem, you will continue to explore the connection between the Pythagorean Theorem and circles .

Consider circle A with its center point located at the origin and point P (5, 0) on the circle as shown .

x8 106

2

6

8

10

�2�2

42�4

�4

�6

�6

�8�10

�8

�10

y

4

P (5, 0)A (0, 0)

0

1. Use the axes to plot three additional points on circle A and label the coordinates for each point .

There are an infinite number of points located on circle A . To determine the coordinates of other points located on circle A, you can use the Pythagorean Theorem .

2. Use the Pythagorean Theorem to determine if point B (4, 3) lies on circle A, and then explain your reasoning .

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13.3 Determining Points on a Circle 991

3. Consider circle D centered at the origin with a diameter of 16 units as shown . Use the Pythagorean Theorem to determine if point H (5, √

___ 38 ) lies on circle D, and then explain

your reasoning .

x8 106

2

6

8

10

�2�2

42�4

�4

�6

�6

�8�10

�8

�10

y

4

P (8, 0)D (0, 0)

H (5, �38)

0

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4. Consider circle E centered at the origin with a diameter of 34 units as shown .

a. Verify that point J (8, 15) lies on circle E . Explain your reasoning .

x16 2012

4

12

16

20

�4�4

84�8

�8

�12

�12

�16�20

�16

�20

y

8

J (8, 15)

0

b. Use symmetry to determine 3 more points on circle E .

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5. Consider circle G with its center point located at (3, 0) and point M (3, 2) on the circle . Determine whether each point lies on circle G, and then explain your reasoning .

J (4 .5, √__

3 ___ 2 )

P (4, √__

3 )

6. Consider Elizabeth’s statement about additional points on circle G .

elizabeth

So, only one of those points was located on circle G. I can use that point and what I know about symmetry in a circle to identify other points on circle G.

Justify Elizabeth’s reasoning and identify additional points on circle G .

x4 53

1

3

4

5

�1�1

21�2

�2

�3

�3

�4�5

�4

�5

y

2M (3, 2)

0

G (3, 0)

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Problem 2 Oh no! The Origin Is nOT at the Center!

Scientists are working on a new circular mirror for an orbiting telescope . They must install eight mounting brackets around the mirror . Once ready, the mirror will be launched into space and astronauts will be required to install the mirror with bolts .

1. Circle C with its center point located at (2, 5) and a radius of 3 units represents the telescope’s new mirror . The location of one of the mounting brackets is shown .

x4 5 6 73

1

�1

3

4

5

6

9

�1 21�2�3

y

C (2, 5)

0

a. Determine the coordinates of the mounting bracket shown .

b. Use symmetry to identify the locations of the other seven mounting brackets on the mirror .

There is no room for error on

this job. So use exact values, not

approximations.

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2. Sal listens to two radio stations . Station WPOP plays “Top 40” hits . Station WREQ, located 50 miles due east of Station WPOP, plays oldies . Each radio station has a circular broadcast range, and Sal lives on the very edge of both of the stations’ ranges . Sal’s house is 24 miles north and 32 miles east of Station WPOP .

a. Let Station WPOP be located at the origin . Plot the location of each station and Sal’s house . Then, graph each station’s broadcast range .

b. Use symmetry to describe the location of the other point that is on the very edge of both of the radio stations’ broadcast ranges .

c. What area is covered by each station’s broadcast range? Show your work .

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d. How many square miles are covered by both radio stations’ broadcast ranges . Show your work and explain your reasoning .


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LeArnIng gOAL


• Derive the equation of a parabola given the focus and the directrix .

KeY TerMS

• locus of points• parabola• focus of a

parabola• directrix of a

parabola• general form of a

parabola

• standard form of a parabola

• axis of symmetry• vertex of a

parabola• concavity

The Parabolaequation of a Parabola

The Golden Gate Bridge, which crosses the entrance to the San Francisco Bay from the Pacific Ocean, is one of the engineering marvels of the United States with its

parabola shaped suspension cables. When it was built, its 4200 foot (1280 meter) suspension span was the longest in the world. Almost 2 billion vehicles have crossed the Golden Gate Bridge since its opening in 1937.

