The document discusses exponential functions and properties of exponents. It provides examples of simplifying expressions using properties like product of powers, power of a power, quotient of powers, and rational exponents. It also covers evaluating exponential functions, graphing them, and identifying linear vs exponential patterns in tables and graphs. The goal is for students to gain a conceptual understanding of working with exponential expressions and functions.
This document discusses composite functions and the order of operations when combining functions.
It provides an example of a mother converting the temperature of her baby's bath water from Celsius to Fahrenheit using two separate functions. The first function converts the Celsius reading to Fahrenheit, and the second maps the Fahrenheit reading to whether the water is too cold, alright, or too hot. Together these functions form a composite function.
Algebraically, a composite function f∘g(x) is defined as applying the inner function f first to the input x, and then applying the outer function g to the output of f. The domain of the inner function must be contained within the range of the outer function. The order of
The document discusses functions and their graphical representations. It defines key terms like domain, range, and one-to-one and many-to-one mappings. It then focuses on quadratic functions, showing that their graphs take characteristic U-shaped or inverted U-shaped forms. The document also examines inequalities involving quadratic expressions and how to determine the range of values satisfying such inequalities by analyzing the graph of the quadratic function.
This document outlines a summer course in linear algebra. It covers topics such as sets and operations on sets, relations and functions, polynomial theorems, and exponential and logarithmic equations. The course will teach students how to solve various types of word problems involving linear equations in two variables. It will also cover matrices, including Gaussian elimination and determinants.
This document provides examples and explanations for identifying key properties of quadratic functions presented in standard form, including:
1. Determining whether the graph opens upward or downward based on the sign of the leading coefficient a.
2. Finding the axis of symmetry by setting b/2a equal to x.
3. Finding the vertex by substituting the x-value from the axis of symmetry into the original function to determine the y-value.
4. Identifying the y-intercept from the constant term c.
Steps are demonstrated for graphing quadratic functions based on these properties, finding minimum/maximum values, and stating the domain and range. Examples analyze functions algebraically and graphically.
Basic galois field arithmatics required for error control codesMadhumita Tamhane
Knowledge of Galois Fields is must for understanding Error Control Codes. This presentation undertakes concepts of Galois Field required for understanding Error Control Codes in very simple manner, explaining its complex mathematical intricacies in a structured manner.
This chapter introduces exponential and logarithmic functions. Exponential functions take the form f(x) = bx and model continuous exponential growth or decay. Their inverse functions are logarithmic functions of the form x = logb y. Key points covered include:
- Exponential growth occurs when b > 1 and decay when 0 < b < 1.
- Logarithmic functions have a domain of positive real numbers and range of all real numbers. Their graphs are reflections of exponential graphs in the line y = x.
- Important properties are established, such as logb(bn) = n, logb1 = 0, and the relationship between exponential and logarithmic forms of an equation.
This document provides an index and overview of programs related to data science concepts in R. The programs cover topics like arithmetic operations on vectors, matrix operations, graphs, loops, and functions. The index lists 8 programs from August to October 2021 covering these topics. For each program, there is a brief description of the concepts covered and examples of R code and output.
The document discusses Venn diagrams and set operations. It provides examples of how to represent different set operations using Venn diagrams, such as (A ∪ B) ∩ C and (A ∩ C) ∪ (B ∩ C). It also discusses set notations and how to represent finite sets, intervals, and inequalities.
This document discusses composite functions and the order of operations when combining functions.
It provides an example of a mother converting the temperature of her baby's bath water from Celsius to Fahrenheit using two separate functions. The first function converts the Celsius reading to Fahrenheit, and the second maps the Fahrenheit reading to whether the water is too cold, alright, or too hot. Together these functions form a composite function.
Algebraically, a composite function f∘g(x) is defined as applying the inner function f first to the input x, and then applying the outer function g to the output of f. The domain of the inner function must be contained within the range of the outer function. The order of
The document discusses functions and their graphical representations. It defines key terms like domain, range, and one-to-one and many-to-one mappings. It then focuses on quadratic functions, showing that their graphs take characteristic U-shaped or inverted U-shaped forms. The document also examines inequalities involving quadratic expressions and how to determine the range of values satisfying such inequalities by analyzing the graph of the quadratic function.
This document outlines a summer course in linear algebra. It covers topics such as sets and operations on sets, relations and functions, polynomial theorems, and exponential and logarithmic equations. The course will teach students how to solve various types of word problems involving linear equations in two variables. It will also cover matrices, including Gaussian elimination and determinants.
This document provides examples and explanations for identifying key properties of quadratic functions presented in standard form, including:
1. Determining whether the graph opens upward or downward based on the sign of the leading coefficient a.
2. Finding the axis of symmetry by setting b/2a equal to x.
3. Finding the vertex by substituting the x-value from the axis of symmetry into the original function to determine the y-value.
4. Identifying the y-intercept from the constant term c.
Steps are demonstrated for graphing quadratic functions based on these properties, finding minimum/maximum values, and stating the domain and range. Examples analyze functions algebraically and graphically.
Basic galois field arithmatics required for error control codesMadhumita Tamhane
Knowledge of Galois Fields is must for understanding Error Control Codes. This presentation undertakes concepts of Galois Field required for understanding Error Control Codes in very simple manner, explaining its complex mathematical intricacies in a structured manner.
This chapter introduces exponential and logarithmic functions. Exponential functions take the form f(x) = bx and model continuous exponential growth or decay. Their inverse functions are logarithmic functions of the form x = logb y. Key points covered include:
- Exponential growth occurs when b > 1 and decay when 0 < b < 1.
- Logarithmic functions have a domain of positive real numbers and range of all real numbers. Their graphs are reflections of exponential graphs in the line y = x.
- Important properties are established, such as logb(bn) = n, logb1 = 0, and the relationship between exponential and logarithmic forms of an equation.
