The document discusses conic sections, which are curves formed by the intersection of a plane and a right circular cone. There are four types of conic sections: circles, ellipses, parabolas, and hyperbolas. Conic sections can be represented by second-degree equations in x and y, and the technique of completing the square is used to determine which equation corresponds to each type of conic section. The document also reviews the distance formula.
This is a powerpoint presentation that discusses about the topic or lesson: Conic Sections. It also includes the definition, types and some terminologies involved in the topic: Conic Sections.
Graphs of the Sine and Cosine Functions LectureFroyd Wess
More: www.PinoyBIX.org
Lesson Objectives
Able to plot the different Trigonometric Graphs
Graph of Sine Function (y = f(x) = sinx)
Graph of Cosine Function (y = f(x) = cosx)
Define the Maximum and Minimum value in a graph
Generalized Trigonometric Functions
Graphs of y = sinbx
Graphs of y = sin(bx + c)
Could find the Period of Trigonometric Functions
Could find the Amplitude of Trigonometric Functions
Variations in the Trigonometric Functions
This is a powerpoint presentation that discusses about the topic or lesson: Conic Sections. It also includes the definition, types and some terminologies involved in the topic: Conic Sections.
Graphs of the Sine and Cosine Functions LectureFroyd Wess
More: www.PinoyBIX.org
Lesson Objectives
Able to plot the different Trigonometric Graphs
Graph of Sine Function (y = f(x) = sinx)
Graph of Cosine Function (y = f(x) = cosx)
Define the Maximum and Minimum value in a graph
Generalized Trigonometric Functions
Graphs of y = sinbx
Graphs of y = sin(bx + c)
Could find the Period of Trigonometric Functions
Could find the Amplitude of Trigonometric Functions
Variations in the Trigonometric Functions
A circle is a simple shape in Euclidean geometry. It is the set of all points in a plane that are at a given distance from a given point, the center; equivalently it is the curve traced out by a point that moves so that its distance from a given point is constant.
Though we had learnt about Simple and Compound Interests at school, because of the technological advantages and new gadgets over the years we have forgotten how to calculate it. This is my sincere effort to refresh the minds of interested persons about its concepts and how to calculate mannually.
MS Report, When we talked about the conic section it involves a double-napped cone and a plane. If a plane intersects a double right circular cone, we get two-dimensional curves of different types. These curves are what we called the conic section.
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Combination of Cubic and Quartic Plane CurveIOSR Journals
The set of complex eigenvalues of unistochastic matrices of order three forms a deltoid.
A cross-section of the set of unistochastic matrices of order three forms a deltoid.
The set of possible traces of unitary matrices belonging to the group SU(3) forms a deltoid.
The intersection of two deltoids parametrizes a family of Complex Hadamard matrices of order six.
The set of all Simson lines of given triangle, form an envelope in the shape of a deltoid. This is known as the
Steiner deltoid or Steiner's hypocycloid after Jakob Steiner who described the shape and symmetry of the
curve in 1856.
The envelope of the area bisectors of a triangle is a deltoid (in the broader sense defined above) with
vertices at the midpoints of the medians. The sides of the deltoid are arcs of hyperbolas that are asymptotic
to the triangle's sides
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
Securing your Kubernetes cluster_ a step-by-step guide to success !KatiaHIMEUR1
Today, after several years of existence, an extremely active community and an ultra-dynamic ecosystem, Kubernetes has established itself as the de facto standard in container orchestration. Thanks to a wide range of managed services, it has never been so easy to set up a ready-to-use Kubernetes cluster.
However, this ease of use means that the subject of security in Kubernetes is often left for later, or even neglected. This exposes companies to significant risks.
In this talk, I'll show you step-by-step how to secure your Kubernetes cluster for greater peace of mind and reliability.
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdfPaige Cruz
Monitoring and observability aren’t traditionally found in software curriculums and many of us cobble this knowledge together from whatever vendor or ecosystem we were first introduced to and whatever is a part of your current company’s observability stack.