13.4

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Problem 1 Parabolas as a Set of Points

You previously studied parabolas as quadratic functions . You analyzed equations and graphed parabolas based on the position of the vertex and additional points determined by using x-values on either side of the axis of symmetry .

In this lesson, you will explore a parabola as a locus of points . A locus of points is a set of points that share a property .

A parabola is the set of all points in a plane that are equidistant from a fixed point and a fixed line . The focus of a parabola is the fixed point . The directrix of a parabola is the fixed line .

1. The center of the circle is represented by point X . The radius of the smallest circle with center X is 1 unit with the radius of each successive circle increasing by 1 unit .

x

y

focus

1

1

–1–2–3–4–5–6–7

23456789

1011

–1–2–3–4–5–6–7–8–9 2 3 4 5 6 7 8 9

x

directrix

a. What is the distance from the star to point X, the focus? How do you know?

b. What is the distance from the star to the given line, the directrix? How do you know?

c. What is the relationship between the distance from the focus to the star and the distance from the directrix to the star?

d. Graph 8 additional points such that the distance from a point to the focus is equal to the distance from the point to the directrix .

e. Draw a parabola by connecting the points with a smooth curve .

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13.4 Equation of a Parabola 999

Problem 2 Equations of Parabolas

1. A parabola is defined such that all points on the parabola are equidistant from the point (0, 2) and the line y 5 22. One point on the parabola is labeled as (x, y). Determine the equation of the parabola by completing the steps.

x42

2

4

6

8

–8–2

–2 86–4

–4

–6

–8

–6

y

y = –2

d1 = d2

d1

d2

(0, 2)(x, y)

0

a. Let d1 represent the distance from (x, y) to (0, 2). Write an equation using the Distance Formula to represent d1. Simplify the equation.

b. Let d2 represent the distance from (x, y) to the line y 5 22. Write an equation using the Distance Formula to represent d2. Simplify the equation.

c. What do you know about the relationship between d1 and d2?

d. Write an equation for the parabola using Question 1, parts (a) through (c). Simplify the equation so that one side of the equation is x2.

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The general form of a parabola centered at the origin is an equation of the form Ax2 1 Dy 5 0 or By2 1 Cx 5 0 .

The standard form of a parabola centered at the origin is an equation of the form x2 5 4py or y2 5 4px, where p represents the distance from the vertex to the focus .

2. Write the equation of the parabola from Question 1 in general form and in standard form .

3. What are the coordinates for the x-intercept(s) of the parabola?

4. What are the coordinates for the y-intercept(s) of the parabola?

5. How many points on the parabola have an x-coordinate of 4? Calculate the coordinates of each point .

6. How many points on the parabola have an x-coordinate of 24? Calculate the coordinates of each point .

7. How many points on the parabola have a y-coordinate of 4 .5? Calculate the coordinates of each point .

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8. How many points on the parabola have a y-coordinate of 24 .5? Calculate the coordinates of each point .

9. Sketch the parabola on the grid shown using the points from Questions 3 through 8 .

10. Describe the symmetry of the parabola .

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Problem 3 Key Characteristics of a Parabola

1. How many lines of symmetry exist for a parabola?

The axis of symmetry of a parabola is a line that passes through the parabola and divides the parabola into two symmetrical parts that are mirror images of each other . The vertex of a parabola is a maximum or minimum point on the curve .

The concavity of a parabola describes the orientation of the curvature of the parabola . A parabola can be concave up, concave down, concave right, or concave left, as shown .

x

y

concave up

x

concave right

y

x

concave down

y

x

concave left

y

2. Consider the parabola represented by the equation y2 5 2x .

a. What are the coordinates for the x-intercept(s) of the parabola?

b. What are the coordinates for the y-intercept(s) of the parabola?

c. How many points on the parabola have an x-coordinate of 8? Calculate the coordinates of each point .

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d. Sketch the parabola using the coordinates from parts (a) through (c) .

e. Is the axis of symmetry along the x-axis or the y-axis? What is the equation for the axis of symmetry?

f. What is the relationship between the axis of symmetry and the parabola’s equation?

g. What are the coordinates of the vertex?

h. Describe the concavity of the parabola .

i. How is the concavity of the parabola related to the orientation of the parabola?

