MHF4U k1+, Advanced Functions, 12, University Virtual High School, Unit Assignment: Trigonometric Functions and Graphs Assignment
Request the complete assignment now.
This document introduces methods for solving quadratic equations beyond factoring, including the square root property, completing the square, and the quadratic formula. It discusses how to determine the number and type of solutions based on the discriminant. The key steps are presented for solving quadratics, graphing quadratic functions as parabolas, and finding the domain and range. Piecewise-defined quadratic functions are also explained.
This document provides examples of factorizing polynomials using various factorization techniques. It contains 30 questions with step-by-step solutions demonstrating how to factorize expressions involving single variables, binomials, trinomials, and polynomials using formulas like difference of squares, perfect square trinomials, and grouping. The techniques shown include finding common factors, using factor trees, recognizing patterns, and applying factorization identities.
De thi-dap-an-tuyen-sinh-vao-lop-10-mon-toan-tinh-hai-duongLinh Nguyễn
Đề thi và đáp án tuyển sinh vào lớp 10 môn Toán tỉnh Hải Dương. Xem thêm các đề thi đáp án các tỉnh khác tại http://www.diemthi60s.com/on-thi-vao-lop-10/
The document provides solutions to recommended problems from a signals and systems textbook. It solves problems related to signal properties such as periodicity, even and odd signals, transformations of signals, and convolutions. Key steps and reasoning are shown for each part of each problem. Graphs and diagrams are included to illustrate signals and solutions.
This document introduces methods for solving quadratic equations beyond factoring, including the square root property, completing the square, and the quadratic formula. It discusses how to determine the number and type of solutions based on the discriminant. The key steps are presented for solving quadratics, graphing quadratic functions as parabolas, and finding the domain and range. Piecewise-defined quadratic functions are also explained.
This document provides examples of factorizing polynomials using various factorization techniques. It contains 30 questions with step-by-step solutions demonstrating how to factorize expressions involving single variables, binomials, trinomials, and polynomials using formulas like difference of squares, perfect square trinomials, and grouping. The techniques shown include finding common factors, using factor trees, recognizing patterns, and applying factorization identities.
De thi-dap-an-tuyen-sinh-vao-lop-10-mon-toan-tinh-hai-duongLinh Nguyễn
Đề thi và đáp án tuyển sinh vào lớp 10 môn Toán tỉnh Hải Dương. Xem thêm các đề thi đáp án các tỉnh khác tại http://www.diemthi60s.com/on-thi-vao-lop-10/
The document provides solutions to recommended problems from a signals and systems textbook. It solves problems related to signal properties such as periodicity, even and odd signals, transformations of signals, and convolutions. Key steps and reasoning are shown for each part of each problem. Graphs and diagrams are included to illustrate signals and solutions.
Math school-books-3rd-preparatory-2nd-term-khawagah-2019khawagah
This document is the introduction to a mathematics textbook for third preparatory year students. It discusses the book's organization and goals. The book is divided into units with lessons, exercises, and tests. It aims to make mathematics enjoyable and practical, helping students understand its importance and appreciate mathematicians. Color images and examples are used to illustrate concepts simply and excitingly to facilitate learning.
The document derives the quadratic formula step-by-step: [1] It begins with the standard quadratic equation ax2 + bx + c = 0 and transforms it through steps such as completing the square and applying the square root property, [2] This results in the familiar quadratic formula of x = (-b ± √(b2 - 4ac)) / 2a, which gives the solutions to any quadratic equation.
The document discusses notation and algebra of functions. It defines a function as a procedure that assigns a unique output to each valid input. Most mathematical functions are represented by formulas like f(x) = x^2 - 2x + 3, where f(x) is the name of the function, x is the input variable, and the formula defines the relationship between input and output. New functions can be formed using basic operations like addition, subtraction, multiplication, and division of existing functions. Examples are provided to demonstrate evaluating functions at given inputs and combining functions algebraically.
