just define the words properly in the chart and add detailed examples! (attached doc) Thanks in adv.

Extra Credit Unit 1 307A

Please define the following words and draw an example. For the last 3

words (18, 19, and 20), you choose the words you want to define.

Word

1. piecewise

function

2. greatest integer

function

3. least integer

function

4. Floor Function

5. Ceiling Function

6.

7.

8.

9.

Box Plot

Coordinate

linear model

correlation

coefficient

10. coefficient of

determination

11. Positive

association

12. negative

association

13. no association

14. ordered pair

15. regression line

16. Slope

17. Outlier

18. line of symmetry

19. reflection

symmetry

20. angle of rotation

Meaning

Example

Introduction:

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just define the words properly in the chart and add detailed examples! (attached doc) Thanks in adv.

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In the field of mathematics, there are numerous terms and concepts that individuals can encounter. Understanding these terms and concepts can be a key factor in mastering the subject matter and achieving success. In this article, we will define several fundamental concepts and provide examples to deepen your comprehension.

Description:

1. Piecewise Function:

A piecewise function is any function built from pieces of different functions over distinct intervals. Each interval has its own function, which is subsequently put together with other functions to form the whole function. A good example of a piecewise function is

f(x) = { x+5, if x<0
x^2, if x ≥0
2. Greatest Integer Function:
The greatest integer function, also known as the step function, is a function that calculates the largest integer less than or equal to the input. For instance, the greatest integer of 3.9 would be 3, and the greatest integer of -2.5 would be -3.
3. Least Integer Function:
The least integer function, on the other hand, calculates the smallest integer greater than or equal to the input. For example, the least integer of 2.4 is 3, and the least integer of -1.1 is -1.
4. Floor Function:
The floor function is another term for the greatest integer function, which has already been discussed. It defines the largest integer that is less than or equal to the input value.
5. Ceiling Function:
The ceiling function is an additional term for the least integer function, which has already been discussed. It is the smallest possible integer that is greater than or equal to the input value.
6. Box Plot:
A box plot, also known as a box-and-whisker plot, is a tool used for showcasing statistical information regarding the distribution of data. The plot shows quartiles of a dataset, including minimum and maximum values as well as outliers.
7. Coordinate:
A coordinate is every one of a set of values that represents the location of a point on a two-dimensional surface. Each location has an x-coordinate and a y-coordinate within the rectangular coordinate system.
8. Linear Model:
A linear model is a mathematical equation used for estimating the linear relationship between independent and dependent variables. The relationship between the independent and dependent variables follows a straight line.
9. Correlation Coefficient:
A correlation coefficient is a statistical measure of the degree to which two variables are linearly related. It ranges from -1 to 1 and indicates the strength and direction of the relationship between two variables.
10. Coefficient of Determination:
The coefficient of determination is a statistic that measures how much of the variation in the dependent variable is accounted for by the independent variable or variables in the model.
11. Positive Association:
Positive association is a relationship in which increases in one variable are linked with increases in the other variable as well. For example, when studying the connection between smoking and lung cancer, there is a positive association between the number of cigarettes smoked and an individual's likelihood of developing lung cancer.
12. Negative Association:
Negative association is a relationship in which increases in one variable are linked to decreases in the other variable as well. For example, an individual's likelihood of developing heart disease decreases as exercise frequency increases, suggesting a negative association.
13. No Association:
No association is a relationship in which there is no clear connection between two variables. For example, there is no link between ice cream consumption and sunscreen sales. Even though both are sold more frequently in the summer, there is no cause-and-effect relationship between the two variables.
14. Ordered Pair:
An ordered pair is a set of two numbers that are used to represent the location of a point on a two-dimensional Cartesian plane. The first number represents the x-coordinate, and the second number represents the y-coordinate.
15. Regression Line:
A regression line is a straight line that shows the relationship between two variables in a linear model. The aim of regression analysis is to obtain the most accurate possible line through a series of data points.
16. Slope:
The slope is the steepness of a straight line, which is determined by the relationship between the change in the y-coordinate and the change in the x-coordinate. The slope of a line is determined by the rise/run formula.
17. Outlier:
An outlier is an observation in a set of data that is significantly different from the other observations. It may be caused by a mistake, measurement error, or a genuine deviation in the data.
18. Line of Symmetry:
A line of symmetry is a line that can be drawn through the center of an object such that the two resulting parts are identical in terms of shape and size.
19. Reflection Symmetry:
Reflection symmetry is a type of symmetry that involves folding an object in half such that one half of the object maps onto the other half. For example, a butterfly has reflection symmetry since both wings appear to be identical.
20. Angle of Rotation:
The angle of rotation is the amount of rotation needed to bring an object or figure back to its original orientation. It is measured in degrees, and the degree of rotation is determined by the number of degrees a figure rotates around a point.
Objectives:
1. Students will be able to define and explain various mathematical functions such as piecewise function, greatest integer function, least integer function, floor function, and ceiling function.
2. Students will be able to interpret and analyze data using box plots and coordinate linear models.
3. Students will be able to describe the correlation between two variables using correlation coefficient, coefficient of determination, positive association, negative association, and no association.
4. Students will be able to apply geometric concepts such as line of symmetry, reflection symmetry, and angle of rotation to solve problems.
Learning Outcomes:
1. Define piecewise function and provide an example of a function that is defined by different formulas on different intervals.
2. Explain the greatest integer function and provide an example of a real-world situation where it can be used.
3. Describe the least integer function and provide an example of how it differs from the greatest integer function.
4. Define and explain the floor function and provide an example of a number that is rounded down to the nearest integer using the floor function.
5. Define and explain the ceiling function and provide an example of a number that is rounded up to the nearest integer using the ceiling function.
6. Construct and interpret box plots using given data sets and explain the significance of outliers in the data.
7. Formulate and interpret linear equations that represent real-world situations using coordinate linear models.
8. Describe the correlation between two variables using correlation coefficient and explain the difference between positive, negative, and no association.
9. Calculate and interpret the coefficient of determination for a given set of data and explain what it measures.
10. Identify and describe the characteristics of a regression line and explain how it is used to make predictions.
11. Calculate and interpret the slope of a line using two given points and explain its significance.
12. Identify outliers in a data set and explain their impact on the overall data analysis.
13. Define and explain the line of symmetry and provide an example of a figure that has a line of symmetry.
14. Describe reflection symmetry and provide an example of a figure that exhibits reflection symmetry.
15. Define angle of rotation and provide an example of a figure that has a certain degree of rotation.
Solution 1:
1. Piecewise function: A piecewise function is a mathematical function defined by multiple sub-functions, each applying to a different piece of the domain. For example,
f(x) =
{ x^2 + 2x - 1 if x > 0