This document provides an index and overview of programs related to data science concepts in R. The programs cover topics like arithmetic operations on vectors, matrix operations, graphs, loops, and functions. The index lists 8 programs from August to October 2021 covering these topics. For each program, there is a brief description of the concepts covered and examples of R code and output.
The document discusses Venn diagrams and set operations. It provides examples of how to represent different set operations using Venn diagrams, such as (A ∪ B) ∩ C and (A ∩ C) ∪ (B ∩ C). It also discusses set notations and how to represent finite sets, intervals, and inequalities.
This document covers representing functions including objectives like defining functions and related terms, determining if a relation is a function, defining piecewise functions, and representing real-life situations
I am Boris M. I am a Computer Science Assignment Help Expert at programminghomeworkhelp.com. I hold MSc. in Programming, McGill University, Canada. I have been helping students with their homework for the past 7 years. I solve assignments related to Computer Science.
Visit programminghomeworkhelp.com or email support@programminghomeworkhelp.com.You can also call on +1 678 648 4277 for any assistance with Computer Science assignments.
This document defines key concepts related to relations and functions including ordered pairs, coordinate planes, relations, functions, graphical representations, domain and range, and the vertical line test. It provides examples and explanations of these terms. Ordered pairs represent points on a graph and functions are defined as sets of ordered pairs where no x-value is repeated, while relations allow repeated x-values. The vertical line test can be used to determine if a relation qualifies as a function by checking if a vertical line passes through only one point for each x-value.
The document discusses finite fields and related algebraic concepts. It begins by defining groups, rings, and fields. It then focuses on finite fields, particularly GF(p) fields consisting of integers modulo a prime p. It discusses finding multiplicative inverses in such fields using the extended Euclidean algorithm. As an example, it finds the inverse of 550 modulo 1759.
The document discusses functions and how to determine if a relationship represents a function using the vertical line test. It defines what constitutes a function and introduces function notation. Examples are provided of evaluating functions for given values of the independent variable and using functions to model and express relationships between variables.
This document contains a multiple choice quiz on discrete mathematics topics. There are 73 questions related to logic, sets, relations, functions, proofs, mathematical induction, counting, recurrence relations, graph theory, trees, discrete structures and automata theory. The questions are in a multiple choice format with a single correct answer out of 4 options for each question.
The document provides examples and explanations for writing and working with equations of lines using slope-intercept form. It includes examples of writing equations given the slope and y-intercept, given two points, using point-slope form, and determining whether lines are parallel or perpendicular. It also provides a multi-step example involving writing an equation to model combinations of small and large vans used to transport a class on a field trip. The examples progress from simple to more complex and include step-by-step solutions and guided practice problems.
Think Like Scilab and Become a Numerical Programming Expert- Notes for Beginn...ssuserd6b1fd
Notes for Scilab Programming. This notes includes the mathematics used behind scilab numerical programming. Illustrated with suitable graphics and examples. Each function is explained well with complete example. Helpful to beginners. GUI programming is also explained.
The document describes several algorithms:
1) Algorithm 5.1.4 tiles a deficient n x n board with trominoes by recursively dividing the board into quarters and placing a tromino in the center.
2) Algorithm 5.2.2 merges two sorted subarrays into a single sorted array.
3) Algorithm 5.2.3 (Mergesort) sorts an array by recursively dividing it in half and merging the sorted halves.
4) Algorithm 5.3.2 finds the closest pair of points in an array of points by recursively dividing the points into halves sorted by x-coordinate and merging while maintaining the distance between closest points.
The document discusses inverse functions and logarithms. It begins by introducing the concept of an inverse function using an example of bacteria population growth over time. It then defines inverse functions formally and discusses their key properties. The document explains that a function must be one-to-one to have an inverse function. It introduces the natural logarithm as the inverse of the exponential function with base e and discusses properties of logarithmic functions like logarithmic laws. Graphs of exponential, logarithmic and natural logarithmic functions are presented.
This document summarizes a machine learning homework assignment with 4 problems:
1) Probability questions about constructing random variables.
2) Questions about Poisson generalized linear models (GLM) including log-likelihood, prediction, and regularization.
3) Questions comparing square loss and logistic loss for an outlier point.
4) Questions about batch normalization including its effect on model expressivity and gradients.
1. The document discusses calculating average values of functions using integral methods. It provides examples of finding average values over different intervals for various functions, including constants, quadratic functions, sine functions, and periodic functions.
2. Key points covered include handling zero crossings, periodicity, and whether the quantity is absolute or algebraically additive. Piecewise methods and multiplying average values over one period are discussed for periodic functions.
3. Several example problems are worked through step-by-step to demonstrate finding average values over different intervals for various functions using integral relations.
This document defines functions and their key properties such as domain, range, and inverse functions. It provides examples of determining whether a function is one-to-one and finding its inverse. Composite functions are discussed, including ensuring the range of the inner function is contained within the domain of the outer function. Self-inverse functions are introduced, where the inverse of a function is equal to the function itself.
The document defines relations and functions. A relation is a set of ordered pairs, while a function is a special type of relation where each x-value is mapped to only one y-value. The domain is the set of x-values and the range is the set of y-values. Functions can be identified using the vertical line test or by mapping the relation to check if any x-values are mapped to multiple y-values. Evaluating functions involves substituting domain values into the function rule to find the corresponding range values.
The document discusses different types of functions and their graphs. It provides examples of constant functions, linear functions, quadratic functions, and absolute value functions. It shows how to graph each type of function by plotting points and describes their domains and ranges. For linear functions, the domain is all real numbers and the range is also all real numbers. For quadratic functions, the graph is a parabola and the range only includes positive values. Absolute value functions have a domain of all real numbers and a range that is positive and excludes negative values below the constant.
This document provides an overview of function notation and how to work with functions. It defines what a function is as a relation that assigns a single output value to each input value. It shows how functions can be represented using standard notation like f(x) and discusses evaluating functions by inputting values. Examples are provided of determining if a relationship represents a function, evaluating functions from tables and graphs, and solving functional equations.