3. Consider the parabola represented by the equation x2 5 9y .

a. What are the coordinates for the x-intercept(s) of the parabola?

b. What are the coordinates for the y-intercept(s) of the parabola?

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c. How many points on the parabola have a y-coordinate of 4? Calculate the coordinates of each point .

d. Sketch the parabola using the coordinates from parts (a) through (c) .

e. Is the axis of symmetry along the x-axis or the y-axis? What is the equation for the axis of symmetry?

f. What is the relationship between the axis of symmetry and the parabola’s equation?

g. What are the coordinates of the vertex?

h. Describe the concavity of the parabola .

i. How is the concavity of the parabola related to the orientation of the parabola?

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4. The standard form of a parabola with its vertex at the origin is an equation of the form x2 5 4py or y2 5 4px .

a. What is the standard form of a parabola that has its axis of symmetry along the y-axis?

b. What is the standard form of a parabola that has its axis of symmetry along the x-axis?

c. What is the equation of the axis of symmetry for a parabola with a vertical orientation?

d. What is the equation of the axis of symmetry for a parabola with a horizontal orientation?

e. Is the concavity of a parabola with a vertical orientation described as concave up/down or concave left/right?

f. Is the concavity of a parabola with a horizontal orientation described as concave up/down or concave left/right?


a. Does the parabola have a horizontal or vertical orientation? How can you tell?

b. Is the axis of symmetry along the x-axis or the y-axis? What is the equation for the axis of symmetry?

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c. Describe the concavity of the parabola . How can you tell?

d. What are the coordinates for the x-intercept(s) of the parabola?

e. What are the coordinates for the y-intercept(s) of the parabola?

f. In order to graph the parabola, more information is needed . What are the points on the parabola that have a y-coordinate of 3?

g. Sketch the parabola .

h. What are the coordinates of the vertex?

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Problem 4 Making Sense of the Constant p

Consider the sketch of the parabola . Let p represent the distance from the vertex to the focus .

x

y

directrix

vertexfocus

Q( x, y)

R

1. Label the vertex, the focus, and the directrix .

2. Label the vertex with its coordinates .

3. Label the axis of symmetry with the equation for its line .

4. Label the distance, p, on the graph .

5. Label the focus with its coordinates .

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6. Label the directrix with the equation for its line . Explain your reasoning .

7. What is the relationship between the directrix and the axis of symmetry?

8. What is the distance from the focus to the directrix? Label this distance on the graph .

9. Write an equation for the parabola .

a. Let d1 represent the distance from point Q on the parabola to the focus . Write an equation using the Distance Formula to represent d1 . Simplify the equation .

b. Line segment QR represents the perpendicular distance from point Q on the parabola to the directrix . Draw in line segment QR . What are the coordinates of point R? Label the coordinates on the graph .

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c. Let d2 represent the distance from point Q to point R . Write an equation using the Distance Formula to represent d2 . Simplify the equation .

d. What do you know about the relationship between d1 and d2?

e. Write an equation for the parabola using parts (a) through (d) . Simplify the equation so that one side of the equation is the squared term .

f. What is the significance of the equation in Question 9, part (e)?

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Problem 5 Using the Constant p to graph a Parabola

1. Consider the parabola represented by the equation y2 5 20x .

a. Identify the vertex of the parabola .

b. Identify the axis of symmetry .

c. Determine the value of p . Show your work .

d. Determine the coordinates of the focus . Justify your reasoning .

e. Determine the equation of the directrix . Justify your reasoning .

f. Graph the parabola using the information from parts (a) through (e) .

g. Describe the concavity of the parabola . Justify your reasoning .

What does the form of the

equation tell you? What do you think

the graph will look like?

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a. Identify the vertex of the parabola .

b. Identify the axis of symmetry .

c. Determine the value of p . Show your work .

d. Determine the coordinates of the focus . Justify your reasoning .

e. Determine the equation of the directrix . Justify your reasoning .

f. Graph the parabola using the information from parts (a) through (e) .

g. Describe the concavity of the parabola . Justify your reasoning .

How does this equation

compare to the one in Question 1?

