The document provides information about graphing and transforming quadratic functions:
- It discusses graphing quadratic functions by making tables of x- and y-values and plotting points. Examples show graphing f(x) = x^2 - 4x + 3 and g(x) = -x^2 + 6x - 8.
- Transformations of quadratic functions are described as translating the graph left/right or up/down, reflecting across an axis, or stretching/compressing vertically or horizontally. Examples demonstrate translating, reflecting, and compressing the graph of f(x) = x^2.
- The vertex form of a quadratic function f(x) = a(x-h)^2 +
This will help you on how to solve quadratic equations by factoring.
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This will help you on how to solve quadratic equations by factoring.
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Polynomial Function and Synthetic DivisionAleczQ1414
This file is about Polynomial Function and Synthetic Division. A project passed to Mrs. Marissa De Ocampo. Submitted by Group 6 of Grade 10-Galilei of Caloocan National Science and Technology High School '15-'16
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Sum and Difference of Cubes. The sum or difference of two cubes can be factored into a product of a binomial times a trinomial. That is, x3 + y3 = (x + y)(x2 − xy + y2) and x3 − y3 = (x − y)(x2 + xy + y2) .
Polynomial Function and Synthetic DivisionAleczQ1414
This file is about Polynomial Function and Synthetic Division. A project passed to Mrs. Marissa De Ocampo. Submitted by Group 6 of Grade 10-Galilei of Caloocan National Science and Technology High School '15-'16
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Sum and Difference of Cubes. The sum or difference of two cubes can be factored into a product of a binomial times a trinomial. That is, x3 + y3 = (x + y)(x2 − xy + y2) and x3 − y3 = (x − y)(x2 + xy + y2) .
The Quadratic Function Derived From Zeros of the Equation (SNSD Theme)SNSDTaeyeon
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This powerpoint presentation discusses or talks about the topic or lesson Direct Variations. It also discusses and explains the rules, concepts, steps and examples of Direct Variations.
This learner's module will discuss or talk about the Graph of Quadratic Functions. It will also discuss on how to draw the Graph of Quadratic Functions using the vertex, axis of symmetry, etc.
The Sum of Two Functions
The Difference of Two functions
The Product of Two Functions
The Quotient of Two Functions
The Product of A constant and a Function
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
for beginners, providing thorough training in areas such as SEO, digital communication marketing, and PPC training in Noida. After finishing the program, students receive the certifications recognised by top different universitie, setting a strong foundation for a successful career in digital marketing.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Unit 8 - Information and Communication Technology (Paper I).pdf
Algebra 2. 9.15. Intro to quadratics
1.
2. Wall Activity What do you know about…? 1. Linear example: y = 3x – 4 2. Inequality Linear example: y > 4x -9 3. Absolute Value example: y = │x - 3│+ 4
6. In Chapters 2 and 3, you studied linear functions of the form f ( x ) = mx + b . A quadratic function is a function that can be written in the form of f ( x ) = a ( x – h ) 2 + k (a ≠ 0). In a quadratic function, the variable is always squared. The table shows the linear and quadratic parent functions.
7. Notice that the graph of the parent function f ( x ) = x 2 is a U-shaped curve called a parabola . As with other functions, you can graph a quadratic function by plotting points with coordinates that make the equation true.
8. Graph f ( x ) = x 2 – 4 x + 3 by using a table. Example 1: Graphing Quadratic Functions Using a Table Make a table. Plot enough ordered pairs to see both sides of the curve. x f ( x )= x 2 – 4 x + 3 ( x , f ( x )) 0 f ( 0 )= ( 0 ) 2 – 4( 0 ) + 3 (0, 3) 1 f ( 1 )= ( 1 ) 2 – 4( 1 ) + 3 (1, 0) 2 f ( 2 )= ( 2 ) 2 – 4( 2 ) + 3 (2,–1) 3 f ( 3 )= ( 3 ) 2 – 4( 3 ) + 3 (3, 0) 4 f ( 4 )= ( 4 ) 2 – 4( 4 ) + 3 (4, 3)
10. Check It Out! Example 1 Graph g ( x ) = – x 2 + 6 x – 8 by using a table. Make a table. Plot enough ordered pairs to see both sides of the curve. x g ( x )= – x 2 +6 x –8 ( x , g ( x )) – 1 g ( –1 )= –( –1 ) 2 + 6(– 1 ) – 8 (–1,–15) 1 g ( 1 )= –( 1 ) 2 + 6( 1 ) – 8 (1, –3) 3 g ( 3 )= –( 3 ) 2 + 6( 3 ) – 8 (3, 1) 5 g ( 5 )= –( 5 ) 2 + 6( 5 ) – 8 (5, –3) 7 g ( 7 )= –( 7 ) 2 + 6( 7 ) – 8 (7, –15)
11. f ( x ) = – x 2 + 6 x – 8 • • • • • Check It Out! Example 1 Continued
12.