The document discusses combinational logic circuits and Boolean algebra. It contains 17 questions covering topics such as:
1. The three basic logic operations - AND, OR, NOT - and their truth tables and logic diagrams.
2. Boolean algebra, functions, truth tables, and logic diagrams.
3. Identities of Boolean algebra, dual expressions, and the consensus theorem.
4. Minimizing logic functions using Boolean algebra identities and Karnaugh maps.
5. Minterms, maxterms, and representing logic functions with sums of products and products of sums.
6. Demorgan's theorem and complementing logic functions.
7. Simplifying logic functions through
The document discusses different methods for minimizing Boolean functions, including algebraic manipulation, tabular methods, and Karnaugh maps. The tabular method involves grouping minterms based on their binary representations and combining terms that differ by one bit. Karnaugh maps provide a visual way to group adjacent minterms and identify prime implicants to find a minimized expression. Both methods aim to cover all minterms with the fewest prime implicants.
Alg II Unit 4-2 Standard Form of a Quadratic Functionjtentinger
This document discusses the standard form of a quadratic function y = ax^2 + bx + c and provides examples of:
- Converting quadratic equations between standard and vertex form
- Using the properties of quadratic functions to find the vertex, axis of symmetry, minimum/maximum values, y-intercept
- Graphing quadratic functions by hand and with a calculator to identify key features
It also provides examples of interpreting real world applications that can be modeled with quadratic functions.
Unix and Shell Programming,
Q P Code: 60305.
Additional Mathematics I
Q P Code: 60306
Computer Organization and Architecture
Q P Code: 62303
Data Structures Using C
Q P Code: 60303
Discrete Mathematical Structures
Q P Code: 60304
Engineering Mathematics - III
Q P Code: 60301
Soft Skill Development
Q P Code: 60307
This document covers representing functions including objectives like defining functions and related terms, determining if a relation is a function, defining piecewise functions, and representing real-life situations
I am Boris M. I am a Computer Science Assignment Help Expert at programminghomeworkhelp.com. I hold MSc. in Programming, McGill University, Canada. I have been helping students with their homework for the past 7 years. I solve assignments related to Computer Science.
Visit programminghomeworkhelp.com or email support@programminghomeworkhelp.com.You can also call on +1 678 648 4277 for any assistance with Computer Science assignments.
This document defines key concepts related to relations and functions including ordered pairs, coordinate planes, relations, functions, graphical representations, domain and range, and the vertical line test. It provides examples and explanations of these terms. Ordered pairs represent points on a graph and functions are defined as sets of ordered pairs where no x-value is repeated, while relations allow repeated x-values. The vertical line test can be used to determine if a relation qualifies as a function by checking if a vertical line passes through only one point for each x-value.
The document discusses finite fields and related algebraic concepts. It begins by defining groups, rings, and fields. It then focuses on finite fields, particularly GF(p) fields consisting of integers modulo a prime p. It discusses finding multiplicative inverses in such fields using the extended Euclidean algorithm. As an example, it finds the inverse of 550 modulo 1759.
The document discusses functions and how to determine if a relationship represents a function using the vertical line test. It defines what constitutes a function and introduces function notation. Examples are provided of evaluating functions for given values of the independent variable and using functions to model and express relationships between variables.
This document contains a multiple choice quiz on discrete mathematics topics. There are 73 questions related to logic, sets, relations, functions, proofs, mathematical induction, counting, recurrence relations, graph theory, trees, discrete structures and automata theory. The questions are in a multiple choice format with a single correct answer out of 4 options for each question.
The document provides examples and explanations for writing and working with equations of lines using slope-intercept form. It includes examples of writing equations given the slope and y-intercept, given two points, using point-slope form, and determining whether lines are parallel or perpendicular. It also provides a multi-step example involving writing an equation to model combinations of small and large vans used to transport a class on a field trip. The examples progress from simple to more complex and include step-by-step solutions and guided practice problems.
Think Like Scilab and Become a Numerical Programming Expert- Notes for Beginn...ssuserd6b1fd
Notes for Scilab Programming. This notes includes the mathematics used behind scilab numerical programming. Illustrated with suitable graphics and examples. Each function is explained well with complete example. Helpful to beginners. GUI programming is also explained.
The document describes several algorithms:
1) Algorithm 5.1.4 tiles a deficient n x n board with trominoes by recursively dividing the board into quarters and placing a tromino in the center.
2) Algorithm 5.2.2 merges two sorted subarrays into a single sorted array.
3) Algorithm 5.2.3 (Mergesort) sorts an array by recursively dividing it in half and merging the sorted halves.
4) Algorithm 5.3.2 finds the closest pair of points in an array of points by recursively dividing the points into halves sorted by x-coordinate and merging while maintaining the distance between closest points.
The document discusses inverse functions and logarithms. It begins by introducing the concept of an inverse function using an example of bacteria population growth over time. It then defines inverse functions formally and discusses their key properties. The document explains that a function must be one-to-one to have an inverse function. It introduces the natural logarithm as the inverse of the exponential function with base e and discusses properties of logarithmic functions like logarithmic laws. Graphs of exponential, logarithmic and natural logarithmic functions are presented.
This document summarizes a machine learning homework assignment with 4 problems:
1) Probability questions about constructing random variables.
2) Questions about Poisson generalized linear models (GLM) including log-likelihood, prediction, and regularization.
3) Questions comparing square loss and logistic loss for an outlier point.
4) Questions about batch normalization including its effect on model expressivity and gradients.
1. The document discusses calculating average values of functions using integral methods. It provides examples of finding average values over different intervals for various functions, including constants, quadratic functions, sine functions, and periodic functions.
2. Key points covered include handling zero crossings, periodicity, and whether the quantity is absolute or algebraically additive. Piecewise methods and multiplying average values over one period are discussed for periodic functions.
3. Several example problems are worked through step-by-step to demonstrate finding average values over different intervals for various functions using integral relations.
