* 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.
Presentation on Fourier Series
contents are:-
Euler’s Formula
Functions having point of discontinuity
Change of interval
Even and Odd functions
Half Range series
Harmonic analysis
Presentation on Fourier Series
contents are:-
Euler’s Formula
Functions having point of discontinuity
Change of interval
Even and Odd functions
Half Range series
Harmonic analysis
Integration Made Easy!
The derivative of a function can be geometrically interpreted as the slope of the curve of the mathematical function f(x) plotted as a function of x. But its implications for the modeling of nature go far deeper than this simple geometric application might imply. After all, you can see yourself drawing finite triangles to discover slope, so why is the derivative so important? Its importance lies in the fact that many physical entities such as velocity, acceleration, force and so on are defined as instantaneous rates of change of some other quantity. The derivative can give you a precise intantaneous value for that rate of change and lead to precise modeling of the desired quantity.
Applied Calculus: Continuity and Discontinuity of Functionbaetulilm
Lecture #: 04: "Continuity and Discontinuity of Function" with in a course on Applied Calculus offered at Faculty of Engineering, University of Central Punjab
By: Prof. Muhammad Rafiq.
Gives idea about function, one to one function, inverse function, which functions are invertible, how to invert a function and application of inverse functions.
When a quantity changes in proportion to itself (for instance, bacteria reproduction or radioactive decay), the growth or decay is exponential in nature. There are many many examples of this to be found.
Identify the transformations to the graph of a quadratic function.
Change a function from general form to vertex form.
Identify the vertex, axis of symmetry, the domain, and the range of the function.
Integration Made Easy!
The derivative of a function can be geometrically interpreted as the slope of the curve of the mathematical function f(x) plotted as a function of x. But its implications for the modeling of nature go far deeper than this simple geometric application might imply. After all, you can see yourself drawing finite triangles to discover slope, so why is the derivative so important? Its importance lies in the fact that many physical entities such as velocity, acceleration, force and so on are defined as instantaneous rates of change of some other quantity. The derivative can give you a precise intantaneous value for that rate of change and lead to precise modeling of the desired quantity.
Applied Calculus: Continuity and Discontinuity of Functionbaetulilm
Lecture #: 04: "Continuity and Discontinuity of Function" with in a course on Applied Calculus offered at Faculty of Engineering, University of Central Punjab
By: Prof. Muhammad Rafiq.
Gives idea about function, one to one function, inverse function, which functions are invertible, how to invert a function and application of inverse functions.
When a quantity changes in proportion to itself (for instance, bacteria reproduction or radioactive decay), the growth or decay is exponential in nature. There are many many examples of this to be found.
Identify the transformations to the graph of a quadratic function.
Change a function from general form to vertex form.
Identify the vertex, axis of symmetry, the domain, and the range of the function.
* Combine functions using algebraic operations.
* Create a new function by composition of functions.
* Evaluate composite functions.
* Find the domain of a composite function.
* Decompose a composite function into its component functions.
* 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.
* 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
* Determine whether a relation represents a function.
* Find the value of a function.
* Determine whether a function is one-to-one.
* Use the vertical line test to identify functions.
* Graph the functions listed in the library of functions.
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.
Write the equation of a circle given the center and radius
Identify the center and radius of a circle in both center-radius and general form
Write the equation of a circle given the center and a point on the circle
* 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
* 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.
* Write the terms of a sequence defined by an explicit formula.
* Write the terms of a sequence defined by a recursive formula.
* Use factorial notation.
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!
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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.
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
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.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
Embracing GenAI - A Strategic ImperativePeter 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.
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.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
2. Concepts and Objectives
⚫ Objectives for this section are
⚫ 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.
3. Quadratic Functions
⚫ A function f is a quadratic function if
where a, b, and c are real numbers, and a 0.
⚫ This is called the general form of the quadratic function.
⚫ The graph of a quadratic function is a parabola whose
shape and position are determined by a, b, and c.
⚫ If a > 0, the parabola opens upward.
⚫ If a < 0, the parabola opens downward.
⚫ |a| determines the width of the parabola.
( )= + +
2
f x ax bx c
4. Vertex Formula
⚫ We can use the general form to find the equation for the
axis of symmetry and the vertex of the parabola:
⚫ If we use the quadratic formula, to
solve for the zeros, we find the value of x
halfway between them is always the equation
for the axis of symmetry.
⚫ To find the y-coordinate of the vertex, plug x into the
general form.
2
4
,
2
b b ac
x
a
− −
=
2
0
ax bx c
+ + =
,
2
b
x
a
= −
5. Vertex Form
⚫ The vertex (or standard) form of a quadratic function is
written
⚫ The graph of this function is the same as that of g(x)
translated h units horizontally and k units vertically.
This means that the vertex of f is at (h, k) and the axis of
symmetry is x = h.
⚫ The vertex form is just a vertical and horizontal shift of
the parent function y = x2.
( ) ( )
2
f x a x h k
= − +
6. Vertex Form (cont.)
⚫ Example: Graph the function and give its domain and
range.