While the dev and ops silo continues to crumble….many organizations still relegate monitoring & observability as the purview of ops, infra and SRE teams. This is a mistake - achieving a highly observable system requires collaboration up and down the stack.
I, a former op, would like to extend an invitation to all application developers to join the observability party will share these foundational concepts to build on:
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
UiPath Test Automation using UiPath Test Suite series, part 3DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 3. In this session, we will cover desktop automation along with UI automation.
Topics covered:
UI automation Introduction,
UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
Generative AI Deep Dive: Advancing from Proof of Concept to ProductionAggregage
Join Maher Hanafi, VP of Engineering at Betterworks, in this new session where he'll share a practical framework to transform Gen AI prototypes into impactful products! He'll delve into the complexities of data collection and management, model selection and optimization, and ensuring security, scalability, and responsible use.
3. Conic Sections
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
4. Conic Sections
A right circular cone
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
5. Conic Sections
A right circular cone and conic sections (wikipedia “Conic Sections”)
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
6. Conic Sections
A Horizontal Section
A right circular cone and conic sections (wikipedia “Conic Sections”)
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
7. Conic Sections
A Horizontal Section
A right circular cone and conic sections (wikipedia “Conic Sections”)
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
8. Conic Sections
A Moderately
Tilted Section
A right circular cone and conic sections (wikipedia “Conic Sections”)
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
9. Conic Sections
A Moderately
Tilted Section
A right circular cone and conic sections (wikipedia “Conic Sections”)
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
10. Conic Sections
A Horizontal Section
A Moderately
Tilted Section
A right circular cone and conic sections (wikipedia “Conic Sections”)
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
Circles and
ellipsis are
enclosed.
11. Conic Sections
A right circular cone and conic sections (wikipedia “Conic Sections”)
A Parallel–Section
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
12. Conic Sections
A right circular cone and conic sections (wikipedia “Conic Sections”)
A Parallel–Section
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
13. Conic Sections
A right circular cone and conic sections (wikipedia “Conic Sections”)
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
An Cut-away
Section
14. Conic Sections
A right circular cone and conic sections (wikipedia “Conic Sections”)
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
An Cut-away
Section
15. Conic Sections
A right circular cone and conic sections (wikipedia “Conic Sections”)
An Cut-away
Section
One way to study a solid is to slice it open. The exposed area
of the sliced solid is called a cross sectional area.
Conic sections are the borders of the cross sectional areas of
a right circular cone as shown.
Parabolas and
hyperbolas are open.
A Horizontal Section
A Moderately
Tilted Section
Circles and
ellipsis are
enclosed.
A Parallel–Section
16. We summarize the four types of conics sections here.
Circles Ellipses
Parabolas Hyperbolas
Conic Sections
18. Conic Sections
Conic sections are the cross sections of right circular cones.
There are four different types of curves:
• circles
• ellipses
19. Conic Sections
Conic sections are the cross sections of right circular cones.
There are four different types of curves:
• circles
• ellipses
• parabolas
20. Conic Sections
Conic sections are the cross sections of right circular cones.
There are four different types of curves:
• circles
• ellipses
• parabolas
• hyperbolas
Where as straight lines are the graphs of 1st degree equations
Ax + By = C, conic sections are the graphs of 2nd degree
equations in x and y.
21. Conic Sections
Conic sections are the cross sections of right circular cones.
There are four different types of curves:
• circles
• ellipses
• parabolas
• hyperbolas
Where as straight lines are the graphs of 1st degree equations
Ax + By = C, conic sections are the graphs of 2nd degree
equations in x and y. In particular, the conic sections that are
parallel to the axes (not tilted) have equations of the form
Ax2
+ By2
+ Cx + Dy = E, where A, B, C, D, and E are
numbers (not both A and B equal to 0).
22. Conic Sections
Conic sections are the cross sections of right circular cones.
