Finding the tangent line to the graph of a function at a single point can be extremely useful when interpreting the information that the function represents. So first to describe what a tangent line is: A tangent line of a function at one point shows the direction that the function is going at that point (Fig.1). Theoretically the tangent line is only touching the curve of the function at one single point, or the point of tangency. To find the equation of the tangent line, certain bits of information are required. One of these bits of information required is the slope of the tangent line. To find the slope of the tangent line of a function at a single point, the equation is used, assuming that “a” is the single point on the equation. The rest of this paper will be used to describe, through graphical methods, why this equation finds the slope of the tangent line.
The slope of any linear equation can be described as rise over run, y over x, the output of a function over the input of a function, or the dependent variable over the independent variable. All of these terms mean the same thing: the Y value on a graph over the X value on the graph. If the equation is examined closely, then it is clear that it represents a slope. The equation has the change of two output values, g(x) – g(a), over the change of two input values, x – a. The equation uses the change of an output, and the change of an input because two points on the graph is the minimum amount of information required to create a line.
Fig.2 and Fig.3 show how the two points on a graph can create an accurate tangent line. Fig.2 shows that two points on the function can create a secant line with a slope that is approximately close to the slope of the tangent line, but it is not accurate enough. Fig.3 shows that as the second point, D, on the function moves closer to the original point, C, the slope of the secant line approaches the slope of the tangent line. This movement shows how the...
...2. miêu tả biểu đồ đường thẳng
 Grammar and vocabulary 
Avoiding repetition
You will receive a higher mark if your writing uses a range of structures and vocabulary correctly rather than a limited number. For example, the candidate who writes:
The number of cases of X disease started at 50 in 1965 and then went up to 200 in 1970 and then went up to 500 in 1980 and then went down to zero in 1990.
will lose marks for being repetitive. You should therefore practise writing reports using a wide variety of terms to describe the different movements in the graphs and different structures to vary your writing.
Describing trends
Trends are changes or movements. These changes are normally expressed in numeric items, for example, population, production volumes or unemployment. There are three basic trends:
  
Expressing movement: nouns and verbs
For each trend there are a number of verbs and nouns to express the movement. We can use a verb of change, for example:
Unemployment levels fell
Or we can use a related noun, for example:
There was a fall in unemployment levels
Direction  Verbs  Nouns 
 Rose (to)
Increased (to)
Went up (to)
Climbed (to)
Boomed  A rise
An increase
Growth
An upward
trend
A boom (a dramatic rise) 
 Fell (to)
Declined (to)
Decreased (to)
Dipped (to)
Dropped (to)
Went down (to)
Slumped (to)
Reduced (to)  A decrease
A decline
A fall
A drop
A slump (a dramatic fall)
A reduction 
...
...Graphs and Function
What is the relation between the graphs and function and how was it applied in the real world?
Graphs are frequently used in national magazines and newspaper to present information about things such as the world’s busiest airports (O’Hare in China is first, Heathrow in London is sixth), about the advertisingdollar receivers in the United States (newspaper are first, radio is fourth) and about NCAA men’s golf team title winner (Yael is first, Houston is second). The function concept is very closely connected to graphs, and functions are the heart of mathematics.
I gathered my information from books especially algebra books and some are from the internet. I went to the library to look for some books and I borrowed some so I have many resources of information.
Many reallife relations between two quantities expressed in the form of equation are functions. To visualize these relationships, geometric figures called graphs are used. Modern technology provides us with graphing utilities needed to draw these graphs as well as enhance man’s knowledge of graphing techniques. Scientist and astronomers identify, visualize, and explore graphical patterns useful in analyzing data about the universe. Economist and businessmen draw mathematical models to find curves of best fit....
...CHAPTER 4 : FUNCTIONS AND THEIR GRAPHS
4.1 Definition of Function
A function from one set X to another set Y is a rule that assigns each element in X to one element in Y.
4.1.1 Notation
If f denotes a function from X to Y, we write
4.1.2 Domain and range
X is known as the domain of f and Y the range of f. (Note that domain and range are sets.)
4.1.3 Object and image
If and , then x and y are known respectively as the objects and images of f. We can write
, , .
We can represent a function in its general form, that is
f(x) = y.
Example 4.1
a. Given that , find f(0), f(1) and f(2).
Example 4.2
a. Given that , find the possible values of a such that
(a) f(a) = 4, (b) f(a) = a.
Solution
a. Given that , find f(0), f(1) and f(2).
b. Given that , find the possible values of a such that
(a) f(a) = 4, (b) f(a) = a.
(a)
(b)
4.2 Graphs of Functions
An equation in x and y defines a function y = f(x) if for each value of x there is only one value
of y.
Example:
y = 3x +1, , .
The graph of a function in the xy plane is the set of all points (x, y) where x is the
domain of f and y is the range of f.
Example
Figure 1 below shows the graph of a linear function, the square root function and a general function....