2x + 3 if x ≤ 0 }

2. Greatest integer function: The greatest integer function, also known as the integer value function or floor function, takes the greatest integer less than or equal to its argument. For example,

f(x) = [x]

[3.5] = 3

3. Least integer function: The least integer function, also known as the ceiling function, takes the smallest integer greater than or equal to its argument. For example,

f(x) = ⌈x⌉

⌈3.5⌉ = 4

4. Floor Function: The floor function is a mathematical function that takes a real number and returns the largest integer less than or equal to that number. For example,

f(x) = ⌊x⌋

⌊3.5⌋ = 3

5. Ceiling Function: The ceiling function is a mathematical function that takes a real number and returns the smallest integer greater than or equal to that number. For example,

f(x) = ⌈x⌉

⌈3.5⌉ = 4

Solution 2:

6. Box plot: A box plot, also known as a box and whisker plot, is a graph used to display the distribution of a set of data. The box represents the middle 50% of the data, while the whiskers represent the minimum and the maximum values. For example,

______________________________________

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|__________________|_________________|

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|__________________|

7. Coordinate linear model: A coordinate linear model is a linear equation used to model the relationship between two variables. For example,

y = mx + b

where m is the slope and b is the y-intercept.

8. Correlation coefficient: The correlation coefficient is a value that measures the strength of the linear relationship between two variables. It ranges from -1 to 1, with values closer to -1 and 1 indicating a strong correlation, and values closer to 0 indicating a weak correlation. For example,

r = -0.75

9. Coefficient of determination: The coefficient of determination, also known as R-squared, is the proportion of the variance in the dependent variable that is explained by the independent variable in a linear regression model. The value of R-squared ranges from 0 to 1, with higher values indicating a better fit. For example,

R² = 0.85

10. Positive association: Positive association is a relationship between two variables in which they both increase or decrease together. For example,

As the temperature increases, so does the demand for ice cream.

11. Negative association: Negative association is a relationship between two variables in which one variable increases while the other decreases. For example,

As the price of a product increases, the demand for it decreases.

12. No association: No association is a relationship between two variables in which there is no observable pattern or relationship between them. For example,

The number of books read by a student has no observable relationship with their shoe size.

13. Ordered pair: An ordered pair is a pair of numbers used to locate a point on a coordinate plane. For example,

(3, 4) represents a point that is three units to the right and four units up from the origin.

14. Regression line: The regression line is a line that best fits the data in a scatter plot and represents the linear relationship between the two variables. For example,

y = 0.5x + 2 is the regression line that best fits the data in a scatter plot.

15. Slope: The slope is a measure of the steepness of a line. It is equal to the change in y (rise) over the change in x (run) between any two points on the line. For example,

The slope of the line passing through the points (2, 3) and (5, 8) is (8 – 3) / (5 – 2) = 5/3.

16. Outlier: An outlier is a data point that is far away from the other data points in a data set. For example,

In a set of exam scores, a score of 90 might be an outlier if most of the other scores are in the range of 60 to 70.

17. Line of symmetry: A line of symmetry is a line that divides a shape into two mirror-image halves. For example,

The line y = x is a line of symmetry for the graph of y = x².

18. Reflection symmetry: Reflection symmetry is a property of shapes that have matching halves when reflected across a line or plane. For example,

The letter M has reflection symmetry across the vertical line of symmetry.

19. Angle of rotation: The angle of rotation is the measure of the amount of rotation of a shape about a point. For example,

A circle has a rotation angle of 360°, while a triangle might have a rotation angle of 120°.

Suggested Resources/Books:

1. “Calculus: Early Transcendentals” by James Stewart – This book provides an in-depth explanation and examples of piecewise functions.

2. “A First Course in Discrete Mathematics” by Ian Anderson – This book explains the concepts of greatest and least integer functions.

3. “Numerical Methods Using MATLAB” by John H. Mathews and Kurtis D. Fink – This book provides a detailed description of the floor function and the ceiling function.

4. “Statistics For Dummies” by Deborah J. Rumsey – This book provides detailed information about the concepts of box plot, correlation coefficient, coefficient of determination, positive association, negative association, no association, ordered pair, regression line, slope, and outlier.

5. “Symmetry: A Journey into the Patterns of Nature” by Marcus Du Sautoy – This book describes the concepts of line of symmetry, reflection symmetry, and angle of rotation.

Similar asked questions:

1. What are the applications of piecewise functions in real life situations?

2. How are greatest integer functions and least integer functions used in computer science?

3. What is the difference between the floor function and the ceiling function?

4. How can box plots help in analyzing data?

5. What is the importance of slope in linear regression?

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