* Recognize characteristics of graphs of polynomial functions.
* Use factoring to find zeros of polynomial functions.
* Identify zeros and their multiplicities.
* Determine end behavior.
* Understand the relationship between degree and turning points.
* Graph polynomial functions.
* Use the Intermediate Value Theorem.
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
This learner's module discusses or talks about the topic of Quadratic Functions. It also discusses what is Quadratic Functions. It also shows how to transform or rewrite the equation f(x)=ax2 + bx + c to f(x)= a(x-h)2 + k. It will also show the different characteristics of Quadratic Functions.
* Identify intervals on which a function increases, decreases, or is constant
* Use graphs to locate relative maxima or minima
* Test for symmetry
* Identify even or odd functions and recognize their symmetries
* Understand and use piecewise functions
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
This learner's module discusses or talks about the topic of Quadratic Functions. It also discusses what is Quadratic Functions. It also shows how to transform or rewrite the equation f(x)=ax2 + bx + c to f(x)= a(x-h)2 + k. It will also show the different characteristics of Quadratic Functions.
* Identify intervals on which a function increases, decreases, or is constant
* Use graphs to locate relative maxima or minima
* Test for symmetry
* Identify even or odd functions and recognize their symmetries
* Understand and use piecewise functions
5.2 Power Functions and Polynomial Functionssmiller5
* Identify power functions.
* Identify end behavior of power functions.
* Identify polynomial functions.
* Identify the degree and leading coefficient of polynomial functions.
* Evaluate a polynomial using the Remainder Theorem.
* Use the Factor Theorem to solve a polynomial equation.
* Use the Rational Zero Theorem to find rational zeros.
* Find zeros of a polynomial function.
* Use the Linear Factorization Theorem to find polynomials with given zeros.
* Use Descartes’ Rule of Signs.
Understanding the "Chain Rule" for Derivatives by Deriving Your Own VersionJames Smith
Because the Chain Rule can confuse students as much as it helps them solve real problems, we put ourselves in the shoes of the mathematicians who derived it, so that students may understand the motivation for the rule; its limitations; and why textbooks present it in its customary form. We begin by finding the derivative of sin2x without using the Chain Rule. That exercise, having shown that even a comparatively simple compound function can be bothersome to differentiate using the definition of the derivative as a limit, provides the motivation for developing our own formula for the derivative of the general compound function g[f(x)]. In the course of that development, we see why the function f must be continuous at any value of x to which the formula is applied. We finish by comparing our formula to that which is commonly given.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
* Recognize characteristics of parabolas.
* Understand how the graph of a parabola is related to its quadratic function.
* Determine a quadratic function’s minimum or maximum value.
* Solve problems involving a quadratic function’s minimum or maximum value.
Had to make this dumb powerpoint for my algebra II class and I put a lot of work into it for some reason... so yeah it's just been sitting on my laptop doing nothing and I thought why not upload this to help other people? So yeah, hope you guys find it useful...
Similar to 5.3 Graphs of Polynomial Functions (20)
* Model exponential growth and decay
* Use Newton's Law of Cooling
* Use logistic-growth models
* Choose an appropriate model for data
* Express an exponential model in base e
* Construct perpendicular and angle bisectors
* Use bisectors to solve problems
* Identify the circumcenter and incenter of a triangle
* Use triangle segments to solve problems
* Identify, write, and analyze conditional statements
* Write the inverse, converse, and contrapositive of a conditional statement
* Write a counterexample to a fake conjecture
* Find the distance between two points
* Find the midpoint of two given points
* Find the coordinates of an endpoint given one endpoint and a midpoint
* Find the coordinates of a point a fractional distance from one end of a segment
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
* Introduce functions and function notation
* Develop skills in constructing and interpreting the graphs of functions
* Learn to apply this knowledge in a variety of situations
* Recognize graphs of common functions.
* Graph functions using vertical and horizontal shifts.
* Graph functions using reflections about the x-axis and the y-axis.
* Graph functions using compressions and stretches.