...this hypothesis would be to measure how much dissolved oxygen is found in the water when no fish is present. Then, measure the water with fish in it, increasing by 2 every time. Record the amount of fish and dissolved oxygen in a chart. When about 20 fish is reached, stop recording and put all the data into a linegraph. Using a linegraph will help see any patterns between dissolved oxygen and number of fish.
4. What are the independent and dependent variables?
The independent variable is the number of fish being observed. The dependent variable is the amount of dissolved oxygen found in the water.
5. What would be your control?
My control would be recording how much dissolved oxygen is found in the water when there are no fish present.
6. What type of graph would be appropriate for this data set? Why?
The type of graph that would be appropriate for this data set would be a linegraph. A linegraph would be appropriate because you would be able to tell if there were any patterns, trends, or relationships in the data or between the number of fish and amount of dissolved oxygen.
7. Graph the data from the table above.
8. Interpret the data from the graph made in Question 7.
...
...objective of linegraphs is to define raw data, making it easily understandable with a visual representation. By plotting data on a linegraph, you assign it a vertical and horizontal value that corresponds to the raw data determining the graph. For instance, if tracking annual sales at a retail store, the data would be defined by the amount of sales in dollars and the months during which these sales took place.
Interpreting Data
The visual representation of data in linegraphs allows users to easily interpret information over a given period of time. Once figures have been plotted on the graph and a line has connected all the points, users can visually analyze data that occurs over time without having to compare specific figures. For instance, monthly temperatures over the period of a year can be easily visualized by looking at the points on a linegraph. The user can quickly ascertain the warmer and colder months based on viewing the graph, without having to compare specific temperatures. This also allows users to quickly and visually identify trends in data.
Uses of LineGraphLinegraphs show the rate of change to a specific data set over a period of time. For instance, linegraphs are commonly used to check the...
...monthly budget. The highest designation of his budget will go to his foods with 45% of his total allowance. Next is for lodging with 30% followed by the projects and fare which will have 10%. The least designation for his budget will be for his savings which has 5% only.
2. BAR GRAPH
The bar graph shows the yearly tourist count for the provinces of region V. the province of Albay got the highest number of tourist with 450 000. It is followed by the provinces of Camarines Sur and Camarines Norte with 400 000 and 350 000 respectively. Sorsogon got 300 000 and Catanduanes with 250 000. The province of Masbate got the lowest number of tourist with 200 000.
3. LINE CHART
Here is a line chart for the number of absentees in class of Mr. Lozada for the 1st semester in 4 of her subjects. English has the most number of absents with 5 meetings. It is then followed by Math and Science with 4 and 3 meeting respectively while Filipino has the least absentees with only 2 meetings.
4. TABLES
KLINE DORMITORY SPORTS EQUIPMENT SPORT  NUMBER OF EQUIPMENT 
VOLLEYBALL  7 
BADMINTON  7 
SOCCER  4 
BASEBALL  12 

This table shows the number of sport equipment for each of the favorite sport of the KLINE scholars. The dormitory has the most sufficient sport equipment with 12. And Soccer is the sport with less number of equipment with only 4 sport equipment.
5.
PICTOGRAPH...
... 
Male 57 59% 
Female 40 41% 
Total 97 100% 
Table 1 reveals the sex profile of the respondents. As reflected on the table, the male has the larger percentage than the female. Out of 97 respondents, 57 or 59% are male while 40 or 41% are female.
To illustrate visually the sex profile, the graph is presented below.
20
Graph 1
[pic]
Gender Profile of the Respondents
Table 2
Analytical Skills
Respondents  S N Computed t Tabular t Decision Remark 
Male 10.84 2.95 57     
    0.33 1.9852 Accept Ho Significant 
5% level of significant and 26 degrees of freedom
Table 2 reveals the level of significant and the degrees of freedom. As reflected on...
...
This pack includes MAT 116 Assignment Functions and their Graphs
Resource: Appendix E, MyMathLab®
Due Date: Day 7 [Individual forum and MyMathLab®]
Complete Appendix E to apply the skills learned in Ch. 7 to a reallife situation.
Use Equation Editor to write mathematical expressions and equations in Appendix E.
Complete the Week Six Assignment: Ch. 7 Quiz in MyMathLab®. This assignment assesses content learned in Ch. 7.
Mathematics  General Mathematics
Find the LCM of 29 and 116
Solve the proportion. x//10
Which property of the real numbers is illustrated by the following statement?
Perform the addition on the 12hour clock. 5 + 11
Find the value of 327 in the mod 7 system
Simplify: 5(3 x + 3)
Evaluate the formula P = 2 l + 2 w when l = 4 in. and w = 4 in
Joe has $10,000 to purchase a used car. If the sales tax is 7% and the fee for title and license plates is $200, what is the maximum amount Joe can spend for a car?
Write 23 in base five
Which property is not true for all numbers on the 12hour clock?
Evaluate (6  6)  5 in the mod 12 system
Solve the inequality: x + 14 < 11
Which property of the real numbers is illustrated by the following statement?
Convert 3214five to base ten
Find the value of y in the mod 9 system.
5 x
Write 629 using Roman numerals
A community college has 3,000 students and 90...
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