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3. Sketch the parabola x2 5 28y . Label the vertex, the axis of symmetry, the focus, and the directrix . Determine the value of p and the concavity of the parabola .

What’s your prediction for the

shape of this parabola?

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4. Sketch the parabola y2 5 210x . Label the vertex, the axis of symmetry, the focus, and the directrix . Determine the value of p and the concavity of the parabola .

5. Analyze each equation and its corresponding graph in Questions 1 through 4 . What do you notice about the sign of the constant p and the concavity of the parabolas?

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Problem 6 Applications of Parabolas

1. The main cables of a suspension bridge are parabolic . The parabolic shape allows the cables to bear the weight of the bridge evenly . The distance between the towers is 900 feet and the height of each tower is about 75 feet .

900 ft

75 ft

Write an equation for the parabola that represents the cable between the two towers .

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2. The cross section of a satellite dish is a parabola . The satellite dish is 5 feet wide at its opening and 1 foot deep . The receiver of the satellite dish should be placed at the focus of the parabola . How far should the receiver be placed from the vertex of the satellite dish?

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3. Many carnivals and amusement parks have mirrors that are parabolic . When you look at your reflection in a parabolic mirror, your image appears distorted and makes you look taller or shorter depending on the shape of the mirror . The focal length of a mirror is the distance from the vertex to the focus of the mirror . Consider a mirror that is 72 inches tall with a vertex that is concave 6 inches from the top and bottom edges of the mirror . What is the focal length of the mirror?

6 inches

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Talk the Talk

Determine the following for each parabola:

• The vertex

• The axis of symmetry

• The value of p

• The focus

• The directrix

• The concavity

Then, graph the parabola .

1. x2 5 18y

2. y2 1 44x 5 0

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3. Complete the table .

Parabola Centered at Origin

Graph

y

xp

p

y

x

p p

Equation of Parabola

Orientation of Parabola

Axis of Symmetry

Coordinates of Vertex

Concavity

Coordinates of Focus

Equation of Directrix


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Scientists explore deep space by using large antennas to listen for distant radio waves. Parabolic antennas amplify faint signals by using the properties of

parabolas to focus them onto a receiver.

The worldwide Deep Space Network also helps to keep track of exploratory spacecraft like the two Voyager spacecrafts, which have nearly left our Solar System!

Learning goaL


• Solve problems using characteristics of parabolas.

13.5Simply ParabolicMore with Parabolas

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Problem 1 going the equidistance

In this activity, we will use the distance formula to determine the equation of points that are equidistant from a given point (the focus) and a given line (the directrix) where the vertex is a point other than the origin .

1. Consider the graph shown .

a. Determine an equation for all the points equidistant from the point (23, 25) and the line y 5 3 .

x86

2

6

8

10

�2�2

42�4

�4

�6

�6

�8�10

�8

�10

�12

y

4

(�3, �5)

d1 � d2

(x, y)

y � 3

d2

d1

0

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b. Complete the following table for the equation you determined in part (a) . Then, graph the equation with the directrix in the grid below . Calculate the value of p, and identify and label the directrix, the focus, the vertex, and the axis of symmetry .

x y

27

23

4

2. Consider the graph shown .

a. Determine the equation for all the points equidistant from the point (7, 5) and the line x 5 1 .

x86

2

6

8

10

10 12�2�2

42�4

�4

�6�8

y

12

14

4 (7, 5)

d1 � d2

(x, y)x � 1

d2

d1

0

13.5 More with Parabolas 1021

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b. Complete the following table for the equation you determined in part (a) . Then, graph the equation with the directrix in the grid . Calculate the value of p, and identify and label the directrix, the focus, the vertex, and the axis of symmetry .

x y

4

6

6

The standard forms of parabolas with vertex at (h, k) are (x 2 h)2 5 4p(y 2 k) and (y 2 k)2 5 4p(x 2 h) .

3. Rewrite the equations from Questions 1 and 2 in this form .

How do these equations

compare with the standard form equation of a circle: (x 2 h)2 1 (y 2 k)2 5 r2?

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4. Transform each equation into standard form and:

• Calculate the value of p, the vertex, the focus, the equation of the directrix, and the equation of the axis of symmetry.

• Sketch its graph with the focus and directrix.

