The document outlines methods for graphing functions that involve addition, subtraction, multiplication, and division of other functions. It provides steps such as graphing the individual functions separately first before combining them based on the operation. Examples are given to illustrate each method, including identifying points where the individual functions are equal to 0 or 1 and investigating asymptotes.
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A PowerPoint presentation on the Derivative as a Function. Includes example problems on finding the derivative using the definition, Power Rule, examining graphs of f(x) and f'(x), and local linearity.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
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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. (D) Addition & Subtraction of
Ordinatesy = f(x) and y = g(x)
y = f(x) + g(x) can be graphed by first graphing
separately and then adding their ordinates together.
3. (D) Addition & Subtraction of
Ordinatesy = f(x) and y = g(x)
y = f(x) + g(x) can be graphed by first graphing
separately and then adding their ordinates together.
NOTE: First locate points on y = f(x) + g(x) corresponding to f(x)=0 and
g(x)=0, then plot further points by addition and subtraction of ordinates
and finally locate the position of stationary points.
4. (D) Addition & Subtraction of
Ordinatesy = f(x) and y = g(x)
y = f(x) + g(x) can be graphed by first graphing
separately and then adding their ordinates together.
NOTE: First locate points on y = f(x) + g(x) corresponding to f(x)=0 and
g(x)=0, then plot further points by addition and subtraction of ordinates
and finally locate the position of stationary points.
1
e.g. y x
x
5. (D) Addition & Subtraction of
Ordinatesy = f(x) and y = g(x)
y = f(x) + g(x) can be graphed by first graphing
separately and then adding their ordinates together.
NOTE: First locate points on y = f(x) + g(x) corresponding to f(x)=0 and
g(x)=0, then plot further points by addition and subtraction of ordinates
and finally locate the position of stationary points.
1 y yx
e.g. y x
x
1
y
x
x
6. (D) Addition & Subtraction of
Ordinatesy = f(x) and y = g(x)
y = f(x) + g(x) can be graphed by first graphing
separately and then adding their ordinates together.
NOTE: First locate points on y = f(x) + g(x) corresponding to f(x)=0 and
g(x)=0, then plot further points by addition and subtraction of ordinates
and finally locate the position of stationary points.
1 y yx
e.g. y x
x
1
y
x
x
7. (D) Addition & Subtraction of
Ordinatesy = f(x) and y = g(x)
y = f(x) + g(x) can be graphed by first graphing
separately and then adding their ordinates together.
NOTE: First locate points on y = f(x) + g(x) corresponding to f(x)=0 and
g(x)=0, then plot further points by addition and subtraction of ordinates
and finally locate the position of stationary points.
1 y yx
e.g. y x
x
1
y
x
x
1
y x
x
8. y = f(x) – g(x) can be graphed by first graphing y = f(x) and y = – g(x)
separately and then adding the ordinates together.
9. y = f(x) – g(x) can be graphed by first graphing y = f(x) and y = – g(x)
separately and then adding the ordinates together.
e.g. y x sin x
10. y = f(x) – g(x) can be graphed by first graphing y = f(x) and y = – g(x)
separately and then adding the ordinates together.
e.g. y x sin x
11. y = f(x) – g(x) can be graphed by first graphing y = f(x) and y = – g(x)
separately and then adding the ordinates together.
e.g. y x sin x
12. y = f(x) – g(x) can be graphed by first graphing y = f(x) and y = – g(x)
separately and then adding the ordinates together.
e.g. y x sin x
13. y = f(x) – g(x) can be graphed by first graphing y = f(x) and y = – g(x)
separately and then adding the ordinates together.
e.g. y x sin x
14. y = f(x) – g(x) can be graphed by first graphing y = f(x) and y = – g(x)
separately and then adding the ordinates together.
e.g. y x sin x
15. (E) Multiplication of Functions
The graph of y = f(x). g(x) can be graphed by first graphing y = f(x) and
y= g(x) separately and then examining the sign of the product. Special
note needs to be made of points where f(x) = 0 or 1, or g(x) = 0 or 1.
NOTE: The regions on the number plane through which the graph must
pass should be shaded in as the first step.
29. (F) Division of Functions
f x
The graph of y can be graphed by;
g x
Step 1: First graph y = f(x) and y = g(x) separately.
Step 2: Mark in vertical asymptotes
31. (F) Division of Functions
f x
The graph of y can be graphed by;
g x
Step 1: First graph y = f(x) and y = g(x) separately.
Step 2: Mark in vertical asymptotes
Step 3: Shade in regions in which the curve must be (same as
multiplication.
39. (F) Division of Functions
f x
The graph of y can be graphed by;
g x
Step 1: First graph y = f(x) and y = g(x) separately.
Step 2: Mark in vertical asymptotes
Step 3: Shade in regions in which the curve must be (same as
multiplication.
Step 4: Investigate the behaviour of the function for large values of x
(find horizontal/oblique asymptotes)
40. (F) Division of Functions
f x
The graph of y can be graphed by;
g x
Step 1: First graph y = f(x) and y = g(x) separately.
Step 2: Mark in vertical asymptotes
Step 3: Shade in regions in which the curve must be (same as
multiplication.
Step 4: Investigate the behaviour of the function for large values of x
(find horizontal/oblique asymptotes)
x 1 x 2
y
x 2 x 1
x2 x 2
2
x x2
2x
1 2
x x2
41. (F) Division of Functions
f x
The graph of y can be graphed by;
g x
Step 1: First graph y = f(x) and y = g(x) separately.
Step 2: Mark in vertical asymptotes
Step 3: Shade in regions in which the curve must be (same as
multiplication.
Step 4: Investigate the behaviour of the function for large values of x
(find horizontal/oblique asymptotes)
x 1 x 2
y
x 2 x 1
x2 x 2
2
x x2
2x
1 2 horizontal asymptote : y 1
x x2