( ) ( )
2
1
4 3
2
f x x
= − − +
7. Vertex Form (cont.)
⚫ Example: Graph the function and give its domain and
range.
Compare to : h = 4 and k = 3 (Notice
the signs!)
Vertex: (4, 3), axis of symmetry x = 4
We can graph this function by graphing the base
function and then shifting it.
( ) ( )
2
1
4 3
2
f x x
= − − +
( ) ( )
2
k
f x a x h
= − +
8. Vertex Form (cont.)
⚫ Example, cont.:
Let’s consider the graph of
⚫ Vertex is at (0, 0)
⚫ Passes through (2, ‒2) and
(4, ‒8).
⚫ (I picked 2 and 4 because of
the half.)
( )= − 2
1
2
g x x
9. Vertex Form (cont.)
⚫ Example, cont.:
To graph f, we just shift everything over 4 units to the
right and 3 units up.
Domain: (‒, )
Range: (‒, 3]
10. Writing a Function From a Graph
⚫ Given a graph of a quadratic function, to write the
equation of the function in general form:
⚫ Identify the vertex, (h, k). It will always be either the
highest or lowest point of the parabola.
⚫ Substitute the values of h and k in the function
⚫ Substitute the values of any point, other than the
vertex, on the parabola for x and f(x), and solve for a.
⚫ Expand and simplify to write in general form.
( ) ( )
2
f x a x h k
= − +
11. Writing a Function From a Graph
⚫ Example: Write an equation for the quadratic function g
in vertex form, and then in general form.
12. Writing a Function From a Graph
⚫ Example: Write an equation for the quadratic function g
in vertex form, and then in general form.
First, we identify the vertex,
which is at (–2, –3). Thus,
h = –2 and k= –3, and our
function looks like this:
( ) ( )
( ) ( )
( ) ( )
2
2
2 3
2 3
g x a x
g x a x
= − + −
= + −
13. Writing a Function From a Graph
⚫ Example: Write an equation for the quadratic function g
in vertex form, and then in general form.
Now, we pick a point on the
graph, for example, (0, –1),
to substitute in for x and
g(x).
( )
2
1 0 2 3
4 2
1
2
a
a
a
− = + −
=
=
14. Writing a Function From a Graph
⚫ Example: Write an equation for the quadratic function g
in vertex form, and then in general form.
So, the vertex form for this
graph is
( ) ( )
2
1
2 3
2
g x x
= + −
15. Writing a Function From a Graph
⚫ Example: Write an equation for the quadratic function g
in vertex form, and then in general form.
To write this in general
form, expand the binomial
and simplify:
( ) ( )
( )
2
2
2
2
1
2 3
2
1
4 4 3
2
1
2 2 3
2
1
2 1
2
g x x
x x
x x
x x
= + −
= + + −
= + + −
= + −
16. General Form to Vertex Form
⚫ Given a quadratic function in general form, to find the
vertex of the parabola (or to write in vertex form):
⚫ Identify a, b, and c from the general form.
⚫ Find h, the x-coordinate of the vertex, by substituting
a and b into .
⚫ Find k, the y-coordinate of the vertex, by evaluating
k .
⚫ Write in vertex form:
2
b
h
a
= −
( )
2
b
f h f
a
= = −
( ) ( )
2
f x a x h k
= − +
17. General Form to Vertex Form
⚫ Example: What is the vertex of the function?
( )= − +
2
6 7
f x x x
18. General Form to Vertex Form
⚫ Example: What is the vertex of the function?
From the general form, a = 1, b = –6, and c = 7.
The vertex is at (3, ‒2).
Vertex form is
( )= − +
2
6 7
f x x x
( )
6
3
2 1
h
−
= − = ( ) ( )
2
3 3 6 3 7
9 18 7 2
k f
= = − +
= − + = −
( ) ( )
2
3 2
f x x
= − −
19. Completing the Square
⚫ Another method we can use is “completing the square”
to transform it into vertex form (see section 2.5), but this
time we do everything on the right-hand side.
⚫ For example, the function is not a
binomial square. We can add 0 in the form of 52 – 52
(5 is half of 10), and group the parts that factor to a
binomial square:
( ) + −
= − +
2
2 2
10 30
5 5
f x x x
( )= − +
2
10 30
f x x x
( )
= − + − +
2 2 2
10 5 5 30
x x
( )
= − +
2
5 5
x
20. Completing the Square (cont.)
⚫ Example: What is the vertex of the function? (same ex.)
( )= − +
2
6 7
f x x x
21. Completing the Square (cont.)
⚫ Example: What is the vertex of the function?
The vertex is at (3, ‒2).
( )= − +
2
6 7
f x x x
( )− +
+
= − 2
2 2
3 7
3
6
x x
( )
= − − +
2
3 9 7
x
( )
= − −
2
3 2
x
6
3
2
=
22. General Form to Vertex Form
⚫ Probably the fastest method to find the vertex is to use
Desmos (surprise!) to graph the function. Desmos will
almost always automatically plot the vertex for you.
⚫ Using our same example:
The graph will show the
vertex as a gray dot.