There are four different types of curves:
• circles
• ellipses
• parabolas
• hyperbolas
Where as straight lines are the graphs of 1st degree equations
Ax + By = C, conic sections are the graphs of 2nd degree
equations in x and y. In particular, the conic sections that are
parallel to the axes (not tilted) have equations of the form
Ax2
+ By2
+ Cx + Dy = E, where A, B, C, D, and E are
numbers (not both A and B equal to 0). We are to match these
2nd degree equations with the different conic sections.
23. Conic Sections
Conic sections are the cross sections of right circular cones.
There are four different types of curves:
• circles
• ellipses
• parabolas
• hyperbolas
Where as straight lines are the graphs of 1st degree equations
Ax + By = C, conic sections are the graphs of 2nd degree
equations in x and y. In particular, the conic sections that are
parallel to the axes (not tilted) have equations of the form
Ax2
+ By2
+ Cx + Dy = E, where A, B, C, D, and E are
numbers (not both A and B equal to 0). We are to match these
2nd degree equations with the different conic sections.
The algebraic technique that enable us to sort out which
equation corresponds to which conic section is called
"completing the square".
24. Conic Sections
Conic sections are the cross sections of right circular cones.
There are four different types of curves:
• circles
• ellipses
• parabolas
• hyperbolas
Where as straight lines are the graphs of 1st degree equations
Ax + By = C, conic sections are the graphs of 2nd degree
equations in x and y. In particular, the conic sections that are
parallel to the axes (not tilted) have equations of the form
Ax2
+ By2
+ Cx + Dy = E, where A, B, C, D, and E are
numbers (not both A and B equal to 0). We are to match these
2nd degree equations with the different conic sections.
The algebraic technique that enable us to sort out which
equation corresponds to which conic section is called
"completing the square". We start with the Distance Formula.
25. The Distance Formula:
Given two points P = (x1, y1) and Q = (x2, y2) in the xy-plane,
the distance r between P and Q is:
Conic Sections
26. The Distance Formula:
Given two points P = (x1, y1) and Q = (x2, y2) in the xy-plane,
the distance r between P and Q is:
r = √(y2 – y1)2
+ (x2 – x1)2
Conic Sections
27. The Distance Formula:
Given two points P = (x1, y1) and Q = (x2, y2) in the xy-plane,
the distance r between P and Q is:
r = √(y2 – y1)2
+ (x2 – x1)2
= √Δy2
+ Δx2
Conic Sections
28. The Distance Formula:
Given two points P = (x1, y1) and Q = (x2, y2) in the xy-plane,
the distance r between P and Q is:
r = √(y2 – y1)2
+ (x2 – x1)2
= √Δy2
+ Δx2
where
Conic Sections
Δy = the difference between the y's = y2 – y1
29. The Distance Formula:
Given two points P = (x1, y1) and Q = (x2, y2) in the xy-plane,
the distance r between P and Q is:
r = √(y2 – y1)2
+ (x2 – x1)2
= √Δy2
+ Δx2
where
Conic Sections
Δy = the difference between the y's = y2 – y1
Δx = the difference between the x's = x2 – x1
30. The Distance Formula:
Given two points P = (x1, y1) and Q = (x2, y2) in the xy-plane,
the distance r between P and Q is:
r = √(y2 – y1)2
+ (x2 – x1)2
= √Δy2
+ Δx2
where
Example A. Find the
distance between (2, –1)
and (–2, 2).
Conic Sections
Δy = the difference between the y's = y2 – y1
Δx = the difference between the x's = x2 – x1
31. The Distance Formula:
Given two points P = (x1, y1) and Q = (x2, y2) in the xy-plane,
the distance r between P and Q is:
r = √(y2 – y1)2
+ (x2 – x1)2
= √Δy2
+ Δx2
where
Example A. Find the
distance between (2, –1)
and (–2, 2).
Δy = (–1) – (2) = –3
Δy=-3
Conic Sections
Δy = the difference between the y's = y2 – y1
Δx = the difference between the x's = x2 – x1
32. The Distance Formula:
Given two points P = (x1, y1) and Q = (x2, y2) in the xy-plane,
the distance r between P and Q is:
r = √(y2 – y1)2
+ (x2 – x1)2
= √Δy2
+ Δx2
where
Example A. Find the
distance between (2, –1)
and (–2, 2).