The document discusses solving systems of 3 linear equations with 3 unknowns. It provides examples of using the elimination method, which involves rewriting the system as two smaller systems, eliminating the same variable from each, solving the resulting system of 2 equations for the remaining 2 variables, then substituting back into one of the original equations to find the third variable. The solution is written as an ordered triple (x, y, z). It demonstrates this process on examples and encourages practicing this method.
1) This document discusses how to solve quadratic equations by graphing, including identifying the terms of a quadratic equation, finding the solutions by graphing, and graphing quadratic functions.
2) The key steps for graphing a quadratic function are to find the axis of symmetry using the standard form equation, find the vertex point, and find two other points to reflect across the axis of symmetry to complete the parabolic graph.
3) An example problem walks through graphing the quadratic equation y = x^2 - 4x by first finding the roots, vertex, and axis of symmetry, and then constructing a table to plot points and graph the parabola.
This document provides information about obtaining fully solved assignments. It instructs students to send their semester and specialization name to the email address "help.mbaassignments@gmail.com" or call the provided phone number. The document notes that email is preferred, and calls should only be made in emergency situations. It also provides the phone number for calling as 08263069601.
Polynomials with common monomial factors.pptxGeraldineArig2
This document contains a lesson on factoring polynomials. It begins with examples of factoring various polynomials and identifying the greatest common factor. It then discusses key terminology related to factoring like common monomial factor and factor. The document provides step-by-step worked examples of factoring polynomials. It concludes with a generalization section and seat work problems for students to practice factoring polynomials.
The document contains instructions for a mathematics exam consisting of 3 sections (A, B, C). Section A has 10 1-mark questions. Section B has 12 4-mark questions. Section C has 7 6-mark questions. Calculators are not permitted. Questions can have either internal choice or no choice. The document provides 10 sample 1-mark questions from Section A to illustrate the format and difficulty level.
The document discusses identifying quadratic, linear, and constant terms in functions. It then provides examples of determining if functions are quadratic or linear and finding the vertex and axis of symmetry of quadratic functions. The document concludes by using a table of data to model a real-world scenario with a quadratic function.
College algebra real mathematics real people 7th edition larson solutions manualJohnstonTBL
This document contains information about the College Algebra Real Mathematics Real People 7th Edition Larson textbook including:
- A link to download the solutions manual and test bank for the textbook
- An overview of the content covered in Chapter 2 on solving equations and inequalities, including linear equations, identities, conditionals, and more.
- 51 example problems from Chapter 2 with step-by-step solutions.
This document provides a sample question paper for Class XII Mathematics. It consists of 3 sections - Section A has 10 one-mark questions, Section B has 12 four-mark questions, and Section C has 7 six-mark questions. The paper is for 3 hours and carries a total of 100 marks. Some questions provide internal choices. Calculators are not permitted. Sample questions include finding inverse functions, evaluating integrals, solving differential equations, and probability questions.
Factoring 15.3 and 15.4 Grouping and Trial and Errorswartzje
This document discusses various methods for factoring trinomials of the form Ax^2 + Bx + C. It begins by outlining three main methods: trial and error, factoring by grouping, and the box method. It then provides examples of using the factoring by grouping method, demonstrating how to find two numbers whose product is AC and sum is B. The document also covers special cases like factoring the difference of squares using the form A^2 - B^2 = (A-B)(A+B), and factoring perfect square trinomials using the form A^2 + 2AB + B^2 = (A+B)^2. In all, it thoroughly explains the step-by
1) Use properties of logarithms to expand the following logarit.docxhirstcruz
1) Use properties of logarithms to expand the following logarithmic expression as much as possible.
Log
b
(√xy
3
/ z
3
)
A. 1/2 log
b
x - 6 log
b
y + 3 log
b
z
B. 1/2 log
b
x - 9 log
b
y - 3 log
b
z
C. 1/2 log
b
x + 3 log
b
y + 6 log
b
z
D. 1/2 log
b
x + 3 log
b
y - 3 log
b
z
2) Solve the following logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, to two decimal places, for the solution.