* Combine transformations.
* Solve polynomial equations by factoring
* Solve equations with radicals and check the solutions
* Solve equations with rational exponents
* Solve equations that are quadratic in form
* Solve absolute value equations
* Determine whether a relation or an equation represents a function.
* Evaluate a function.
* Use the vertical line test to identify functions.
* Identify the domain and range of a function from its graph
* Identify intercepts from a function’s graph
* Solve counting problems using the Addition Principle.
* Solve counting problems using the Multiplication Principle.
* Solve counting problems using permutations involving n distinct objects.
* Solve counting problems using combinations.
* Find the number of subsets of a given set.
* Solve counting problems using permutations involving n non-distinct objects.
* Use summation notation.
* Use the formula for the sum of the first n terms of an arithmetic series.
* Use the formula for the sum of the first n terms of a geometric series.
* Use the formula for the sum of an infinite geometric series.
* Solve annuity problems.
* Find the common ratio for a geometric sequence.
* List the terms of a geometric sequence.
* Use a recursive formula for a geometric sequence.
* Use an explicit formula for a geometric sequence.
* Find the common difference for an arithmetic sequence.
* Write terms of an arithmetic sequence.
* Use a recursive formula for an arithmetic sequence.
* Use an explicit formula for an arithmetic sequence.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
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.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
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
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
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.
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.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
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.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
1. 5.3 Graphs of Polynomial
Functions
Chapter 5 Polynomial and Rational Functions
2. Concepts and Objectives
⚫ Objectives for this section are
⚫ Recognize characteristics of graphs of polynomial
functions.
⚫ Use factoring to find zeros of polynomial functions.
⚫ Identify zeros and their multiplicities.
⚫ Determine end behavior.
⚫ Understand the relationship between degree and
turning points.
⚫ Graph polynomial functions.
⚫ Use the Intermediate Value Theorem.
3. Graph Characteristics
⚫ Polynomial functions of degree 2 or more have graphs
that do not have sharp corners; these types of graphs are
called smooth curves.
⚫ Polynomial graphs are also continuous because they
have no breaks.
4. Using Factoring to Find Zeros
⚫ If f is a polynomial function, the values of x for which
f(x) = 0 are called zeros of f. If the equation of the
polynomial function can be factored, we can set each
factor equal to zero and solve for the zeros.
⚫ Sometimes the factors will already be given to you;
otherwise, you will either need to factor it or use
Desmos to locate the x-intercepts (the zeros).
⚫ Example: Find the x-intercepts of ( ) ( )( )
= − +
3 1
f x x x x
0
x = 3 0
3
x
x
− =
=
1 0
1
x
x
+ =
= −
5. Using Factoring to Find Zeros
⚫ Example: Find the x-intercepts of
Set each factor equal to 0 and solve:
( ) 6 4 2
3 2
f x x x x
= − +
Re-write as equation
Factor out the GCF
Factor the trinomial
( )
2 4 2
3 1 0
x x x
− + =
( )( )
2 2 2
1 2 0
x x x
− − =
6 4 2
3 2 0
x x x
− + =
2
0
0
x
x
=
=
2
2
1 0
1
1
x
x
x
− =
=
=
2
2
2 0
2
2
x
x
x
− =
=
=
6. Using Factoring to Find Zeros
⚫ Example (cont):
You can also graph the function in Desmos, although
you lose some of the accuracy of the .
2
7. Finding the y-intercepts
⚫ Finding the y-intercepts of a polynomial graph is much
simpler: just plug in 0 for x and evaluate.
⚫ Example: Find the y-intercepts of
Substituting 0 for x makes everything 0 except for
the –6, so the y-intercept is (0, –6).
⚫ Example: Find the y-intercepts of
So the y-intercept is (0, 12)
( ) 3 2
4 6
g x x x x
= + + −
( ) ( ) ( )
2
2 2 3
h x x x
= − +
( ) ( ) ( )
( )
( )( )
2
0 0 2 2 0 3
4 3 12
h = − +
= =
8. Zeros and Multiplicities
⚫ The number of time a given factor appears in the
factored form of the equation of a polynomial is called
the multiplicity. For example, the polynomial
g has multiplicity 5.
