- The document discusses revising and sketching parabolic functions of the form y=ax+b, including how the parameters a and b affect the graph shape and position.
- It introduces parabolic equations in standard form y=a(x-p)+q and turning point form, identifying characteristics like the turning point, axes of symmetry, intercepts, and asymptotes.
- Examples are provided to demonstrate how to determine these characteristics, sketch the graph, and state the domain and range for parabolic functions given in equation form.
Properties of bivariate and conditional Gaussian PDFsAhmad Gomaa
Properties of bi-variate Gaussian pdf
Properties of conditional Gaussian pdf
Effect of correlation on bi-variate and conditional Gaussian pdf
Analytic expressions of bivariate and conditional Gaussian pdfs
3-D and 2-D contour plots of Gaussian pdfs
Conditional mean and variance
Matlab code of density functions plots
This Presentation Elliptical Curve Cryptography give a brief explain about this topic, it will use to enrich your knowledge on this topic. Use this ppt for your reference purpose and if you have any queries you'll ask questions.
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.
Properties of bivariate and conditional Gaussian PDFsAhmad Gomaa
Properties of bi-variate Gaussian pdf
Properties of conditional Gaussian pdf
Effect of correlation on bi-variate and conditional Gaussian pdf
Analytic expressions of bivariate and conditional Gaussian pdfs
3-D and 2-D contour plots of Gaussian pdfs
Conditional mean and variance
Matlab code of density functions plots
This Presentation Elliptical Curve Cryptography give a brief explain about this topic, it will use to enrich your knowledge on this topic. Use this ppt for your reference purpose and if you have any queries you'll ask questions.
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.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
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!
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.
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
2. Lesson outcomes:
• Revise parabolas of basic form y= ax + b
• Sketch parabolas
• Introduce parabolic equations in standard and
turning point form
• Identify and find shape, turning point, axes of
symmetry, intercepts and asymptotes
2
3. Revision
• Functions of the general form y= ax + b are called parabolic functions,
where a and b are constants.
• The effects of a and b on y= ax + b:
b affects the vertical shift:
• b >0 , y is shifted vertically upwards by b units
• The turning point of y is above the x-axis
• b is the y –intercept
2
2
y= x
2
y= x + 1
2
Graph
has
shifted 1
unit
upwards
b=0
b=1
4. • b < 0, y is shifted vertically downwards by b units.
• The turning point of y is below the x-axis
-1
0
y = x - 1
2
y= x
2
b=0
b= - 1
5.
6. Effects of a on the graph shape
• For a > 0; the graph of y is a “smile” and has a minimum turning
point (0;b). As the value of a becomes larger, the graph becomes
narrower.
• For a < 0; the graph of y is a “frown” and has a maximum turning
point (0;q). As the value of a becomes smaller, the graph becomes
narrower.
b >0
b =
0
b <0
8. Functions of the form
y=a(x+p)+q
p shifts the graph horizontally
• For p > 0, the graph is shifted to the left by p units.
• For p < 0, the graph is shifted to the right by p units.
The value of p also affects whether the turning point is to the left of the
y-axis (p>0) or to the right of the y-axis (p<0).
• The axis of symmetry is the line x=−p
9. q has the same effect as ‘b’
• For q >0, y shifts q units upwards
• For q < 0 y shifts q units downwards
10. Characteristics for
• The domain is {x:x∈R} because f(x)=y is defined for all x values
• The range depends on whether the value for a is positive or
negative. If a>0 we have:
• ( since a pefect square is always + )
• ( a is + )
• Thus f(x) q
• The range is therefore {y :y≥ q, y ∈ R} if a > 0. Similarly, if a < 0,
the range is {y : y ≤ q ,y ∈ R}
11. Example: DOMAIN AND RANGE
State the domain and range for g(x)= −2(x−1) + 3.
Determine the domain
The domain is {x:x∈R} because there is no value of x for which g(x) is
undefined.
Determine the range
The range of g(x) can be calculated from:
(x-1) 0
-2(x-1) 0
-2 (x-1) +3 3
g(x) 3
Therefore the range is {g(x):g(x)≤3} or in interval notation (−∞;3].
12. Intercepts
The y-intercept: let x=0.
example,
the y-intercept of g(x)= (x−1)+ 5
g(0)=(x−1) +5=(0−1) +5=6
This gives the point (0;6).