2 log x = log 25
A. {12}
B. {5}
C. {-3}
D. {25}
3) Write the following equation in its equivalent logarithmic form.
2
-4
= 1/16
A. Log
4
1/16 = 64
B. Log
2
1/24 = -4
C. Log
2
1/16 = -4
D. Log
4
1/16 = 54
4) Use properties of logarithms to condense the following logarithmic expression. Write the expression as a single logarithm whose coefficient is 1.
log
2
96 – log
2
3
A. 5
B. 7
C. 12
D. 4
5) Use the exponential growth model, A = A
0
e
kt
, to show that the time it takes a population to double (to grow from A
0
to 2A
0
) is given by t = ln 2/k.
A. A
0
= A
0
e
kt
; ln = e
kt
; ln 2 = ln e
kt
; ln 2 = kt; ln 2/k = t
B. 2A
0
= A
0
e; 2= e
kt
; ln = ln e
kt
; ln 2 = kt; ln 2/k = t
C. 2A
0
= A
0
e
kt
; 2= e
kt
; ln 2 = ln e
kt
; ln 2 = kt; ln 2/k = t
D. 2A
0
= A
0
e
kt
; 2 = e
kt
; ln 1 = ln e
kt
; ln 2 = kt; ln 2/k = t
oe
6) Find the domain of following logarithmic function.
f(x) = log (2 - x)
A. (∞, 4)
B. (∞, -12)
C. (-∞, 2)
D. (-∞, -3)
7) An artifact originally had 16 grams of carbon-14 present. The decay model A = 16e -0.000121t describes the amount of carbon-14 present after t years. How many grams of carbon-14 will be present in 5715 years?
A. Approximately 7 grams
B. Approximately 8 grams
C. Approximately 23 grams
D. Approximately 4 grams
8) Use properties of logarithms to expand the following logarithmic expression as much as possible.
log
b
(x
2
y) / z
2
A. 2 log
b
x + log
b
y - 2 log
b
z
B. 4 log
b
x - log
b
y - 2 log
b
z
C. 2 log
b
x + 2 log
b
y + 2 log
b
z
D. log
b
x - log
b
y + 2 log
b
z
9) The exponential function f with base b is defined by f(x) = __________, b > 0 and b ≠ 1. Using interval notation, the domain of this function is __________ and the range is __________.
A. bx; (∞, -∞); (1, ∞)
B. bx; (-∞, -∞); (2, ∞)
C. bx; (-∞, ∞); (0, ∞)
D. bx; (-∞, -∞); (-1, ∞)
10) Approximate the following using a calculator; round your answer to three decimal places.
3
√5
A. .765
B. 14297
C. 11.494
D. 11.665
11) Write the following equation in its equivalent exponential form.
4 = log
2
16
A. 2 log
4
= 16
B. 2
2
= 4
C. 4
4
= 256
D. 2
4
= 16
12) Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.
3
1-x
= 1/27
A. {2}
B. {-7}
C. {4}
D. {3}
13) Use properties of logarithms to expand the followin.
NUMERICAL METHODS WITH MATLAB : bisection,mueller's,newton-raphson,false poin...Parhamsagharchi
This document describes solving a nonlinear equation to determine the time when a Saturn V rocket reaches the speed of sound using various numerical methods. Specifically, it compares the bisection method, linear interpolation method, Newton-Raphson method, Mueller's method, and the x=g(x) method. For each method, it provides the MATLAB script used to solve the equation and displays the solution time and number of iterations. The root found for all methods was approximately 70.878 seconds.
The document provides examples and explanations for solving different types of equations, including:
1) Polynomial equations through factoring or the quadratic formula.
2) Rational equations by clearing denominators.
3) Radical equations by squaring both sides to remove radicals.
4) Absolute value equations by recognizing that |x-c|=r implies x=c±r.
The document also discusses solving power equations, finding zeros and domains of functions, and using properties of absolute values.
1) Use properties of logarithms to expand the following logarithm.docxdorishigh
1) Use properties of logarithms to expand the following logarithmic expression as much as possible.