⚫ The multiplicity of a zero and whether the multiplicity is
even or odd determines what the graph does at a zero.
⚫ A zero of multiplicity one crosses the x-axis.
⚫ A zero of even multiplicity turns or “bounces” at the x-axis
⚫ A zero of odd multiplicity greater than one crosses the x-
axis and “wiggles”.
⚫ The sum of the multiplicities is equal to the degree.
( ) ( )
5
4
x x
= −
9. Zeros and Multiplicities
⚫ Example: Use the graph of function of degree 6 to
identify the zeros of the function and their possible
multiplicities.
10. Zeros and Multiplicities
⚫ Example: Use the graph of function of degree 6 to
identify the zeros of the function and their possible
multiplicities.
From the left, the first zero is at
x = –3. The graph bounces, so it
is an even multiplicity.
11. Zeros and Multiplicities
⚫ Example: Use the graph of function of degree 6 to
identify the zeros of the function and their possible
multiplicities.
From the left, the first zero is at
x = –3. The graph bounces, so it
is an even multiplicity.
Next is a zero at x = –1. The
graph goes straight through, so
we assume a multiplicity of 1.
12. Zeros and Multiplicities
⚫ Example: Use the graph of function of degree 6 to
identify the zeros of the function and their possible
multiplicities.
From the left, the first zero is at
x = –3. The graph bounces, so it
is an even multiplicity.
Next is a zero at x = –1. The
graph goes straight through, so
we assume a multiplicity of 1.
Finally, we have a zero at x = 4
that “wiggles”, so we assume an
odd multiplicity.
13. Zeros and Multiplicities
⚫ Example: Use the graph of function of degree 6 to
identify the zeros of the function and their possible
multiplicities.
Because the degree of the
function is 6, the multiplicites
have to add up to 6.
So, even + 1 + odd = 6 means
that –3 has multiplicity 2, –1 has
multiplicity 1, and 4 has
multiplicity 3.
14. Turning Points
⚫ The point where a graph changes direction (“bounces”
or “wiggles”) is called a turning point of the function.
⚫ A function of degree n will have at most n – 1 turning
points, with at least one turning point between each
pair of adjacent zeros.
15. Intermediate Value Theorem
⚫ This means that if we plug in two numbers for x, and the
answers have different signs (one positive and one
negative), the function has to have crossed the x-axis
somewhere between the two values.
If f (x) defines a polynomial function with only real
coefficients, and if for real numbers a and b, the
values f (a) and f (b) are opposite in sign, then
there exists at least one real zero between a and b.
17. Intermediate Value Theorem
⚫ Example: Show that has a real
zero between 2 and 3.
Since the function went from negative at 2 to
positive at 3, it must have crossed 0 somewhere
between.
( )= − − +
3 2
2 1
f x x x x
( ) ( )
3 2
2 2 2 2 2 1
8 8 2 1 1
f = − − +
= − − + = −
( ) ( )
3 2
3 3 2 3 3 1
27 18 3 1 7
f = − − +
= − − + =
18. Intermediate Value Theorem
⚫ If f (a) and f (b) are not opposite in sign, it does not
necessarily mean that there is no zero between a and b.
Consider the function, , at –1 and 3:
( )= − −
2
2 1
f x x x
f (–1) = 2 > 0 and f (3)= 2 >0
This would imply that there is no
zero between –1 and 3, but we can
see that f has two zeros between
those points.
19. Putting It All Together
⚫ Given a graph of a polynomial function, we can use
everything we’ve learned to write a formula for the
function.
⚫ To do this:
1. Identify the x-intercepts to find the factors.
2. Determine the multiplicity of the zeros.
3. Write a polynomial of least degree that fits.
4. Use any other point on the graph (the y-intercept
may be easiest) to write the exact function.