This document defines functions and their key properties such as domain, range, and inverse functions. It provides examples of determining whether a function is one-to-one and finding its inverse. Composite functions are discussed, including ensuring the range of the inner function is contained within the domain of the outer function. Self-inverse functions are introduced, where the inverse of a function is equal to the function itself.
The document defines relations and functions. A relation is a set of ordered pairs, while a function is a special type of relation where each x-value is mapped to only one y-value. The domain is the set of x-values and the range is the set of y-values. Functions can be identified using the vertical line test or by mapping the relation to check if any x-values are mapped to multiple y-values. Evaluating functions involves substituting domain values into the function rule to find the corresponding range values.
The document discusses different types of functions and their graphs. It provides examples of constant functions, linear functions, quadratic functions, and absolute value functions. It shows how to graph each type of function by plotting points and describes their domains and ranges. For linear functions, the domain is all real numbers and the range is also all real numbers. For quadratic functions, the graph is a parabola and the range only includes positive values. Absolute value functions have a domain of all real numbers and a range that is positive and excludes negative values below the constant.
This document provides an overview of function notation and how to work with functions. It defines what a function is as a relation that assigns a single output value to each input value. It shows how functions can be represented using standard notation like f(x) and discusses evaluating functions by inputting values. Examples are provided of determining if a relationship represents a function, evaluating functions from tables and graphs, and solving functional equations.
The document discusses combinational logic circuits and Boolean algebra. It contains 17 questions covering topics such as:
1. The three basic logic operations - AND, OR, NOT - and their truth tables and logic diagrams.
2. Boolean algebra, functions, truth tables, and logic diagrams.
3. Identities of Boolean algebra, dual expressions, and the consensus theorem.
4. Minimizing logic functions using Boolean algebra identities and Karnaugh maps.
5. Minterms, maxterms, and representing logic functions with sums of products and products of sums.
6. Demorgan's theorem and complementing logic functions.
7. Simplifying logic functions through
The document discusses different methods for minimizing Boolean functions, including algebraic manipulation, tabular methods, and Karnaugh maps. The tabular method involves grouping minterms based on their binary representations and combining terms that differ by one bit. Karnaugh maps provide a visual way to group adjacent minterms and identify prime implicants to find a minimized expression. Both methods aim to cover all minterms with the fewest prime implicants.
Alg II Unit 4-2 Standard Form of a Quadratic Functionjtentinger
This document discusses the standard form of a quadratic function y = ax^2 + bx + c and provides examples of:
- Converting quadratic equations between standard and vertex form
- Using the properties of quadratic functions to find the vertex, axis of symmetry, minimum/maximum values, y-intercept
- Graphing quadratic functions by hand and with a calculator to identify key features
It also provides examples of interpreting real world applications that can be modeled with quadratic functions.
Unix and Shell Programming,
Q P Code: 60305.
Additional Mathematics I
Q P Code: 60306
Computer Organization and Architecture
Q P Code: 62303
Data Structures Using C
Q P Code: 60303
Discrete Mathematical Structures
Q P Code: 60304
Engineering Mathematics - III
Q P Code: 60301
Soft Skill Development
Q P Code: 60307
This document provides instructions on how to evaluate algebraic expressions using a Casio fx-350ES PLUS calculator. It includes an example of evaluating the expression a2+b3 when a=4 and b=2. The steps shown are to assign the values 4 and 2 to the variables a and b using the calculator's storage function, then using the variables to calculate the expression. Exercises are provided to evaluate other algebraic expressions using the given calculator.
This document provides instructions on how to evaluate algebraic expressions using a Casio fx-350ES PLUS calculator. It includes an example of evaluating the expression a2+b3 when a=4 and b=2. The steps shown are to assign the values 4 and 2 to the variables a and b using the calculator's storage function, then using the variables to calculate the expression. Exercises are provided to evaluate other algebraic expressions using the given calculator.
This document provides information about determinants of square matrices:
- It defines the determinant of a matrix as a scalar value associated with the matrix. Determinants are computed using minors and cofactors.
- Properties of determinants are described, such as how determinants change with row/column operations or identical rows/columns.
- Examples are provided to demonstrate computing determinants by expanding along rows or columns and using cofactors and minors.
- Applications of determinants include finding the area of triangles and solving systems of linear equations.
1. The document discusses polynomials, including algebraic expressions, variables, constants, terms, evaluation of expressions, laws of exponents, classification by degree and number of terms, and standard form.
2. Operations on polynomials are covered, including removing grouping symbols, addition, subtraction, and multiplying polynomials.
3. Polynomials can be classified based on degree, number of terms, and type of numerical coefficients. They can be in the standard form of highest degree term to constant term.
This document discusses functions and their representations. It begins by defining relations and functions, and providing examples of each. It then discusses different types of functions including linear, quadratic, constant, identity, absolute value, and piecewise functions. Examples are provided for each type. The document ends with exercises asking the reader to determine if given relations are functions, and to identify what type of function is being described.
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or
Call us at : 08263069601
APLICACIONES DE LA DERIVADA EN LA CARRERA DE (Mecánica, Electrónica, Telecomu...WILIAMMAURICIOCAHUAT1
El cálculo diferencial proporciona información sobre el comportamiento de las funciones
matemáticas. Todos estos problemas están incluidos en el alcance de la optimización de funciones y pueden resolverse aplicando cálculo
The document discusses rules for operations involving exponents, including:
- Multiplying powers with the same base by adding the exponents
- Dividing powers with the same base by subtracting the exponents
- Raising a power to a power by multiplying the exponents
Examples are provided to illustrate multiplying, dividing, and raising powers to powers according to these rules. A quiz with problems applying the rules concludes the document.
1. The document provides instructions for simplifying square root expressions using properties like the product and quotient properties. It also covers adding, subtracting, and rationalizing denominators of square root expressions.
2. Examples are given to practice simplifying expressions, adding/subtracting roots, and rationalizing denominators. A word problem asks students to find the width of a square poster using its given area.