• Determine if the parabola is concave up or down, left or right.

• Graph and label the parabola.

a. ( y 1 1)2 5 12(x 2 3)

b. y2 1 8y 1 8x 1 16 5 0


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c. 4x2 2 40x 1 48y 1 4 5 0

5. Write an equation in standard form for each parabola. Then, graph and label the parabola.

a. A parabola with a vertex at (3, 2) and a focus at (3, 4).

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b. A parabola with a vertex at (4, 1) and a directrix at x 5 2 .

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6. Complete the table .

Parabola

Graph

y

x

(h, k)p

p

y

x

(h, k)

p p

Equation of Parabola

(x 2 h)2 5 4p( y 2 k) ( y 2 k)2 5 4p(x 2 h)

Orientation of Parabola

Axis of Symmetry

Coordinates of Vertex

Coordinates of Focus

Equation of Directrix

Concavity

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Problem 2 Archway

The finish line of a 5K race is an archway of balloons . The archway is formed by two parabolas, one representing the top of the archway and one representing the bottom of the archway as shown .

9 ft

The width of the archway on the ground is 60 feet . The height of the top of the archway is 18 feet . The height of the bottom of the archway is 9 feet . The framework of the archway consists of vertical posts 10 feet apart with posts connecting the tops and bottoms of adjacent vertical posts . Calculate the sum of the lengths of the posts by answering each question .

1. Graph the archway on the coordinate plane . Let the x-axis represent the ground . Let the vertex of each arch lie on the y-axis .

2. Determine the coordinates of each point described . Then, label each point on the coordinate plane .

a. The vertex of the top of the archway .

b. The vertex of the bottom of the archway .

c. The points where the archway touches the ground .

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3. Calculate an equation for the parabola representing the top of the archway .

4. Calculate an equation for the parabola representing the bottom of the archway .

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5. What are the x-coordinates of each vertical post?

6. Determine the coordinates representing the endpoints of the five vertical posts . Label each in the coordinate plane .

7. Calculate the length of each vertical post .

8. Determine the coordinates representing the endpoints of the four non-vertical posts .

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9. Calculate the length of each non-vertical post .

10. What is the total length of all the posts in the archway?


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Chapter 13 Summary©

Car

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13

KeY TerMS

• locus of points (13 .4)• parabola (13 .4)• focus of a parabola (13 .4)• directrix of a parabola (13 .4)

• general form of a parabola (13 .4)

• standard form of a parabola (13 .4)

• axis of symmetry (13 .4)• vertex of a parabola (13 .4)• concavity (13 .4)

Classifying Polygons on the Coordinate PlaneYou can classify polygons inscribed in circles in the coordinate plane, using midpoints and theorems .

Example

nABC is inscribed in circle E with coordinates A (25, 0), B (5, 0), and C (3, 4) . Classify nABC by sides and by angles .

Slope of ___

AC : 25 2 3 _______ 0 2 4

5 28 ___ 24

5 2

Slope of ___

AB : 25 2 5 _______ 0 2 0

5 210 _____ 0 5 undefined

Slope of ___

BC : 5 2 3 ______ 0 2 4

5 2 ___ 24

5 2 1 __ 2

Since the slope of ___

AC and ___

BC are negative reciprocals, the segments are perpendicular and form a right angle . Therefore the triangle is a right triangle .

Length of ___

AC : √___________________

(25 2 3)2 1 (0 2 4)2 5 √____________

(28)2 1 (24)2 5 √________

64 1 16 5 √___

80

Length of ___

AB : √___________________

(25 2 5)2 1 (0 2 0)2 5 √______

(210)2 5 √____

100 5 10

Length of ___

BC : √_________________

(5 2 3)2 1 (0 2 4)2 5 √__________

22 1 (24)2 5 √_______

4 1 16 5 √___

20

Since all three sides have a different length, the triangle is scalene .

13.1

x8 106

2

6

8

10

�2�2

42�4

�4

�6

�6

�8�10

�8

�10

y

4

BEA

C

0

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Using the Standard Form of the equation of a CircleThe standard form of the equation of a circle is (x 2 h)2 1 (y 2 k)2 5 r2 where the center of the circle is the point (h, k) and the radius is of length r .