Logb (√xy3 / z3)
A. 1/2 logb x - 6 logb y + 3 logb z
B. 1/2 logb x - 9 logb y - 3 logb z
C. 1/2 logb x + 3 logb y + 6 logb z
D. 1/2 logb x + 3 logb y - 3 logb z
2) Solve the following logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, to two decimal places, for the solution.
2 log x = log 25
A. {12}
B. {5}
C. {-3}
D. {25}
3) Write the following equation in its equivalent logarithmic form.
2-4 = 1/16
A. Log4 1/16 = 64
B. Log2 1/24 = -4
C. Log2 1/16 = -4
D. Log4 1/16 = 54
4) Use properties of logarithms to condense the following logarithmic expression. Write the expression as a single logarithm whose coefficient is 1.
log2 96 – log2 3
A. 5
B. 7
C. 12
D. 4
5) Use the exponential growth model, A = A0ekt, to show that the time it takes a population to double (to grow from A0 to 2A0 ) is given by t = ln 2/k.
A. A0 = A0ekt; ln = ekt; ln 2 = ln ekt; ln 2 = kt; ln 2/k = t
B. 2A0 = A0e; 2= ekt; ln = ln ekt; ln 2 = kt; ln 2/k = t
C. 2A0 = A0ekt; 2= ekt; ln 2 = ln ekt; ln 2 = kt; ln 2/k = t
D. 2A0 = A0ekt; 2 = ekt; ln 1 = ln ekt; ln 2 = kt; ln 2/k = toe
6) Find the domain of following logarithmic function.
f(x) = log (2 - x)
A. (∞, 4)
B. (∞, -12)
C. (-∞, 2)
D. (-∞, -3)
7) An artifact originally had 16 grams of carbon-14 present. The decay model A = 16e -0.000121t describes the amount of carbon-14 present after t years. How many grams of carbon-14 will be present in 5715 years?
A. Approximately 7 grams
B. Approximately 8 grams
C. Approximately 23 grams
D. Approximately 4 grams
8) Use properties of logarithms to expand the following logarithmic expression as much as possible.
logb (x2 y) / z2
A. 2 logb x + logb y - 2 logb z
B. 4 logb x - logb y - 2 logb z
C. 2 logb x + 2 logb y + 2 logb z
D. logb x - logb y + 2 logb z
9) The exponential function f with base b is defined by f(x) = __________, b > 0 and b ≠ 1. Using interval notation, the domain of this function is __________ and the range is __________.
A. bx; (∞, -∞); (1, ∞)
B. bx; (-∞, -∞); (2, ∞)
C. bx; (-∞, ∞); (0, ∞)
D. bx; (-∞, -∞); (-1, ∞)
10) Approximate the following using a calculator; round your answer to three decimal places.
3√5
A. .765
B. 14297
C. 11.494
D. 11.665
11) Write the following equation in its equivalent exponential form.
4 = log2 16
A. 2 log4 = 16
B. 22 = 4
C. 44 = 256
D. 24 = 16
12) Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.
31-x = 1/27
A. {2}
B. {-7}
C. {4}
D. {3}
13) Use properties of logarithms to expand the following logarithmic expression as much as possible.
logb (x2y)
A. 2 logy x + logx y
B. 2 logb x + logb y
C. logx - logb y
D. logb x – ...
This document discusses quadratic equations and their properties. It defines quadratic equations as equations of the form y=ax^2 +bx + c, where the highest power is 2. It explains that quadratic equations can be solved using the quadratic formula, x = -b ± √(b^2 - 4ac) / 2a. The number of solutions depends on the discriminant, b^2 - 4ac. If it is greater than 0, there are two solutions, if equal to 0 there is one solution, and if less than 0 there are no solutions. Examples are provided to demonstrate solving quadratic equations.