3. A quiz is outlined to assess understanding of simplifying roots, rationalizing denominators, and adding/subtracting roots. Students are instructed to complete worksheets and ask any remaining questions.
QUADRATIC FUNCTIONS AND EQUATIONS IN ONE VARIABLE(STUDENTS).pdfSim Ping
This document discusses quadratic functions and equations involving one variable with the highest power being 2. It defines quadratic expressions and determines whether expressions are quadratic. It covers determining the values of a, b, and c for quadratic expressions. It discusses key features of quadratic functions including the turning point (maximum/minimum), the coordinate of the turning point, the y-intercept, x-intercepts, and the equation of the axis of symmetry. Examples are provided to illustrate these concepts and have students practice identifying these features from graphs and expressions.
This MATLAB program models a 300km transmission line using short line and nominal pi approximations to calculate the sending end voltage, current, power factor and power. The line parameters, receiving end voltage and load are used as inputs. The program calculates the parameters for both methods for lengths from 10-600km and plots the values to compare the results. The theoretical calculations are also shown. The program verifies that the nominal pi method gives more accurate results for longer lines as expected.
Mauricio opened a bank account with $20 and deposits $10 each week. His account balance can be modeled as a linear function f(x) = 20 + 10x, where x is the number of weeks and f(x) is the balance in dollars. The function shows that after 0 weeks the balance is $20, after 1 week it is $30, after 2 weeks $40, and so on, increasing by $10 each week.
This learner's module discusses about the topic of Radical Expressions. It also discusses the definition of Rational Exponents. It also teaches about writing an expression in a radical or exponential form. It also teaches about how to simplify expressions involving rational exponents.
The document discusses quadratic functions, including how to recognize a quadratic function based on its standard form, how to plot graphs of quadratic functions based on either tabulated values or a given function, and examples of determining if a function is quadratic and plotting quadratic graphs. Key aspects covered are the standard form of a quadratic function as f(x) = ax^2 + bx + c where a ≠ 0, recognizing quadratic functions, tabulating x and f(x) values for a given quadratic function, and using Geometer's Sketchpad software to plot the graphs.
SAMPLE QUESTIONExercise 1 Consider the functionf (x,C).docxanhlodge
SAMPLE QUESTION:
Exercise 1: Consider the function
f (x,C)=
sin(C x)
Cx
(a) Create a vector x with 100 elements from -3*pi to 3*pi. Write f as an inline or anonymous function
and generate the vectors y1 = f(x,C1), y2 = f(x,C2) and y3 = f(x,C3), where C1 = 1, C2 = 2 and
C3 = 3. Make sure you suppress the output of x and y's vectors. Plot the function f (for the three
C's above), name the axis, give a title to the plot and include a legend to identify the plots. Add a
grid to the plot.
(b) Without using inline or anonymous functions write a function+function structure m-file that does
the same job as in part (a)
SAMPLE LAB WRITEUP:
MAT 275 MATLAB LAB 1 NAME: __________________________
LAB DAY and TIME:______________
Instructor: _______________________
Exercise 1
(a)
x = linspace(-3*pi,3*pi); % generating x vector - default value for number
% of pts linspace is 100
f= @(x,C) sin(C*x)./(C*x) % C will be just a constant, no need for ".*"
C1 = 1, C2 = 2, C3 = 3 % Using commans to separate commands
y1 = f(x,C1); y2 = f(x,C2); y3 = f(x,C3); % supressing the y's
plot(x,y1,'b.-', x,y2,'ro-', x,y3,'ks-') % using different markers for
% black and white plots
xlabel('x'), ylabel('y') % labeling the axis
title('f(x,C) = sin(Cx)/(Cx)') % adding a title
legend('C = 1','C = 2','C = 3') % adding a legend
grid on
Command window output:
f =
@(x,C)sin(C*x)./(C*x)
C1 =
1
C2 =
2
C3 =
3
(b)
M-file of structure function+function
function ex1
x = linspace(-3*pi,3*pi); % generating x vector - default value for number
% of pts linspace is 100
C1 = 1, C2 = 2, C3 = 3 % Using commans to separate commands
y1 = f(x,C1); y2 = f(x,C2); y3 = f(x,C3); % function f is defined below
plot(x,y1,'b.-', x,y2,'ro-', x,y3,'ks-') % using different markers for
% black and white plots
xlabel('x'), ylabel('y') % labeling the axis
title('f(x,C) = sin(Cx)/(Cx)') % adding a title
legend('C = 1','C = 2','C = 3') % adding a legend
grid on
end
function y = f(x,C)
y = sin(C*x)./(C*x);
end
Command window output:
C1 =
1
C2 =
2
C3 =
3
More instructions for the lab write-up:
1) You are not obligated to use the 'diary' function. It was presented only for you convenience. You
should be copying and pasting your code, plots, and results into some sort of "Word" type editor that
will allow you to import graphs and such. Make sure you always include the commands to generate
what is been asked and include the outputs (from command window and plots), unless the pr.
This document provides an overview of 14 labs covering topics in digital signal processing using MATLAB. The labs progress from basic introductions to MATLAB and signals and systems concepts to more advanced topics like filters, the z-transform, the discrete Fourier transform, image processing, and signal processing toolboxes. Lab 1 focuses on introducing basic MATLAB operations and functions for defining variables, vectors, matrices, and m-files.
Similar to Alg1 power points_-_unit_7_-_exponents_and_exponential_functions (20)
Alg1 power points_-_unit_7_-_exponents_and_exponential_functions
1. Unit Essential Questions
How can you simplify expressions involving exponents?
What are the key features and essential components of exponential
functions?
2. MACC.912.A-SSE.A.2: Use the structure of an expression to
identify ways to rewrite it.
MACC.912.F-IF.C.8b: Use the properties of exponents to
interpret expressions for exponential functions.
