The document discusses functions and function notation. It begins by reviewing the slope-intercept form of a linear equation. It then defines different types of functions including polynomial, exponential, logarithmic, and trigonometric functions. It explains that a function relates an input to a unique output and can be represented by f(x) = mx + b or other notations like g(x) or h(x). The document demonstrates how to evaluate functions by plugging in values for x and concludes by assigning homework problems on functions.
This topic is based on research in Computer Science | Pattern Recognition | Probability and Statistics.
Here, we discuss Regression Line with a simple example. Basics of Line equation is demonstrated with a real-world example.
Eg: A railway track is paved in a line connecting various cities. It can be associated with the google maps with cities connected by the flight route / railway route.
Eg: Politicians demanding alternative route for railway constructions
This video is contributed by
https://sites.google.com/view/amarnath-r
Sample Program:
https://sites.google.com/view/amarnath-r/introduction-to-regression-analysis
This topic is based on research in Computer Science | Pattern Recognition | Probability and Statistics.
Here, we discuss Regression Line with a simple example. Basics of Line equation is demonstrated with a real-world example.
Eg: A railway track is paved in a line connecting various cities. It can be associated with the google maps with cities connected by the flight route / railway route.
Eg: Politicians demanding alternative route for railway constructions
This video is contributed by
https://sites.google.com/view/amarnath-r
Sample Program:
https://sites.google.com/view/amarnath-r/introduction-to-regression-analysis
1. Graph the exponential function by hand. Identify any asymptotes.docxjackiewalcutt
1. Graph the exponential function by hand. Identify any asymptotes and intercepts and determine whether the graph of the function is increasing or decreasing. (Enter NONE in any unused answer blanks.)
Equation of horizontal asymptote:
Equation of vertical asymptote:
Value of y-intercept
Value of x-intercept
The function is
2. Use the graph of y = 2x to match the function with its graph.
A
B
C
D
y = 2x – 4
y = 2x – 5
y = 2x + 4
y = 2–x
3. Use the graph of f to describe the transformation that yields the graph of g. Then sketch the graphs of f and g by hand.
f(x) = −2x, g(x) = 5 − 2x
The graph of g(x) = 5 − 2x is a vertical shift five units downward of f(x) = −2x.
The graph of g(x) = 5 − 2x is a horizontal shift five units to the left of f(x) = −2x.
The graph of g(x) = 5 − 2x is a vertical shift five units upward of f(x) = −2x.
The graph of g(x) = 5 − 2x is a horizontal shift five units to the right of f(x) = −2x.
Sketch the graphs of f and g.
4. Use a graphing utility to construct a table of values for the function. (Round your answers to three decimal places.)
x
f(x) = 5x − 3
-1
0
1
2
3
Sketch the graph of the function.
Identify any asymptotes of the graph. (Enter NONE in any unused answer blanks.)
vertical asymptote
x
=
horizontal asymptote
y
=
5. Use a graphing utility to construct a table of values for the function. (Round your answers to three decimal places.)
x
g(x) = 4 − e−3x
-4
-3
-2
-1
0
Sketch the graph of the function.
Identify any asymptotes of the graph. (Enter NONE in any unused answer blanks.)
vertical asymptote
x
=
horizontal asymptote
y
=
6. Fill in the blank.
If
x = ey,
then y = .
7. For what value of x is
ln x = ln 9?
x =
8. Write the logarithmic equation in exponential form. For example, the exponential form of
log5 25 = 2 is 52 = 25.
log2 512 = 9
=
9. Write the logarithmic equation in exponential form. For example, the exponential form of log5 25 = 2 is 52 = 25.
log
1
100,000,000
= -8
=
10. Write the exponential equation in logarithmic form. For example, the logarithmic form of
23 = 8 is log2(8) = 3.
43/2 = 8
log
=
11. Use a calculator to evaluate the function at the indicated value of x. Round your result to three decimal places.
f(x) = log10(x) x = 4/5
12. Solve the equation for x.
log10(102) = x
13. Write the logarithmic equation in exponential form. For example, the exponential form of ln(5) = 1.6094... is e1.6094... = 5. (Do not use ... in your answer.)
=
14. Write the exponential equation in logarithmic form. For example, the logarithmic form of
e2 = 7.3890 is ln 7.3890 = 2.
(Do not use ... in your answer.)
e2.2 = 9.0250
ln
=
15. Use the properties of natural logarithms to rewrite the expression.
5 ln(e5)
16. Use the properties of logarithms to rewrite and simplify the logarithmic expression.
ln
9
e9
17. Use the properties of logarithms to expand the expression as a ...
A function of two variables is defined similar to a function of one variable. It has a domain (in the plane) and a range. The graph of such a function is a surface in space and we try to sketch some.
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...
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
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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.
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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.
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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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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.
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1. September 14, 2009
Warm Up
Do you remember the
slopeintercept form of
a linear equation?
Use that form to find
the equation of the
graph to the right.
5. Function Notation
f (x) means "function of x"
NOT f times x !
y = mx + b
f (x) = mx + b
} mean same thing
Doesn't have to be "f". For example:
g(x) = mx + b
still mean the same thing
h(x) = mx + b