The x-intercept: let y=0.
g(x)=(x−1) +5
0= (x-1) + 5
-5 = (x-1)
which has no real solutions. Therefore, the
graph of g(x) lies above the x-axis and
does not have any x-intercepts.
13. Turning points
The turning point of the function f(x)=a(x+p) +q is determined by examining
the range of the function:
If a>0, f(x) has a minimum turning point and the range is [q;∞):
• The minimum value of f(x) is q.
• If f(x)=q, then a(x+p) =0, and therefore x=−p.
• This gives the turning point (−p;q).
If a<0, f(x) has a maximum turning point and the range is (−∞;q]:
• The maximum value of f(x) is q.
• If f(x)=q, then a(x+p) =0, and therefore x=−p.
• This gives the turning point (−p;q).
• Therefore the turning point of the quadratic function f(x)=a(x+p) +q is
(−p;q)
14. Determine the turning point of
y = 3x -6x -1
Step 1:
Y= 3 ( x - 2x ) – 1 use (–b/2) = (-2/2) = 1
Y= 3 (x - 2x +1 -1) -1 add and subtract the value next to bx and
Y= 3 (( x - 1 ) -1) -1 factorize the underlined equation
Y= 3 (x -1) -3-1 simplify
y= 3 ( x-1 ) -4
Step 2: determine the turning point (-p; q)
P= -1
q= -4
Thus the turning point is (-(-1); -4) = (1; -4)
Write the equation in the form y=a(x+p)
+q
15. Axis of symmetry
The axis of symmetry for f(x)=a(x+p) +q is the vertical line x=−p. The
axis of symmetry passes through the turning point (−p;q) and is parallel
to the y-axis.
16. Sketching graphs of the form
f(x)=a(x+p)+q
In order to sketch graphs of the form f(x)= a(x+p)
+q, we need to determine five characteristics:
• sign of a (+ or - )
• turning point (-p; q)
• y-intercept (x=0)
• x-intercept(s) (if they exist) (y=0)
• domain and range
17. Sketch the graph of y=−(1/2)(x+1)−3.
Mark the intercepts, turning point and the axis of symmetry. State the
domain and range of the function.
Examine the equation of the form y=a(x+p)+q
• We notice that a<0, therefore the graph is a “frown” and has a
maximum turning point.
• Determine the turning point (−p;q)
From the equation we know that the turning point is (-p; q) = (−1;−3).
• Determine the axis of symmetry x=−p
From the equation we know that the axis of symmetry is x=−1
• Determine the y-intercept
The y-intercept is obtained by letting x=0:
y=−(1/2)((0)+1) −3=(−1/2)−3= -
3.5
This gives the point (0;−3.5).
18. • Determine the x-intercepts
The x-intercepts are obtained by letting y=0:
0= -(1/2) (x + 1) -3
3x (-2)= (x+1)
which has no real solutions. Therefore, there are no x-intercepts
and the graph lies below the x-axis.
• Plot the points and sketch the graph.
State the domain and
range
Domain: {x:x∈R}
Range: {y:y≤−3,y∈R}
19. Sketch the graph of y=(1/2)x −4x + (7/2).
• Examine the equation of the form y=ax +bx+c
We notice that a>0, therefore the graph is a “smile” and has a
minimum turning point
• determine the turning point and the axis of symmetry
Check that the equation is in standard form and identify the
coefficients.
a=(1/2) ;b=−4; c=(7/2)
Calculate the x-value of the turning point using
x=−b/2a =−(−4 /2(1/2)) =4
Therefore the axis of symmetry is x=4.
Substitute x=4 into the original equation to obtain the
corresponding y-value.
y=−4.5
This gives the point (4;−4.5).
20. • Determine the y-intercept
The y-intercept is obtained by letting x=0:
y=(1/2)(0)−4(0)+(7/2) =7/2
This gives the point (0;7/2)
• Determine the x-intercepts
The x-intercepts are obtained by letting y=0:
0=(1/2)x −4x+(7/2) = x −8x+7=(x−1)(x−7)
Therefore x=1 or x=7. This gives the points (1;0) and (7;0)
• Plot the points and sketch the graph
Domain: {x:x∈R}
Range: {y:y ≥ −4.5 ,y∈R}