9.1 apply the distance and midpoint formulas · Example 3: Use the given distance d between the two...

9.1 apply the distance and midpoint formulas

DISTANCE FORMULA

MIDPOINT FORMULA

To find the midpoint between two points 2211

,, yxandyx , we

Example 1: Find the distance between the two points. Then, find the midpoint of the line segment joining the two points. Round all non-integer answers to two decimal places.

a. 9,3.24,5.1 and

Distance: ______________ Midpoint: ,

What are the differences between:

Scalene Triangle

Isosceles Triangle

Equilateral Triangle

Example 2: The vertices of a triangle are given. Classify the triangle as scalene, isosceles, or equilateral.

1,06,21,4 It may be helpful to label the points!

Distance: __________

Type of Triangle:


___________________


Example 3: Use the given distance d between the two points to find the value of x. 8912,7,2 dx

Example 4: Write an equation for the perpendicular bisector of the line segment joining the two points:

6,54,3 DC Remember—to make an equation for a line, we need a point and a slope!

Point: Find the midpoint of the segment.

Slope: Find the slope of the segment, then apply the opposite reciprocal.

Place the Point and Slope into 11

xxmyy

9.2 Graph and Write Equations for Parabolas ( 2x )

1.

pyxFS

yxEquation

4:..

8:

2

2

Step 1: Get the squared variable or 2 alone. yx 82

Step 2: Identify the vertex. If there are no numbers being added to or subtracted

from the variables, you can assume the vertex is at 0,0 . Vertex ,

Step 3: Identify the p value. To do so, set whatever is in front of the non-squared variable equal to 4p. Solve for p.

p48

Step 4: Determine the direction of opening. If p is positive, the graph will open up. If p is negative, it will open down.

Opens _________

Step 5: Find the focus. The focus is a point found by adding p to the y value of the vertex. It is always located within the arc of the parabola. (always add)

Focus ,

Step 6: Find the directrix. The directrix is a horizontal line .____y To find the

number, subtract the p value from the y value of the vertex. (always subtract) _________y

Step 7: Determine the axis of symmetry. The axis of symmetry will be a vertical line .____x It will be equal to the x-coordinate of the vertex.

_________x

Step 8: Graph this information along with two additional points to complete the graph.

x y

2. 036: 2 yxEquation

x y

3.

kyphxFS

yxEquation

4:..

382:

2

2

x y

9.2 Day 2: Graph and Write Equations for Parabolas ( 2y )

1.

pxyFS

xyEquation

4:..

32:

2

2

Step 1: Get the squared variable or 2 alone. xy 322

Step 2: Identify the vertex. If there are no numbers being added to or

subtracted from the variables, you can assume the vertex is at 0,0 . Vertex ,

Step 3: Identify the p value. To do so, set whatever is in front of the non-squared variable equal to 4p. Solve for p.

p432

Step 4: Determine the direction of opening. If p is positive, the graph will open right. If p is negative, the graph will open left.

Opens _________

Step 5: Find the focus. The focus is a point found by adding p to the x value of the vertex.

Focus ,

Step 6: Find the directrix. The directrix is a vertical line .____x To find

the number, subtract the p value from the x value of the vertex.

_________x

Step 7: Determine the axis of symmetry. The axis of symmetry will be a horizontal line .____y It will be equal to the y-coordinate of the vertex.

_________y

Step 8: Graph this information along with two additional points to complete the graph.

x y

2. xyEquation 243: 2

x y

3.

hxpkyFS

xyEquation

4:..

4203:

2

2

x y

9.2 Day 3 (continued)

Example 1: Write the standard form of the equation of the parabola with the given focus 4,0 and vertex at 0,0 .

Graph the given information to help you determine whether this would

be an pxyorpyx 44 22 equation.

Determine the p value.

Place the p value into the standard form and simplify the equation.

Example 2: Write the standard form of the equation of the parabola with the given focus

0,

2

5and vertex at 0,0 .





Example 3: Write the standard form of the equation of the parabola with the given directrix 5x and vertex at 0,0 .





9.3 Graph and Write equations of circles

Notice that 2x and 2y are on the left side of the equation, joined by a plus sign, and that

the 2x is listed first.

Also notice that the 2x and 2y are alone (no numbers in front).

Example 1: 4055: 22 xyEquation

Step 1: Get the squared variables or 2 on the left side of the equal sign

with the 2x listed first.

4055 22 xy

Step 2: Identify the center. If there are no numbers being added to or

subtracted from the variables, you can assume the center is at 0,0 . Center ,

Step 3: Identify the radius. To do so, take the square root of the number on the right side of the equal sign. (no decimals)

r = ___________

Step 4: Graph the center along with several points, as determined by the radius.

x y

What is the difference between the standard equation of a circle and the standard equation of the parabolas?

Example 2:

6451:

:..

22

222

xyEquation

rkyhxFS

x y

Example 3: Write the standard form of the equation of the circle with the given radius 25r and whose center is at

12,7 .

222:.. rkyhxFS

Equation: __________________________________

Example 4: Write the standard form of the equation of the circle that passes through the given point 14,8 and

whose center is the origin.