This document provides notes and formulas for mathematics topics covered in Form 1 through Form 4 in Malaysian secondary schools. It includes formulas and explanations for topics like solid geometry, circle theorems, polygons, factorisation, expansion of algebraic expressions, indices, algebraic fractions, linear equations, simultaneous equations, quadratic expressions, sets, statistics, trigonometry, angles of elevation and depression, and lines and planes. The document is intended to serve as a single reference for key mathematics concepts and formulas for secondary school students.
The document provides information and practice questions about linear functions and slope:
- It instructs the reader to prepare for an upcoming final exam and coordinate test.
- Several practice questions are provided about slope, equations of lines, graphs, and using linear functions to model real-world scenarios.
- The questions cover topics like determining the slope between two points, writing equations in point-slope form, interpreting slope in context, and using linear models to solve word problems.
Lecture 15 section 5.4 graph of sin & cosnjit-ronbrown
This document provides examples and procedures for graphing sine and cosine functions of the form y=a*sin(b(x-c))+d and y=a*cos(b(x-c))+d. It explains that the parameters a, b, c, and d control the amplitude, period, phase shift, and vertical shift of the graphs. Several examples are worked through step-by-step to demonstrate how to graph functions with different parameters. The key steps are to identify the function parameters, determine the period and phase shift, find the x-coordinates of important points, and connect those points to sketch the graph over one full cycle.
Math school-books-3rd-preparatory-2nd-term-khawagah-2019khawagah
This document is the introduction to a mathematics textbook for third preparatory year students. It discusses the book's organization and goals. The book is divided into units with lessons, exercises, and tests. It aims to make mathematics enjoyable and practical, helping students understand its importance and appreciate mathematicians. Color images and examples are used to illustrate concepts simply and excitingly to facilitate learning.
The document derives the quadratic formula step-by-step: [1] It begins with the standard quadratic equation ax2 + bx + c = 0 and transforms it through steps such as completing the square and applying the square root property, [2] This results in the familiar quadratic formula of x = (-b ± √(b2 - 4ac)) / 2a, which gives the solutions to any quadratic equation.
The document discusses notation and algebra of functions. It defines a function as a procedure that assigns a unique output to each valid input. Most mathematical functions are represented by formulas like f(x) = x^2 - 2x + 3, where f(x) is the name of the function, x is the input variable, and the formula defines the relationship between input and output. New functions can be formed using basic operations like addition, subtraction, multiplication, and division of existing functions. Examples are provided to demonstrate evaluating functions at given inputs and combining functions algebraically.
The document discusses solving systems of 3 linear equations with 3 unknowns. It provides examples of using the elimination method, which involves rewriting the system as two smaller systems, eliminating the same variable from each, solving the resulting system of 2 equations for the remaining 2 variables, then substituting back into one of the original equations to find the third variable. The solution is written as an ordered triple (x, y, z). It demonstrates this process on examples and encourages practicing this method.
1) This document discusses how to solve quadratic equations by graphing, including identifying the terms of a quadratic equation, finding the solutions by graphing, and graphing quadratic functions.
2) The key steps for graphing a quadratic function are to find the axis of symmetry using the standard form equation, find the vertex point, and find two other points to reflect across the axis of symmetry to complete the parabolic graph.
3) An example problem walks through graphing the quadratic equation y = x^2 - 4x by first finding the roots, vertex, and axis of symmetry, and then constructing a table to plot points and graph the parabola.
This document provides information about obtaining fully solved assignments. It instructs students to send their semester and specialization name to the email address "help.mbaassignments@gmail.com" or call the provided phone number. The document notes that email is preferred, and calls should only be made in emergency situations. It also provides the phone number for calling as 08263069601.
Polynomials with common monomial factors.pptxGeraldineArig2
This document contains a lesson on factoring polynomials. It begins with examples of factoring various polynomials and identifying the greatest common factor. It then discusses key terminology related to factoring like common monomial factor and factor. The document provides step-by-step worked examples of factoring polynomials. It concludes with a generalization section and seat work problems for students to practice factoring polynomials.