Δy = (–1) – (2) = –3
Δx = (2) – (–2) = 4
Δy=-3
Δx=4
Conic Sections
Δy = the difference between the y's = y2 – y1
Δx = the difference between the x's = x2 – x1
33. The Distance Formula:
Given two points P = (x1, y1) and Q = (x2, y2) in the xy-plane,
the distance r between P and Q is:
r = √(y2 – y1)2
+ (x2 – x1)2
= √Δy2
+ Δx2
where
Example A. Find the
distance between (2, –1)
and (–2, 2).
Δy = (–1) – (2) = –3
Δx = (2) – (–2) = 4
r = √(–3)2
+ 42
= √25 = 5
Δy=-3
Δx=4
r=5
Conic Sections
Δy = the difference between the y's = y2 – y1
Δx = the difference between the x's = x2 – x1
34. The Distance Formula:
Given two points P = (x1, y1) and Q = (x2, y2) in the xy-plane,
the distance r between P and Q is:
r = √(y2 – y1)2
+ (x2 – x1)2
= √Δy2
+ Δx2
where
Example A. Find the
distance between (2, –1)
and (–2, 2).
Δy = (–1) – (2) = –3
Δx = (2) – (–2) = 4
r = √(–3)2
+ 42
= √25 = 5
Δy=-3
Δx=4
r=5
Conic Sections
The geometric definition of all four types of conic sections are
distance relations between points.
Δy = the difference between the y's = y2 – y1
Δx = the difference between the x's = x2 – x1
35. The Distance Formula:
Given two points P = (x1, y1) and Q = (x2, y2) in the xy-plane,
the distance r between P and Q is:
r = √(y2 – y1)2
+ (x2 – x1)2
= √Δy2
+ Δx2
where
Example A. Find the
distance between (2, –1)
and (–2, 2).
Δy = (–1) – (2) = –3
Δx = (2) – (–2) = 4
r = √(–3)2
+ 42
= √25 = 5
Δy=-3
Δx=4
r=5
Conic Sections
The geometric definition of all four types of conic sections are
distance relations between points. We start with the circles.
Δy = the difference between the y's = y2 – y1
Δx = the difference between the x's = x2 – x1
36. Circles
A circle is the set of all the points that have equal distance r,
called the radius, to a fixed point C which is called the center.
37. r
r
Circles
A circle is the set of all the points that have equal distance r,
called the radius, to a fixed point C which is called the center.
C
38. r
r
The radius and the center completely determine the circle.
Circles
center
A circle is the set of all the points that have equal distance r,
called the radius, to a fixed point C which is called the center.
39. r
The radius and the center completely determine the circle.
Circles
Let (h, k) be the center of a
circle and r be the radius.
(h, k)
A circle is the set of all the points that have equal distance r,
called the radius, to a fixed point C which is called the center.
40. r
The radius and the center completely determine the circle.
Circles
(x, y)
Let (h, k) be the center of a
circle and r be the radius.
Suppose (x, y) is a point on
the circle, then the distance
between (x, y) and the center
is r.
(h, k)
A circle is the set of all the points that have equal distance r,
called the radius, to a fixed point C which is called the center.
41. r
The radius and the center completely determine the circle.
Circles
(x, y)
Let (h, k) be the center of a
circle and r be the radius.
Suppose (x, y) is a point on
the circle, then the distance
between (x, y) and the center
is r. Hence,
(h, k)
r = √ (x – h)2
+ (y – k)2
A circle is the set of all the points that have equal distance r,
called the radius, to a fixed point C which is called the center.
42. r
The radius and the center completely determine the circle.
Circles
(x, y)
Let (h, k) be the center of a
circle and r be the radius.
Suppose (x, y) is a point on
the circle, then the distance
between (x, y) and the center
is r. Hence,
(h, k)
r = √ (x – h)2
+ (y – k)2
or
r2
= (x – h)2
+ (y – k)2
A circle is the set of all the points that have equal distance r,
called the radius, to a fixed point C which is called the center.