4. KEY CONCEPTS AND
VOCABULARY
PRODUCT OF POWERS
PROPERTY
POWER OFA POWER
PROPERTY
For every nonzero number a and integers
m and n,
Example:
For every nonzero number a and
integers m and n,
Example:
POWER OF A PRODUCT PROPERTY
For every nonzero numbers a and b and integer m,
Example:
am
×an
= am+n
am
( )
n
= am×n
x2
×x8
= x2+8
= x10 x3
( )
6
= x3×6
= x18
ab( )m
= am
bm
2x( )4
= 24
x4
=16x4
7. EXAMPLE 3:
SIMPLIFYING POWER OF A PRODUCT
Simplify.
a)
b)
c)
-3xy( )2
a2
b9
( )
4
2x2
yz3
( )
5
9x2
y2
a8
b36
32x10
y5
z15
8. Express the area as a monomial.
a) b)
EXAMPLE 4:
SIMPLIFYING USING GEOMETRIC FORMULAS
r = xyz2
h = x3
y2
b = 2x2
y
px2
y2
z4
x5
y3
9. Simplify.
a)
b)
c)
EXAMPLE 5:
SIMPLIFYING MORE CHALLENGING
EXPONENTIAL EXPRESSIONS
5
6
x3æ
èç
ö
ø÷
2
4y3
( ) 3
4
xy4æ
èç
ö
ø÷ -3x2
y2
( )
3x2
( )
2
2xy( )2
é
ë
ù
û
3
25
36
x6
-9x3
y9
576x10
y6
10. RATE YOUR UNDERSTANDING
MULTIPLICATION PROPERTIES OF
EXPONENTS
MACC.912.A-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
MACC.912.F-IF.C.8b: Use the properties of exponents to interpret expressions for exponential
functions.
RATING LEARNING SCALE
4
I am able to
• simplify expressions using the multiplication properties of
exponents in more challenging problems that I have never
previously attempted
3
I am able to
• simplify expressions using the multiplication properties of
exponents
2
I am able to
• simplify expressions using the multiplication properties of
exponents with help
1
I am able to
• identify the multiplications properties of exponents
TARGET
11. MACC.912.A-SSE.A.2: Use the structure of an expression to
identify ways to rewrite it.
MACC.912.F-IF.C.8b: Use the properties of exponents to
interpret expressions for exponential functions.
13. KEY CONCEPTS AND
VOCABULARY
QUOTIENT OF POWERS
PROPERTY
POWER OFA QUOTIENT
PROPERTY
For every nonzero number a and integers m and n,
Example:
For every nonzero numbers a and b and integer m,
Example:
ZERO EXPONENT PROPERTY NEGATIVE EXPONENT PROPERTY
For every nonzero number a,
Example:
For every nonzero number a and integer n,
Example:
am
an = am-n
a
b
æ
èç
ö
ø÷
m
=
am
bm
a0
=1 a-n
=
1
an
x5
x2 = x5-2
= x3 x
2
æ
èç
ö
ø÷
4
=
x4
24
=
x4
16
2xyz
p
æ
èç
ö
ø÷
0
= 1 3-4
=
1
34
18. Simplify.
a)
b)
c)
EXAMPLE 5:
SIMPLIFYING MORE CHALLENGING
EXPONENTIAL EXPRESSIONS
16x2
y-1
( )
0
4x0
y-4
z( )
-3
80
c2
d3
f
4c-3
d-4
æ
èç
ö
ø÷
-2
3x3
y2
( )
3
6x2
y-3
( )
-2
64z3
y12
16
c10
d14
f 2
972x13
19. RATE YOUR UNDERSTANDING
DIVISION PROPERTIES OF EXPONENTS
MACC.912.A-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
MACC.912.F-IF.C.8b: Use the properties of exponents to interpret expressions for exponential
functions.
RATING LEARNING SCALE
4
I am able to
• divide expressions using the properties of exponents for
more challenging problems that I have never previously
attempted
3
I am able to
• divide expressions using the properties of exponents
• simplify expressions containing negative and zero exponents
2
I am able to
• divide expressions using the properties of exponents with
help
• simplify expressions containing negative and zero exponents
with help
1
I am able to
• understand the division properties of exponents
TARGET
20. MACC.912.N-RN.A.1: Explain how the definition of the
meaning of rational exponents follows from extending the
properties of integer exponents to those values, allowing for
a notation for radicals in terms of rational exponents.
MACC.912.N-RN.A.2: Rewrite expressions involving
radicals and rational exponents using the properties of
exponents.
22. KEY CONCEPTS AND
VOCABULARY
RATIONAL EXPONENTS
If the nth root of a b is a real number and m and n are
positive integers, then
anda
1
n = an a
m
n = an
( )
m
24. Convert to exponential form.
a)
b)
c)
d)
EXAMPLE 2:
CONVERTING TO EXPONENTIAL FORM
x4
a5
b35
(2x)73
x
1
4
a
1
5
b
3
5
2
7
3 x
7
3
25. EXAMPLE 3:
CONVERTING TO RADICAL FORM
Convert to radical form.
a) b)
c) d)
t
1
2 x
3
7
3x
3
2 (4a)
3
5
t x37
3 x3
(4a)35
26. Evaluate.
a) b)
c) d)
EXAMPLE 4:
EVALUATING AN EXPRESSION WITH A
RATIONAL EXPONENT
64
1
6 8
2
3
1
81
æ
èç
ö
ø÷
1
4
625
3
4
2 4
1
3
125
27. Solve.
a) b)
c) d)
EXAMPLE 5:
SOLVING EXPONENTIAL EQUATIONS BY
REWRITING IN EXPONENTIAL FORM
8x
= 64 3x
= 27
3x
= 243 12x
=144
2 3
5 2
28. The frequency f in hertz of the nth key on a piano is .
If a middle C is the 40th key, what is the frequency of a middle C?
EXAMPLE 6:
MODELING EXPONENTIAL EXPRESSIONS IN
REAL-WORLD SITUATIONS
f = 440 2
1
2
æ
è
ç
ö
ø
÷
n-49
19.45 hertz
29. RATE YOUR UNDERSTANDING
RATIONAL EXPONENTS
MACC.912.N-RN.A.1: Explain how the definition of the meaning of rational exponents follows from extending
the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational
exponents.