The document contains instructions for a mathematics exam consisting of 3 sections (A, B, C). Section A has 10 1-mark questions. Section B has 12 4-mark questions. Section C has 7 6-mark questions. Calculators are not permitted. Questions can have either internal choice or no choice. The document provides 10 sample 1-mark questions from Section A to illustrate the format and difficulty level.
The document discusses identifying quadratic, linear, and constant terms in functions. It then provides examples of determining if functions are quadratic or linear and finding the vertex and axis of symmetry of quadratic functions. The document concludes by using a table of data to model a real-world scenario with a quadratic function.
College algebra real mathematics real people 7th edition larson solutions manualJohnstonTBL
This document contains information about the College Algebra Real Mathematics Real People 7th Edition Larson textbook including:
- A link to download the solutions manual and test bank for the textbook
- An overview of the content covered in Chapter 2 on solving equations and inequalities, including linear equations, identities, conditionals, and more.
- 51 example problems from Chapter 2 with step-by-step solutions.
This document provides a sample question paper for Class XII Mathematics. It consists of 3 sections - Section A has 10 one-mark questions, Section B has 12 four-mark questions, and Section C has 7 six-mark questions. The paper is for 3 hours and carries a total of 100 marks. Some questions provide internal choices. Calculators are not permitted. Sample questions include finding inverse functions, evaluating integrals, solving differential equations, and probability questions.
Factoring 15.3 and 15.4 Grouping and Trial and Errorswartzje
This document discusses various methods for factoring trinomials of the form Ax^2 + Bx + C. It begins by outlining three main methods: trial and error, factoring by grouping, and the box method. It then provides examples of using the factoring by grouping method, demonstrating how to find two numbers whose product is AC and sum is B. The document also covers special cases like factoring the difference of squares using the form A^2 - B^2 = (A-B)(A+B), and factoring perfect square trinomials using the form A^2 + 2AB + B^2 = (A+B)^2. In all, it thoroughly explains the step-by
1) Use properties of logarithms to expand the following logarit.docxhirstcruz
1) Use properties of logarithms to expand the following logarithmic expression as much as possible.
Log
b
(√xy
3
/ z
3
)
A. 1/2 log
b
x - 6 log
b
y + 3 log
b
z
B. 1/2 log
b
x - 9 log
b
y - 3 log
b
z
C. 1/2 log
b
x + 3 log
b
y + 6 log
b
z
D. 1/2 log
b
x + 3 log
b
y - 3 log
b
z
2) Solve the following logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, to two decimal places, for the solution.
2 log x = log 25
A. {12}
B. {5}
C. {-3}
D. {25}
3) Write the following equation in its equivalent logarithmic form.
2
-4
= 1/16
A. Log
4
1/16 = 64
B. Log
2
1/24 = -4
C. Log
2
1/16 = -4
D. Log
4
1/16 = 54
4) Use properties of logarithms to condense the following logarithmic expression. Write the expression as a single logarithm whose coefficient is 1.
log
2
96 – log
2
3
A. 5
B. 7
C. 12
D. 4
5) Use the exponential growth model, A = A
0
e
kt
, to show that the time it takes a population to double (to grow from A
0
to 2A
0
) is given by t = ln 2/k.
A. A
0
= A
0
e
kt
; ln = e
kt
; ln 2 = ln e
kt
; ln 2 = kt; ln 2/k = t
B. 2A
0
= A
0
e; 2= e
kt
; ln = ln e
kt
; ln 2 = kt; ln 2/k = t
C. 2A
0
= A
0
e
kt
; 2= e
kt
; ln 2 = ln e
kt
; ln 2 = kt; ln 2/k = t
D. 2A
0
= A
0
e
kt
; 2 = e
kt
; ln 1 = ln e
kt
; ln 2 = kt; ln 2/k = t
oe
6) Find the domain of following logarithmic function.
f(x) = log (2 - x)
A. (∞, 4)
B. (∞, -12)
C. (-∞, 2)
D. (-∞, -3)
7) An artifact originally had 16 grams of carbon-14 present. The decay model A = 16e -0.000121t describes the amount of carbon-14 present after t years. How many grams of carbon-14 will be present in 5715 years?