43. r
The radius and the center completely determine the circle.
Circles
(x, y)
Let (h, k) be the center of a
circle and r be the radius.
Suppose (x, y) is a point on
the circle, then the distance
between (x, y) and the center
is r. Hence,
(h, k)
r = √ (x – h)2
+ (y – k)2
or
r2
= (x – h)2
+ (y – k)2
This is called the standard form of circles.
A circle is the set of all the points that have equal distance r,
called the radius, to a fixed point C which is called the center.
44. r
The radius and the center completely determine the circle.
Circles
(x, y)
Let (h, k) be the center of a
circle and r be the radius.
Suppose (x, y) is a point on
the circle, then the distance
between (x, y) and the center
is r. Hence,
(h, k)
r = √ (x – h)2
+ (y – k)2
or
r2
= (x – h)2
+ (y – k)2
This is called the standard form of circles. Given an equation
of this form, we can easily identify the center and the radius.
A circle is the set of all the points that have equal distance r,
called the radius, to a fixed point C which is called the center.
46. r2
= (x – h)2
+ (y – k)2
must be “ – ”
Circles
47. r2
= (x – h)2
+ (y – k)2
r is the radius must be “ – ”
Circles
48. r2
= (x – h)2
+ (y – k)2
r is the radius must be “ – ”
(h, k) is the center
Circles
49. r2
= (x – h)2
+ (y – k)2
r is the radius must be “ – ”
(h, k) is the center
Circles
Example B. Write the equation
of the circle as shown.
50. r2
= (x – h)2
+ (y – k)2
r is the radius must be “ – ”
(h, k) is the center
Circles
Example B. Write the equation
of the circle as shown.
The center is (–1, 3) and the
radius is 5. (–1, 3)
51. r2
= (x – h)2
+ (y – k)2
r is the radius must be “ – ”
(h, k) is the center
Circles
Example B. Write the equation
of the circle as shown.
The center is (–1, 3) and the
radius is 5.
Hence the equation is:
52
= (x – (–1))2
+ (y – 3)2
(–1, 3)
52. r2
= (x – h)2
+ (y – k)2
r is the radius must be “ – ”
(h, k) is the center
Circles
Example B. Write the equation
of the circle as shown.
The center is (–1, 3) and the
radius is 5.
Hence the equation is:
52
= (x – (–1))2
+ (y – 3)2
or
25 = (x + 1)2
+ (y – 3 )2
(–1, 3)
53. Example C. Identify the center and
the radius of 16 = (x – 3)2
+ (y + 2)2
.
Label the top, bottom, left and right
most points. Graph it.
Circles
54. Example C. Identify the center and
the radius of 16 = (x – 3)2
+ (y + 2)2
.
Label the top, bottom, left and right
most points. Graph it.
Put 16 = (x – 3)2
+ (y + 2)2
into the
standard form:
42
= (x – 3)2
+ (y – (–2))2
Circles
55. Example C. Identify the center and
the radius of 16 = (x – 3)2
+ (y + 2)2
.
Label the top, bottom, left and right
most points. Graph it.
Put 16 = (x – 3)2
+ (y + 2)2
into the
standard form:
42
= (x – 3)2
+ (y – (–2))2
Hence r = 4, center = (3, –2)
Circles
56. Example C. Identify the center and
the radius of 16 = (x – 3)2
+ (y + 2)2
.
Label the top, bottom, left and right
most points. Graph it.
Put 16 = (x – 3)2
+ (y + 2)2
into the
standard form:
42
= (x – 3)2
+ (y – (–2))2
Hence r = 4, center = (3, –2)
(3,-2)
Circles
57. Example C. Identify the center and
the radius of 16 = (x – 3)2
+ (y + 2)2
.
Label the top, bottom, left and right
most points. Graph it.
Put 16 = (x – 3)2
+ (y + 2)2
into the
standard form:
42
= (x – 3)2
+ (y – (–2))2
Hence r = 4, center = (3, –2)
(3,-2)
Circles
(3, 2)
(3,-6)
58. Example C. Identify the center and
the radius of 16 = (x – 3)2
+ (y + 2)2
.