20. Putting It All Together
⚫ Example: Write a formula for the graph.
What we know:
• The graph has zeros at –3, 2,
and 5.
• –3 and 5 have multiplicity 1,
• 2 has an even multiplicity.
Since we are assuming a
polynomial of least degree, we
will set the multiplicity at 2.
• The y-intercept is at (0, –2)
21. Putting It All Together
⚫ Example: Write a formula for the graph.
This gives us a starting point of
Now we plug in the y-intercept:
( ) ( )( ) ( )
2
3 2 5
f x a x x x
= + − −
( )( ) ( )
( )( )( )
2
0 3 0 2 0 5 2
3 4 5 2
60 2
1
30
a
a
a
a
+ − − = −
− = −
− = −
=
22. Putting It All Together
⚫ Example: Write a formula for the graph.
So the graph appears to represent
the function
( ) ( )( ) ( )
2
1
3 2 5
30
f x x x x
= + − −
23. Local and Global Extrema
⚫ With quadratics, we were able to find the maximum or
minimum value of the function by finding the vertex,
either algebraically or by graphing.
⚫ To do this algebraically for functions of higher degree
requires more advanced techniques from calculus, so for
our purposes, we will estimate the locations of turning
points using graphs.
⚫ Each turning point represents a local maximum or
minimum. Sometimes a turning point is the highest or
lowest point on the entire graph. In these cases, the
turning point is the global maximum/minimum.
24. Using Local Extrema
⚫ An open-top box is to be constructed by cutting out
squares from each corner of a 14 cm by 20 cm sheet
of plastic and then folding up the sides. Find the size of
the squares that should be cut out to maximize the
volume enclosed by the box.
25. Using Local Extrema
⚫ An open-top box is to be constructed by cutting out
squares from each corner of a 14 cm by 20 cm sheet of
plastic and then folding up the sides. Find the size of the
squares that should be cut out to maximize the volume
enclosed by the box.
1. Draw a picture! It’s a
lot easier to see
what’s going on if
you sketch it out.
x
x
20 cm
14
cm
26. Using Local Extrema (cont.)
2. Notice that after the squares are cut out, it leaves a
(20-2x) cm by (14-2x) cm rectangle for the base of the
box, and the box will be x cm tall.
x
x
20 cm
14
cm
20-2x
14-2x
3. This gives the volume
4. The three zeros are 0,
10, and 7. A height of 0
doesn’t make sense, so
we will consider only
10 and 7.
( ) ( )( )
20 2 14 2
V x x x x
= − −
27. Using Local Extrema (cont.)
5. If we use x = 7, the side that is 14-2x becomes 0, so we
have to restrict the domain of our function to 0 < x < 7.
6. Now, let’s look at the graph of this function in Desmos:
28. Using Local Extrema (cont.)
5. If we use x = 7, the side that is 14-2x becomes 0, so we
have to restrict the domain of our function to 0 < x < 7.
6. Now, let’s look at the graph of this function in Desmos:
7.
7. Looking at the graph
between 0 and 7, we
can see that the
maximum value
around x = 2.75.
29. Using Local Extrema (cont.)
5. If we use x = 7, the side that is 14-2x becomes 0, so we
have to restrict the domain of our function to 0 < x < 7.
6. Now, let’s look at the graph of this function in Desmos:
7.
7. Looking at the graph
between 0 and 7, we
can see that the
maximum value
around x = 2.75.
8. Clicking on the local
maximum, we get a
more precise figure.
30. Using Local Extrema (cont.)
9. The local maximum gives us the value of (2.7, 339).
10. This means that if we cut a square of 2.7 cm from each
end, we will get a maximum volume of 339 cm2.
31. Classwork
⚫ College Algebra 2e
⚫ 5.3: 8-28 (×4); 5.2: 26-38 (even); 5.1: 66-72 (even)
⚫ 5.3 Classwork Check
⚫ Quiz 5.2
⚫ I have reposted the notes from 5.2 in case you need to
refresh your memory.