MACC.912.N-RN.A.2: Rewrite expressions involving radicals and rational exponents using the properties of
exponents.
RATING LEARNING SCALE
4
I am able to
• evaluate and rewrite expressions involving radicals and rational
exponents in real-world situations or more challenging problems
that I have never previously attempted
3
I am able to
• evaluate and rewrite expressions involving radicals and rational
exponents
• solve equations involving expressions with rational exponents
2
I am able to
• evaluate and rewrite expressions involving radicals and rational
exponents with help
• solve equations involving expressions with rational exponents with
help
1
I am able to
• understand that I can use rational exponents to represent radicals
TARGET
30. MACC.912.F-IF.C.7e: Graph exponential and logarithmic functions,
showing intercepts and end behavior, and trigonometric functions,
showing period, midline, and amplitude.
MACC.912.F-LE.A.2: Construct linear and exponential functions,
including arithmetic and geometric sequences, given a graph, a
description of a relationship, or two input-output pairs. (include
reading these from a table).
MACC.912.F-IF.C.9: Compare properties of two functions each
represented in a different way (algebraically, graphically,
numerically in tables, or by verbal descriptions).
32. KEY CONCEPTS AND
VOCABULARY
EXPONENTIAL FUNCTIONS
If a ≠ 0 and b >0, then exponential functions are of the
form
y = abx
Notice: the variable x is an exponent
33. Does the table represent a linear or an exponential function?
Explain your reasoning.
a) b)
EXAMPLE 1:
IDENTIFYING LINEAR AND EXPONENTIAL
FUNCTIONS FROM A TABLE OF VALUES
x 1 2 3 4
y 3 6 12 24
x 1 2 3 4
y 10 13 16 19
Exponential
Common ratio of 2
Linear
Common difference of 3
34. Is the function linear or exponential? Explain your
reasoning.
a) b)
EXAMPLE 2:
IDENTIFYING LINEAR AND EXPONENTIAL
FUNCTIONS GIVEN A FUNCTION RULE
y = 3x + 4 y = 4
1
3
æ
èç
ö
ø÷
x
Linear
Equation is in
slope-intercept form
Exponential
Variable is in the exponent
35. EXAMPLE 3:
EVALUATING AN EXPONENTIAL FUNCTION
Evaluate for the given value.
a)
b)
c)
f (x) =15×(2)x
; f (3)
f (x) = -10×
1
3
æ
èç
ö
ø÷
x
; f (-1)
f (x) = 200×(5)x
; f (-2)
120
–30
8
36. Graph the exponential function.
a) b)
EXAMPLE 4:
GRAPHING AN EXPONENTIAL FUNCTION
f (x) = 2×(3)x
f (x) = 4 ×
1
2
æ
èç
ö
ø÷
x
37. Determine if the graph is exponential, linear, or neither.
a) b) c)
EXAMPLE 5:
IDENTIFYING AN EXPONENTIAL GRAPH
Linear Neither Exponential
38. Order the functions from least to greatest for f(100).
a) b) c)
EXAMPLE 6:
COMPARING LINEAR GROWTH TO
EXPONENTIAL GROWTH
f(x)
f (x) = 3x
x 1 2 3 4
y 2 4 8 16
3 – Lowest value
at f(100)
1 – Highest value
at f(100)
2 – Middle value
at f(100)
39. RATE YOUR UNDERSTANDING
EXPONENTIAL FUNCTIONS
MACC.912.F-IF.C.7e: Graph exponential and logarithmic functions, showing intercepts and end behavior, and
trigonometric functions, showing period, midline, and amplitude.
MACC.912.F-LE.A.2: Construct linear and exponential functions, including arithmetic and geometric sequences,
given a graph, a description of a relationship, or two input-output pairs. (include reading these from a table).
MACC.912.F-IF.C.9: Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions).
RATING LEARNING SCALE
4
I am able to
• evaluate and graph exponential functions in real-world situations or
more challenging problems that I have never previously attempted
3
I am able to
• evaluate and graph exponential functions
• identify data and graphs that represents exponential behavior
2
I am able to
• evaluate and graph exponential functions with help
• identify data and graphs that represents exponential behavior with
help
1
I am able to
• understand that exponential functions are non-linear and model an
initial amount that is multiplied by the same number
TARGET
40. MACC.912.F-IF.C.8b: Use the properties of exponents to interpret
expressions for exponential functions.
MACC.912.F-LE.A.1a: Prove that linear functions grow by equal
differences over equal intervals, and that exponential functions grow by
equal factors over equal intervals.
MACC.912.F-LE.A.3: Observe using graphs and tables that a quantity
increasing exponentially eventually exceeds a quantity increasing linearly,
quadratically, or (more generally) as a polynomial function.
MACC.912.A-SSE.B.3c: Use the properties of exponents to transform
expressions for exponential functions.
41. WARM UP
Explain why each expression is not in simplest form.
1)
2)
3)
4)
53
x2
x4
y-2
x-1
y2
z0
(2x)3
53 = 125
Negative Exponent
Negative Exponent
z0 = 1
Distribute Power
42. KEY CONCEPTS AND
VOCABULARY
EXPONENTIAL GROWTH AND DECAY
y = abx when b is between 0 and 1
Example: Graph y = 100(0.5)x
Exponential Decay
y = abx when b>1
Example: Graph y = 1(2)x
Exponential Growth
Exponential decay model:
a = initial amount, r = rate, t = time
Exponential growth model:
a = initial amount, r = rate, t = time
y = a(1- r)t
y = a(1+ r)t
Compound Interest – the interest earned or paid on both
the initial investment and previously earned interest.