Label the top, bottom, left and right
most points. Graph it.
Put 16 = (x – 3)2
+ (y + 2)2
into the
standard form:
42
= (x – 3)2
+ (y – (–2))2
Hence r = 4, center = (3, –2)
(3,-2)
Circles
(3, 2)
(–1,-2) (7,-2)
(3,-6)
59. Example C. Identify the center and
the radius of 16 = (x – 3)2
+ (y + 2)2
.
Label the top, bottom, left and right
most points. Graph it.
Put 16 = (x – 3)2
+ (y + 2)2
into the
standard form:
42
= (x – 3)2
+ (y – (–2))2
Hence r = 4, center = (3, –2)
(3,-2)
Circles
(3, 2)
(–1,-2) (7,-2)
(3,-6)
60. Example C. Identify the center and
the radius of 16 = (x – 3)2
+ (y + 2)2
.
Label the top, bottom, left and right
most points. Graph it.
Put 16 = (x – 3)2
+ (y + 2)2
into the
standard form:
42
= (x – 3)2
+ (y – (–2))2
Hence r = 4, center = (3, –2)
Circles
When equations are not in the standard form, we have to
rearrange them into the standard form. We do this by
"completing the square".
(3,-2)
(3, 2)
(–1,-2) (7,-2)
(3,-6)
61. Example C. Identify the center and
the radius of 16 = (x – 3)2
+ (y + 2)2
.
Label the top, bottom, left and right
most points. Graph it.
Put 16 = (x – 3)2
+ (y + 2)2
into the
standard form:
42
= (x – 3)2
+ (y – (–2))2
Hence r = 4, center = (3, –2)
Circles
When equations are not in the standard form, we have to
rearrange them into the standard form. We do this by
"completing the square". To complete the square means to
add a number to an expression so the sum is a perfect
square.
(3,-2)
(3, 2)
(–1,-2) (7,-2)
(3,-6)
62. Example C. Identify the center and
the radius of 16 = (x – 3)2
+ (y + 2)2
.
Label the top, bottom, left and right
most points. Graph it.
Put 16 = (x – 3)2
+ (y + 2)2
into the
standard form:
42
= (x – 3)2
+ (y – (–2))2
Hence r = 4, center = (3, –2)
Circles
When equations are not in the standard form, we have to
rearrange them into the standard form. We do this by
"completing the square". To complete the square means to
add a number to an expression so the sum is a perfect
square. This procedure is the main technique in dealing with
2nd degree equations.
(3,-2)
(3, 2)
(–1,-2) (7,-2)
(3,-6)
64. (Completing the Square)
If we are given x2
+ bx, then adding (b/2)2
to the expression
makes the expression a perfect square,
Circles
65. (Completing the Square)
If we are given x2
+ bx, then adding (b/2)2
to the expression
makes the expression a perfect square, i.e. x2
+ bx + (b/2)2
is the perfect square (x + b/2)2
.
Circles
66. (Completing the Square)
If we are given x2
+ bx, then adding (b/2)2
to the expression
makes the expression a perfect square, i.e. x2
+ bx + (b/2)2
is the perfect square (x + b/2)2
.
Circles
Example D. Fill in the blank to make a perfect square.
a. x2
– 6x + (–6/2)2
67. (Completing the Square)
If we are given x2
+ bx, then adding (b/2)2
to the expression
makes the expression a perfect square, i.e. x2
+ bx + (b/2)2
is the perfect square (x + b/2)2
.
Circles
Example D. Fill in the blank to make a perfect square.
a. x2
– 6x + (–6/2)2
68. (Completing the Square)
If we are given x2
+ bx, then adding (b/2)2
to the expression
makes the expression a perfect square, i.e. x2
+ bx + (b/2)2
is the perfect square (x + b/2)2
.