222:.. rkyhxFS

Equation: __________________________________

9.4 Graph and Write Equations OF ELLIPSES

An ellipse is the set of all points P in a plane such that the sum of the distances between P and two fixed points, called the foci, is a constant.

Standard form of an ellipse with horizontal major axis.

1

2

2

2

2

b

ky

a

hx

In each problem, we will graph the center, the

vertices, the co-vertices, and the foci.

1. 1:..

100254:

2

2

2

2

22

b

ky

a

hxFS

yxEquation

100254 22 yx

Step 1: Get the right side of the equation equal to 1.

Step 2: Identify the center. If there are no numbers being added to or subtracted from

the variables, you can assume the center is at 0,0 . If there are numbers being added

or subtracted, take the opposite of each as the x and y coordinates of the center.

Center ,

Step 3: Determine which denominator contains the 2a . The

2a is the larger

denominator. The purpose of the a value is to help us create the major axis and the

vertices that are on each end of the major axis. Today, the 2a is under the x variable,

which means that the major axis will be horizontal like the x-axis. Find the 2a and

determine how the major axis will look. Take the square root of 2a to determine a.

Major axis: ____________________

2a = _________

a = _________

Step 4: The vertices are points at either end of the major axis. If the 2a is under the x-

variable, the vertices can be found by adding a to the x value of the center.

Vertices , ,

Step 5: Determine which denominator contains the 2b . It will always be the smaller of

the two denominators. It will help us determine the minor axis and the co-vertices that

are on each end. Today, the 2b is under the y variable, which means that the minor

axis will be vertical like the y -axis. Find the 2b and use it to determine .b

2b = _________

b = _________

Step 6: The co-vertices are points at either end of the minor axis. Since the 2b is under

the y -variable, the vertices can be found by adding b to the y value of the center.

Co-

vertices , ,

Step 7: Determine the foci. Foci are two points located on the major axis. Because, today, the major axis is horizontal like the x-axis, we will find these two points by adding c to the x value of the center. C is determined by the

formula: 222 bac

58.421

21

425

25

2

2

222

orc

c

c

c

Foci , ,

Step 8: Graph the center, vertices, a line along the major axis, co-vertices, a line along the minor axis, and foci. Use the major and minor axis lines to help you sketch the ellipse.

2. 1:..

144169:

2

2

2

2

22

b

ky

a

hxFS

yxEquation

Center , Major axis:_______________

a = _________

Vertices , ,

b = _________

Co-vertices , ,

c = _________

Foci , ,

3. 3614522 yx

4. 832122 yx


a = _________

Vertices , ,


Vertices , ,

b = _________

Co-vertices , ,

Co-vertices , ,

c = _________

Foci , ,

Foci , ,

9.4 Graph and Write Equations OF ELLIPSES (day 2)

An ellipse is the set of all points P in a plane such that the sum of the distances between P and two fixed points, called the foci, is a constant.

Standard form of an ellipse with vertical major axis.

1

2

2

2

2

b

hx

a

ky

In each problem, we will graph the center, the

vertices, the co-vertices, and the foci.

1. 1:..

100425:

2

2

2

2

22

b

hx

a

kyFS

yxEquation

100425 22 yx


Step 2: Identify the center. If there are no numbers being added to or subtracted from

the variables, you can assume the center is at 0,0 . If there are numbers being added

or subtracted, take the opposite of each as the x and y coordinates of the center.

Center ,


2a is the larger

denominator. The purpose of the a value is to help us create the major axis and the

vertices that are on each end of the major axis. Today, the 2a is under the y variable,

which means that the major axis will be vertical like the y -axis. Find the 2a and

determine the how the major axis will look. Take the square root of 2a to determine a.

Major axis: ____________________

2a = _________

a = _________

Step 4: The vertices are points at either end of the major axis. If the 2a is under the y -

variable, the vertices can be found by adding a to the y value of the center.

Vertices , ,

Step 5: Determine which denominator contains the 2b . It will always be the smaller of

the two denominators. It will help us determine the minor axis and the co-vertices that

are on each end. Today, the 2b is under the x variable, which means that the minor

axis will be horizontal like the x -axis. Find the 2b and use it to determine .b

2b = _________

b = _________

Step 6: The co-vertices are points at either end of the minor axis. Since the 2b is under

the x -variable, the vertices can be found by adding b to the x value of the center.

Co-

vertices , ,

Step 7: Determine the foci. Foci are two points located on the major axis. Because, today, the major axis is vertical like the y-axis, we will find these two

points by adding c to the y value of the center. C is determined by the formula: 222 bac

58.421

21

425

25

2

2

222

orc

c

c

c

Foci , ,

Step 8: Graph the center, vertices, a line along the major axis, co-vertices, a line along the minor axis, and foci. Use the major and minor axis lines to help you sketch the ellipse.

2.

136

3

4

222

yx

3. 949 22 yx


a = _________

Vertices , ,


Vertices , ,

b = _________

Co-vertices , ,

Co-vertices , ,

c = _________

Foci , ,

Foci , ,

9.5 Graph and Write Equations OF hyperbolas

A hyperbola is the set of all point P such that the difference of the distances between P and two fixed points, again called the foci, is a constant.