Example

A circle is expressed by the equation (x 2 2)2 1 (y 1 3)2 5 16 . So, the center of the circle is the point (2, 23) and the circle has a radius of 4 .

Transforming the general Form of the equation of a Circle into Standard FormThe general form of the equation of a circle is Ax2 1 Cy2 1 Dx 1 Ey 1 F 5 0 where A, C, D, and E are constants, A 5 C, and x fi y . A method called complete the square can be used to transform the equation of a circle in general form into standard form .

Example

x2 1 y2 1 10x 1 4y 2 12 5 0

(x2 1 10x) 1 (y2 1 4y) 5 12 Rewrite the equation equal to the constant .

x2 1 10x 1 25 1 y2 1 4y 1 4 5 12 1 25 1 4 Complete the square in each parentheses .

(x 1 5)2 1 (y 1 2)2 5 41 Factor each trinomial .

Determining if a Point Is on a CircleGiven the coordinates of the center of a circle, and a point on the circle, the Pythagorean Theorem can be used to determine if another given point is on the circle .

Example

Given circle A with center point (25, 2), point B (22, 2) on the circle, and a point X (23, 0) . Using the Pythagorean Theorem the length of ___

AX is √__

8 . Since AX is not equal to 3, point X is not on the circle .

c2 5 22 1 22

c2 5 4 1 4

c2 5 8

c 5 √__

8

13.2

13.2

13.3

x

1

3

4

5

�1�2�3�4�5�6�7�8

y

6

7

8

1

2

X

A B

0

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Chapter 13 Summary 1033

Determining the equation of a ParabolaA parabola is the set of all points in a plane equidistant from a fixed point, called the focus, and a fixed line, called the directrix .

Example

If a parabola has a focus at (2, 0) and a directrix of x 5 22, the equation of the parabola is y2 5 8x .

Distance from focus to a point on the parabola: √

_________________ (2 2 x)2 1 (0 2 y)2

Distance from directrix to the same point on the parabola: √

__________________ (22 2 x)2 1 (y 2 y)2

√_________________

(2 2 x)2 1 (0 2 y)2 5 √__________________

(22 2 x)2 1 (y 2 y)2

√______________

(2 2 x)2 1 (2y)2 5 √_________

(22 2 x)2

(2 2 x)2 1 (2y)2 5 (22 2 x)2

4 2 4x 1 x2 1 y2 5 4 1 4x 1 x2

y2 5 8x

Determining Information About a Parabola with a Vertex at the OriginYou can determine information about a parabola with a vertex at the origin from its equation and graph .

Examples

When the equation of a parabola is in the form x2 5 4py, then the parabola has the following characteristics:

• vertical and concave up or down

• axis of symmetry is y–axis, or x 5 0

• vertex at (0, 0)

• has a focus at (0, p)

• directrix is at y 5 2p

When the equation of a parabola is in the form y2 5 4px, then the parabola has the following characteristics:

• horizontal and concave left or right

• axis of symmetry is x–axis, or y 5 0

• vertex at (0, 0)

• has a focus at (p, 0)

• directrix is at x 5 2p

13.4

x

1

3

4

5

�1�2

(�2, y) (x, y)

�3�4�5 1 2

(2, 0)

3 4 5

y

�1

�2

�3

�4

�5

2

0

13.4

y

xp

p

y

x

p p

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Determining Information About a Parabola with a Vertex not at the OriginYou can determine information about a parabola with a vertex not at the origin from its equation and graph .

Examples

When the equation of a parabola is in the form (x 2 h)2 5 4p(y 2 k), then the parabola has the following characteristics:

• vertical and concave up or down

• axis of symmetry is the line x 5 h

• vertex at (h, k)

• has a focus at (h, k 1 p)

• directrix is at y 5 k 2 p

When the equation of a parabola is in the form (y 2 k)2 5 4p(x 2 h), then the parabola has the following characteristics:

• horizontal and concave left or right

• axis of symmetry is the line y 5 k

• vertex at (h, k)

• has a focus at (h 1 p, k)

• directrix is at x 5 h 2 p

13.5

y

x

(h, k)p

p

y

x

(h, k)

p p

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