Circles
Example D. Fill in the blank to make a perfect square.
a. x2
– 6x + (–6/2)2
= x2
– 6x + 9 = (x – 3)2
69. (Completing the Square)
If we are given x2
+ bx, then adding (b/2)2
to the expression
makes the expression a perfect square, i.e. x2
+ bx + (b/2)2
is the perfect square (x + b/2)2
.
Circles
Example D. Fill in the blank to make a perfect square.
a. x2
– 6x + (–6/2)2
= x2
– 6x + 9 = (x – 3)2
b. y2
+ 12y + (12/2)2
70. (Completing the Square)
If we are given x2
+ bx, then adding (b/2)2
to the expression
makes the expression a perfect square, i.e. x2
+ bx + (b/2)2
is the perfect square (x + b/2)2
.
Circles
Example D. Fill in the blank to make a perfect square.
a. x2
– 6x + (–6/2)2
= x2
– 6x + 9 = (x – 3)2
b. y2
+ 12y + (12/2)2
71. (Completing the Square)
If we are given x2
+ bx, then adding (b/2)2
to the expression
makes the expression a perfect square, i.e. x2
+ bx + (b/2)2
is the perfect square (x + b/2)2
.
Circles
Example D. Fill in the blank to make a perfect square.
a. x2
– 6x + (–6/2)2
= x2
– 6x + 9 = (x – 3)2
b. y2
+ 12y + (12/2)2
= y2
+ 12y + 36 = ( y + 6)2
72. (Completing the Square)
If we are given x2
+ bx, then adding (b/2)2
to the expression
makes the expression a perfect square, i.e. x2
+ bx + (b/2)2
is the perfect square (x + b/2)2
.
Circles
Example D. Fill in the blank to make a perfect square.
a. x2
– 6x + (–6/2)2
= x2
– 6x + 9 = (x – 3)2
b. y2
+ 12y + (12/2)2
= y2
+ 12y + 36 = ( y + 6)2
The following are the steps in putting a 2nd
degree equation
into the standard form.
73. (Completing the Square)
If we are given x2
+ bx, then adding (b/2)2
to the expression
makes the expression a perfect square, i.e. x2
+ bx + (b/2)2
is the perfect square (x + b/2)2
.
Circles
Example D. Fill in the blank to make a perfect square.
a. x2
– 6x + (–6/2)2
= x2
– 6x + 9 = (x – 3)2
b. y2
+ 12y + (12/2)2
= y2
+ 12y + 36 = ( y + 6)2
The following are the steps in putting a 2nd
degree equation
into the standard form.
1. Group the x2
and the x-terms together, group the y2
and y
terms together, and move the number term the the other
side of the equation.
74. (Completing the Square)
If we are given x2
+ bx, then adding (b/2)2
to the expression
makes the expression a perfect square, i.e. x2
+ bx + (b/2)2
is the perfect square (x + b/2)2
.
Circles
Example D. Fill in the blank to make a perfect square.
a. x2
– 6x + (–6/2)2
= x2
– 6x + 9 = (x – 3)2
b. y2
+ 12y + (12/2)2
= y2
+ 12y + 36 = ( y + 6)2
The following are the steps in putting a 2nd
degree equation
into the standard form.
1. Group the x2
and the x-terms together, group the y2
and y
terms together, and move the number term the the other
side of the equation.
2. Complete the square for the x-terms and for the y-terms.
Make sure add the necessary numbers to both sides.
75. Example E. Use completing the square to find the center
and radius of x2
– 6x + y2
+ 12y = –36. Find the top, bottom,
left and right most points. Graph it.
Circles
76. Example E. Use completing the square to find the center
and radius of x2
– 6x + y2
+ 12y = –36. Find the top, bottom,
left and right most points. Graph it.
We use completing the square to put the equation into the
standard form:
Circles
77. Example E. Use completing the square to find the center
and radius of x2
– 6x + y2
+ 12y = –36. Find the top, bottom,
left and right most points. Graph it.
We use completing the square to put the equation into the
standard form:
x2
– 6x + + y2
+ 12y + = –36
Circles
78. Example E. Use completing the square to find the center
and radius of x2
– 6x + y2
+ 12y = –36. Find the top, bottom,
left and right most points. Graph it.