23. General Form to Vertex Form
⚫ Probably the fastest method to find the vertex is to use
Desmos (surprise!) to graph the function. Desmos will
almost always automatically plot the vertex for you.
⚫ Using our same example:
The graph will show the
vertex as a gray dot. Click
to get the coordinates.
24. Practice
⚫ Find the vertex using the method of your choice. Then
give the axis of symmetry and the domain and range.
1. f(x) = x2 +8x + 5
2. g(x) = x2 – 5x + 8
3. h(x) = 3x2 + 12x – 5
25. Practice (cont.)
1. f(x) = x2 +8x + 5
The vertex is at (‒4, ‒11).
The axis of symmetry is x = –4.
The domain is (–∞, ∞), and the range is [–11, ∞)
( ) ( )
+
= + − +
2
2 2
8 4 4 5
f x x x
( )
= + − +
2
4 16 5
x
( )
= + −
2
4 11
x
( )
( ) ( )
2
8
4
2 1
4 8 4 5
16 32 5 11
h
k
= − = −
= − + − +
= − + = −
26. Practice (cont.)
2. g(x) = x2 – 5x + 8
The vertex is at , and the axis is .
The domain is (–∞, ∞), and the range is .
−
5 7
,
2 4
5
2
x = −
7
,
4
( )
2
5 5
2 1 2
5 5
5 8
2 2
25 25 25 50 32 7
8
4 2 4 4 4 4
h
k
−
= − =
= − +
= − + = − + =
27. Practice (cont.)
3. h(x) = –3x2 – 12x – 5
The vertex is at (‒2, 7), and the axis is x = –2.
The domain is (–∞, ∞), and the range is (–∞, 7].
( )
( ) ( )
2
12
2
2 3
3 2 12 2 5
12 24 5 7
h
k
−
= − = −
−
= − − − − −
= − + − =
Remember, the –3 flips the graph, so
the vertex is at the top.
28. Practice (cont.)
3. h(x) = –3x2 – 12x – 5 (Completing the Square method)
The vertex is at (‒2, 7), and the axis is x = –2.
The domain is (–∞, ∞), and the range is (–∞, 7].
( ) ( )
2
4 5
3
h x x x
= + −
−
( ) ( )
2 2 2
3
4 2 2
3 5
x x
= +
+
− + −
( )
2
3 2 12 5
x
= − + + −
( )
3 2 7
x
= − + +
Notice that I factored
the –3 out of the first
two expressions. This means that I’m
adding and
subtracting –3(22).
29. Practice (cont.)
3. h(x) = –3x2 – 12x – 5 (Desmos)
The vertex is at (‒2, 7), and the axis is x = –2.
The domain is (–∞, ∞), and the range is (–∞, 7].
30. Maximum and Minimum Values
⚫ The output of the quadratic function is the maximum or
minimum value of the function, depending on the
orientation of the parabola.
⚫ There are many real-world scenarios that involve
finding the maximum or minimum value of a quadratic
function.
⚫ If you are solving a problem involving quadratics,
and they ask for the maximum/minimum value, find
the vertex.
⚫ Depending on the problem, you will need either the
x-coordinate or the y-coordinate to solve the
problem.
31. Maximum and Minimum Values
⚫ Example: A backyard farmer wants to enclose a
rectangular space for a new garden within her fenced
backyard. She has purchased 80 feet of wire fencing to
enclose three sides, and she will use a section of the
backyard fence as the fourth side.
⚫ Find a formula for the area enclosed by the fence if
the sides of the fencing perpendicular to the existing
fence have length L.
⚫ What dimensions should she make her garden to
maximize the enclose area?
32. Maximum and Minimum Value
⚫ Example (cont.)
(We want to do this because the area will be A = LW.)
She only has 80 feet of
fencing available, so the
three sides would be
L + W + L = 80, or
2L + W = 80.
This means we can write W
in terms of L:
W = 80 – 2L
33. Maximum and Minimum Value
⚫ Example (cont.)
This is our quadratic function.
To find the maximum area, we find the vertex.
The area of the garden will
thus be
A = LW = L(80 – 2L)
or
2
2
80 2
2 80
A L L
L L
= −
= − +
34. Maximum and Minimum Value
⚫ Example (cont.):
( )
( ) ( )
2
80
20
2 2
2 20 80 20
800 1600 800
h
k
= − =
−
= − +
= − + =
( ) ( )( )
( )
2 2 2
2
2 40 20 2 20
2 20 800
20, 800
A L L
L
h k
= − − + − −
= − − +
= =
35. Maximum and Minimum Value
⚫ Example (cont.):
If you don’t see the parabola immediately, you may
have to zoom out and/or move the graph. If you still
don’t see it, make sure you typed the equation
correctly!
36. Maximum and Minimum Value
⚫ Example (cont.):
⚫ Whichever method you used, we have our vertex at
(20, 800). So what does it mean?
⚫ Remember our function was a function of L, which
was the length. So 20 represents the L at the
maximum area, which is 800.
⚫ To finish this out, the two perpendicular sides should
be 20 feet, the side between them will be
80 – 2(20) = 40 feet, and the maximum area for the
garden will thus be 800 square feet.