Standard form of a hyperbola with horizontal transverse axis.

1

2

2

2

2

b

ky

a

hx

In each problem, we will graph the center, the vertices, a

line through the transverse axis, the conjugate axis, the foci, and the asymptotes.

1. 1:..

100425:

2

2

2

2

22

b

ky

a

hxFS

yxEquation

100425 22 yx


Step 2: Identify the center and graph it. Center ,


2a is always underneath

the positively squared variable. The purpose of the a value is to help us create the

transverse axis and the vertices that are on each end of the transverse axis. Today, the 2a is under the x variable, which means that the transverse axis will be horizontal like

the x-axis. Find the 2a and take the square root of

2a to determine a.

2a = _________

a = _________

Step 4: The vertices are points at either end of the transverse axis. If the 2a is under

the x-variable, the vertices can be found by adding a to the x value of the center.

Draw the vertices and connect them to create the transverse axis.

Vertices , ,

Step 5: Determine which denominator contains the2b . It will always be below the

negatively square variable. Today, the 2b is under the y variable. Find the

2b and use

it to determine .b

2b = _________

b = _________

Step 6: The b value will help us create the two points that connect the conjugate

axis. Since the 2b is under the y -variable, the points can be found by adding b to

the y value of the center.

Conjugate axis points:

, ,

Step 7: Determine the foci. Foci are two points located beyond the transverse axis. Because, today, the transverse axis is horizontal like the x-axis, we will find these two points by adding c to the x value of the center. C is determined by the

formula: 222 bac

Foci , ,

Step 8: Determine the asymptotes. What’s unique about hyperbolas is the

presence of asymptotes. For today’s hyperbolas, in which the 2a is under the x-

variable, the equation will be a

b . We will start from the center of the hyperbola and

apply this equation as two different slopes: a

b and

a

b . We will draw the

asymptotes as dashed lines. They will help us create the shape of the branches.

2

5

2

5

2

5

and

a

b

Step 9: Graph the center, the vertices, a line through the transverse axis, the conjugate axis, the foci, and the asymptotes. Then sketch the shape of the branches.

2. 196494 22 yx

Center ,

a = _________

Vertices , ,

b = _________

Conjugate axis points , ,

c = _________

Foci , ,

Asymptotes: _________

3. 364 22 yx

Center ,

a = _________ Vertices , ,

b = _________ Conjugate axis points , ,

c = _________ Foci , ,


9.5 Graph and Write Equations OF hyperbolas (Day 2)

Standard form of a hyperbola with vertical transverse axis.

1

2

2

2

2

b

hx

a

ky

In each problem, we will graph the center, the vertices, a

line through the transverse axis, the conjugate axis, the foci, and the asymptotes.

1. 64164: 22 xyEquation

64164 22 xy


Step 2: Identify the center and graph it. Center ,

Step 3: Determine which denominator contains the 2a . Today, the

2a is under the y

variable, which means that the transverse axis will be vertical like the y-axis.

2a = _________

a = _________

Step 4: The vertices are points at either end of the transverse axis. If the 2a is under

the y-variable, the vertices can be found by adding a to the y value of the center.

Vertices , ,

Step 5: Determine which denominator contains the2b . It will always be below the

negatively square variable. Today, the 2b is under the x variable.

2b = _________

b = _________

Step 6: The b value will help us create the two points that connect the conjugate

axis. Since the 2b is under the x -variable, the points can be found by adding b to

the x value of the center.

Conjugate axis points:

, ,

Step 7: Determine the foci. Because, today, the transverse axis is vertical like the y-axis, we will find these two points by adding c to the y value of the center. C

is determined by the formula: 222 bac

Foci , ,

Step 8: Determine the asymptotes. For today’s hyperbolas, in which the 2a is

under the y-variable, the equation will be b

a .

b

a

Step 9: Graph the center, the vertices, a line through the transverse axis, the conjugate axis, the foci, and the asymptotes. Then sketch the shape of the branches.

2. 36369 22 xy

Center ,

a = _________ Vertices , ,

b = _________ Conjugate axis points , ,

c = _________ Foci , ,


3. 1002510 22 xy

Center ,

Vertices , ,

Conjugate axis points , ,

Foci , ,


9.6 classify conic sections

Any conic can be described by a general second degree equation, where A, B, C, D, E, and F are just coefficients. Given any conic equation, we can figure out whether it is a Parabola, a

Circle, an Ellipse, or a Hyperbola.

022 FEyDxCyBxyAx

1. 014222 yxyx

CTS:

1. Isolate the x’s and y’s with parenthesis. If you have both the squared and linear version of the variable, then you’ll need to complete the square.

2. Make sure that the coefficient on the squared

variable is 1. If not, pull out a GCF.

3. Find the c value with the equation

2

2

b.

4. Carefully balance the equation. 5. Factor. 6. Based on what you know about the conic, rewrite it

in its standard form.

2. 0442 22 xyx

3. 044164 22 yxyx

4. 06242 xyy

9.1 apply the distance and midpoint formulas · Example 3: Use the given distance d between the two...

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