13. Use the graph of f ( x ) = x 2 as a guide, describe the transformations and then graph each function. Example 2A: Translating Quadratic Functions g ( x ) = ( x – 2) 2 + 4 Identify h and k . g ( x ) = ( x – 2 ) 2 + 4 Because h = 2 , the graph is translated 2 units right . Because k = 4 , the graph is translated 4 units up . Therefore, g is f translated 2 units right and 4 units up. h k
14. Use the graph of f ( x ) = x 2 as a guide, describe the transformations and then graph each function. Example 2B: Translating Quadratic Functions Because h = –2 , the graph is translated 2 units left . Because k = –3 , the graph is translated 3 units down . Therefore, g is f translated 2 units left and 4 units down. g ( x ) = ( x + 2) 2 – 3 Identify h and k . g ( x ) = ( x – (–2) ) 2 + (–3) h k
15. Using the graph of f ( x ) = x 2 as a guide, describe the transformations and then graph each function. g ( x ) = x 2 – 5 Identify h and k . g ( x ) = x 2 – 5 Because h = 0 , the graph is not translated horizontally. Because k = –5 , the graph is translated 5 units down . Therefore, g is f is translated 5 units down . Check It Out! Example 2a k
16. Use the graph of f ( x ) = x 2 as a guide, describe the transformations and then graph each function. Because h = –3 , the graph is translated 3 units left . Because k = –2 , the graph is translated 2 units down . Therefore, g is f translated 3 units left and 2 units down. g ( x ) = ( x + 3) 2 – 2 Identify h and k . g ( x ) = ( x – (–3) ) 2 + (–2) Check It Out! Example 2b h k
19. Using the graph of f(x ) = x 2 as a guide, describe the transformations and then graph each function . Example 3A: Reflecting, Stretching, and Compressing Quadratic Functions Because a is negative, g is a reflection of f across the x -axis. g x 2 1 4 x Because |a| = , g is a vertical compression of f by a factor of .
20. Using the graph of f(x ) = x 2 as a guide, describe the transformations and then graph each function . g ( x ) =(3 x ) 2 Example 3B: Reflecting, Stretching, and Compressing Quadratic Functions Because b = , g is a horizontal compression of f by a factor of .
21. Using the graph of f(x ) = x 2 as a guide, describe the transformations and then graph each function . Check It Out! Example 3a g ( x ) =(2 x ) 2 Because b = , g is a horizontal compression of f by a factor of .
22. Using the graph of f(x ) = x 2 as a guide, describe the transformations and then graph each function . Check It Out! Example 3b Because a is negative, g is a reflection of f across the x -axis. Because |a| = , g is a vertical compression of f by a factor of . g ( x ) = – x 2
23. If a parabola opens upward, it has a lowest point. If a parabola opens downward, it has a highest point. This lowest or highest point is the vertex of the parabola . The parent function f ( x ) = x 2 has its vertex at the origin. You can identify the vertex of other quadratic functions by analyzing the function in vertex form. The vertex form of a quadratic function is f ( x ) = a ( x – h ) 2 + k, where a, h, and k are constants.
24. Because the vertex is translated h horizontal units and k vertical from the origin, the vertex of the parabola is at ( h , k ). When the quadratic parent function f ( x ) = x 2 is written in vertex form, y = a ( x – h ) 2 + k , a = 1 , h = 0 , and k = 0. Helpful Hint
25. The parent function f ( x ) = x 2 is vertically stretched by a factor of and then translated 2 units left and 5 units down to create g. Use the description to write the quadratic function in vertex form. Example 4: Writing Transformed Quadratic Functions Step 1 Identify how each transformation affects the constant in vertex form. Translation 2 units left: h = –2 Translation 5 units down: k = –5 Vertical stretch by : 4 3 a 4 3
26. Example 4: Writing Transformed Quadratic Functions Step 2 Write the transformed function. g ( x ) = a ( x – h ) 2 + k Vertex form of a quadratic function Simplify. = ( x – (–2) ) 2 + (–5) Substitute for a, –2 for h, and –5 for k. = ( x + 2) 2 – 5 g ( x ) = ( x + 2) 2 – 5
27. Check Graph both functions on a graphing calculator. Enter f as Y 1 , and g as Y 2 . The graph indicates the identified transformations. f g
28. Check It Out! Example 4a Use the description to write the quadratic function in vertex form. Step 1 Identify how each transformation affects the constant in vertex form. Translation 2 units right: h = 2 Translation 4 units down: k = –4 The parent function f ( x ) = x 2 is vertically compressed by a factor of and then translated 2 units right and 4 units down to create g. Vertical compression by : a =
29. Step 2 Write the transformed function. g ( x ) = a ( x – h ) 2 + k Vertex form of a quadratic function Simplify. Check It Out! Example 4a Continued = ( x – 2) 2 + ( –4 ) = ( x – 2) 2 – 4 Substitute for a, 2 for h, and –4 for k. g ( x ) = ( x – 2) 2 – 4
30. Check Graph both functions on a graphing calculator. Enter f as Y 1 , and g as Y 2 . The graph indicates the identified transformations. Check It Out! Example 4a Continued f g
31. The parent function f ( x ) = x 2 is reflected across the x -axis and translated 5 units left and 1 unit up to create g. Check It Out! Example 4b Use the description to write the quadratic function in vertex form. Step 1 Identify how each transformation affects the constant in vertex form. Translation 5 units left: h = –5 Translation 1 unit up: k = 1 Reflected across the x -axis: a is negative
32. Step 2 Write the transformed function. g ( x ) = a ( x – h ) 2 + k Vertex form of a quadratic function Simplify. = –( x – (–5) 2 + (1) = –( x +5) 2 + 1 Substitute –1 for a , – 5 for h, and 1 for k. Check It Out! Example 4b Continued g ( x ) = – ( x +5) 2 + 1
33. Check Graph both functions on a graphing calculator. Enter f as Y 1 , and g as Y 2 . The graph indicates the identified transformations. Check It Out! Example 4b Continued f g
34. Lesson Quiz: Part I 1. Graph f ( x ) = x 2 + 3 x – 1 by using a table.
35. Lesson Quiz: Part II 2. Using the graph of f ( x ) = x 2 as a guide, describe the transformations, and then graph g(x ) = ( x + 1) 2 . g is f reflected across x -axis, vertically compressed by a factor of , and translated 1 unit left.
36. Lesson Quiz: Part III 3. The parent function f ( x ) = x 2 is vertically stretched by a factor of 3 and translated 4 units right and 2 units up to create g . Write g in vertex form. g ( x ) = 3( x – 4) 2 + 2