We use completing the square to put the equation into the
standard form:
x2
– 6x + + y2
+ 12y + = –36 ;complete squares
x2
– 6x + 9 + y2
+ 12y + 36 = –36 + 9 + 36
Circles
79. Example E. Use completing the square to find the center
and radius of x2
– 6x + y2
+ 12y = –36. Find the top, bottom,
left and right most points. Graph it.
We use completing the square to put the equation into the
standard form:
x2
– 6x + + y2
+ 12y + = –36 ;complete squares
x2
– 6x + 9 + y2
+ 12y + 36 = –36 + 9 + 36
Circles
80. Example E. Use completing the square to find the center
and radius of x2
– 6x + y2
+ 12y = –36. Find the top, bottom,
left and right most points. Graph it.
We use completing the square to put the equation into the
standard form:
x2
– 6x + + y2
+ 12y + = –36 ;complete squares
x2
– 6x + 9 + y2
+ 12y + 36 = –36 + 9 + 36
( x – 3 )2
+ (y + 6)2
= 9
Circles
81. Example E. Use completing the square to find the center
and radius of x2
– 6x + y2
+ 12y = –36. Find the top, bottom,
left and right most points. Graph it.
We use completing the square to put the equation into the
standard form:
x2
– 6x + + y2
+ 12y + = –36 ;complete squares
x2
– 6x + 9 + y2
+ 12y + 36 = –36 + 9 + 36
( x – 3 )2
+ (y + 6)2
= 9
( x – 3 )2
+ (y + 6)2
= 32
Circles
82. Example E. Use completing the square to find the center
and radius of x2
– 6x + y2
+ 12y = –36. Find the top, bottom,
left and right most points. Graph it.
We use completing the square to put the equation into the
standard form:
x2
– 6x + + y2
+ 12y + = –36 ;complete squares
x2
– 6x + 9 + y2
+ 12y + 36 = –36 + 9 + 36
( x – 3 )2
+ (y + 6)2
= 9
( x – 3 )2
+ (y + 6)2
= 32
Hence the center is (3 , –6),
and radius is 3.
Circles
83. Example E. Use completing the square to find the center
and radius of x2
– 6x + y2
+ 12y = –36. Find the top, bottom,
left and right most points. Graph it.
We use completing the square to put the equation into the
standard form:
x2
– 6x + + y2
+ 12y + = –36 ;complete squares
x2
– 6x + 9 + y2
+ 12y + 36 = –36 + 9 + 36
( x – 3 )2
+ (y + 6)2
= 9
( x – 3 )2
+ (y + 6)2
= 32
Hence the center is (3 , –6),
and radius is 3.
Circles
(3, –6)
84. Example E. Use completing the square to find the center
and radius of x2
– 6x + y2
+ 12y = –36. Find the top, bottom,
left and right most points. Graph it.
We use completing the square to put the equation into the
standard form:
x2
– 6x + + y2
+ 12y + = –36 ;complete squares
x2
– 6x + 9 + y2
+ 12y + 36 = –36 + 9 + 36
( x – 3 )2
+ (y + 6)2
= 9
( x – 3 )2
+ (y + 6)2
= 32
Hence the center is (3 , –6),
and radius is 3.
Circles
(3, –6)
(3, –3)
(3, –9)
(6, –6)(0, –6)
85. Example E. Use completing the square to find the center
and radius of x2
– 6x + y2
+ 12y = –36. Find the top, bottom,
left and right most points. Graph it.
We use completing the square to put the equation into the
standard form:
x2
– 6x + + y2
+ 12y + = –36 ;complete squares
x2
– 6x + 9 + y2
+ 12y + 36 = –36 + 9 + 36
( x – 3 )2
+ (y + 6)2
= 9
( x – 3 )2
+ (y + 6)2
= 32
Hence the center is (3 , –6),
and radius is 3.
Circles
(3, –6)
(3, –3)
(3, –9)
(6, –6)(0, –6)