43. Without graphing, determine whether the function represents
exponential growth or decay. Then find the y-intercept.
a) b) c) y = 0.455(3)x
EXAMPLE 1:
IDENTIFYING EXPONENTIAL GROWTH OR
DECAY
y = 3×
2
3
æ
èç
ö
ø÷
x
y =
1
3
×(2)x
Decay Growth Growth
y-intercept =
(0, 3)
y-intercept =
(0, 1/3)
y-intercept =
(0, 0.455)
44. Write an exponential function to model each situation. Find
each amount after the specified time.
a) A population of 2,155,000 grows 1.3% per year for 10
years.
b) A population of 824,000 decreases 1.4% per year for 18
years.
c) A new car that sells for $27,000 depreciates 25% each
year for 4 years.
EXAMPLE 2:
WRITING AND SOLVING EXPONENTIAL
MODELS
y = 2,155,000(1+ 0.013)10
y = 2,452,120
y = 824,000(1- 0.014)18
y = 639,310
y = 27,000(1- 0.25)4
y = $8542.97
45. EXAMPLE 3:
REAL-WORLD APPLICATIONS OF EXPONENTIAL
GROWTH AND DECAY
a) You invested $1000 in a savings account at the end of 6th
grade. The account pays 5% annual interest. How much
money will be in the account after 6 years?
b) Each year the local country club sponsors a tennis
tournament. Play starts with 128 participants. During
each round, half of the players are eliminated. How
many players remain after 5 rounds?
$1340.10
4 Players
46. A mother and father were negotiating an allowance with their
child. They offered him two scenarios that he could choose from.
The first scenario offered the child $20 every week. The second
scenario offered $0.01 the first week and the amount would
double every week after. Which scenario should the son choose?
Explain by graphing each function.
EXAMPLE 4:
COMPARING EXPONENTIAL GROWTH AND
LINEAR GROWTH
The son should choose the $0.01 option.
After around 15 weeks, the allowance is
significantly more than the linear $20 a
week option.
47. RATE YOUR UNDERSTANDING
EXPONENTIAL GROWTH AND DECAY
MACC.912.F-IF.C.8b: Use the properties of exponents to interpret expressions for exponential functions.
MACC.912.F-LE.A.1a: Prove that linear functions grow by equal differences over equal intervals, and that
exponential functions grow by equal factors over equal intervals.
MACC.912.F-LE.A.3: Observe using graphs and tables that a quantity increasing exponentially eventually
exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
MACC.912.A-SSE.B.3c: Use the properties of exponents to transform expressions for exponential functions.
RATING LEARNING SCALE
4
I am able to
• model exponential growth and decay in real-world situations
or more challenging problems that I have never previously
attempted
3
I am able to
• model exponential growth and decay
2
I am able to
• model exponential growth and decay with help
1
I am able to
• identify growth and decay looking at a model
TARGET
48. MACC.912.F-IF.A.3: Recognize that sequences are functions,
sometimes defined recursively, whose domain is a subset of the integers.
For example, the Fibonacci sequence is defined recursively by f(0) =
f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
MACC.912.F-BF.A.2: Write arithmetic and geometric sequences both
recursively and with an explicit formula, use them to model situations,
and translate between the two forms.
MACC.912.F-LE.A.1c: Recognize situations in which a quantity grows
or decays by a constant percent rate per unit interval relative to
another.
49. WARM UP
Describe the pattern. Find the next 2 terms in each
sequence.
1) 2, 4, 8, 16,… 2) -–1, –4, –16, –64
Multiply 2 to each terms
32, 64
Multiply 4 to each term
–256, –1024
50. KEY CONCEPTS AND
VOCABULARY
Geometric Sequence – has a common ratio, r, between a term and its
preceding term that is constant.
Common Ratio - the name of the ratio in a Geometric Sequence.
16, 8, 4, 2, 1,… is a geometric sequence with
First term: a1 = 16
Common ratio: r = 1/2
Explicit Formula
The nth term of a geometric sequence with first term a1 and common ratio
r is given by:
an = a1rn – 1, for n > 1
51. For the following sequences, describe the patterns and identify
the next 3 terms.
a) 1, 4, 16, 48,… b) 2, –4, 8, –16,…
EXAMPLE 1:
EXTENDING GEOMETRIC SEQUENCES
Multiply 4 to each term
192, 768, 3072
Multiply –2 to each term
32, –64, 128
52. For the following sequences, identify whether it is a geometric
sequence. If it is, find the common ratio.
a) 4, –8, 16, –32,… b) 3, 9, –27, –81, 243,…
EXAMPLE 2:
IDENTIFYING A GEOMETRIC SEQUENCE
Yes, r = –2 No
53. EXAMPLE 3:
WRITING AN EQUATION FOR A GEOMETRIC
SEQUENCE
Write an equation for the nth term of the sequence. Then
find a6.
a) 5, 2, 0.8, 0.32,… b) –2, –6, –18,…
an = 5(0.4)n-1
a6 = 0.0512
an = -2(3)n-1
a6 = -486
54. Identify if the sequence is arithmetic, geometric, or neither.
Explain.
a) 1/2, 1/4, 1/8,…
b) 2, –4, 8, 16, –32,…
c) 15, 12, 9, 6,…
EXAMPLE 4:
CLASSIFYING THE SEQUENCE
Geometric
Has a common ration of 1/2
Neither
No Common ratio or difference
Arithmetic
Common difference of –3
55. RATE YOUR UNDERSTANDING
GEOMETRIC SEQUENCES
MACC.912.F-IF.A.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a
subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) =
f(n) + f(n-1) for n ≥ 1.
MACC.912.F-BF.A.2: Write arithmetic and geometric sequences both recursively and with an explicit formula,
use them to model situations, and translate between the two forms.
MACC.912.F-LE.A.1c: Recognize situations in which a quantity grows or decays by a constant percent rate per
unit interval relative to another.
RATING LEARNING SCALE
4
I am able to
• use an explicit formula for a geometric sequence to solve
real world problems or more challenging problems that I
have never previously attempted
3
I am able to
• identify and apply geometric sequences
2
I am able to
• identify and apply geometric sequences with help
1
I am able to
• define a geometric sequence
TARGET