A. Approximately 7 grams
B. Approximately 8 grams
C. Approximately 23 grams
D. Approximately 4 grams
8) Use properties of logarithms to expand the following logarithmic expression as much as possible.
log
b
(x
2
y) / z
2
A. 2 log
b
x + log
b
y - 2 log
b
z
B. 4 log
b
x - log
b
y - 2 log
b
z
C. 2 log
b
x + 2 log
b
y + 2 log
b
z
D. log
b
x - log
b
y + 2 log
b
z
9) The exponential function f with base b is defined by f(x) = __________, b > 0 and b ≠ 1. Using interval notation, the domain of this function is __________ and the range is __________.
A. bx; (∞, -∞); (1, ∞)
B. bx; (-∞, -∞); (2, ∞)
C. bx; (-∞, ∞); (0, ∞)
D. bx; (-∞, -∞); (-1, ∞)
10) Approximate the following using a calculator; round your answer to three decimal places.
3
√5
A. .765
B. 14297
C. 11.494
D. 11.665
11) Write the following equation in its equivalent exponential form.
4 = log
2
16
A. 2 log
4
= 16
B. 2
2
= 4
C. 4
4
= 256
D. 2
4
= 16
12) Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.
3
1-x
= 1/27
A. {2}
B. {-7}
C. {4}
D. {3}
13) Use properties of logarithms to expand the followin.
NUMERICAL METHODS WITH MATLAB : bisection,mueller's,newton-raphson,false poin...Parhamsagharchi
This document describes solving a nonlinear equation to determine the time when a Saturn V rocket reaches the speed of sound using various numerical methods. Specifically, it compares the bisection method, linear interpolation method, Newton-Raphson method, Mueller's method, and the x=g(x) method. For each method, it provides the MATLAB script used to solve the equation and displays the solution time and number of iterations. The root found for all methods was approximately 70.878 seconds.
The document provides examples and explanations for solving different types of equations, including:
1) Polynomial equations through factoring or the quadratic formula.
2) Rational equations by clearing denominators.
3) Radical equations by squaring both sides to remove radicals.
4) Absolute value equations by recognizing that |x-c|=r implies x=c±r.
The document also discusses solving power equations, finding zeros and domains of functions, and using properties of absolute values.
1) Use properties of logarithms to expand the following logarithm.docxdorishigh
1) Use properties of logarithms to expand the following logarithmic expression as much as possible.
Logb (√xy3 / z3)
A. 1/2 logb x - 6 logb y + 3 logb z
B. 1/2 logb x - 9 logb y - 3 logb z
C. 1/2 logb x + 3 logb y + 6 logb z
D. 1/2 logb x + 3 logb y - 3 logb z
2) Solve the following logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, to two decimal places, for the solution.
2 log x = log 25
A. {12}
B. {5}
C. {-3}
D. {25}
3) Write the following equation in its equivalent logarithmic form.
2-4 = 1/16
A. Log4 1/16 = 64
B. Log2 1/24 = -4
C. Log2 1/16 = -4
D. Log4 1/16 = 54
4) Use properties of logarithms to condense the following logarithmic expression. Write the expression as a single logarithm whose coefficient is 1.
log2 96 – log2 3
A. 5
B. 7
C. 12
D. 4
5) Use the exponential growth model, A = A0ekt, to show that the time it takes a population to double (to grow from A0 to 2A0 ) is given by t = ln 2/k.
A. A0 = A0ekt; ln = ekt; ln 2 = ln ekt; ln 2 = kt; ln 2/k = t
B. 2A0 = A0e; 2= ekt; ln = ln ekt; ln 2 = kt; ln 2/k = t
C. 2A0 = A0ekt; 2= ekt; ln 2 = ln ekt; ln 2 = kt; ln 2/k = t
D. 2A0 = A0ekt; 2 = ekt; ln 1 = ln ekt; ln 2 = kt; ln 2/k = toe
6) Find the domain of following logarithmic function.
f(x) = log (2 - x)
A. (∞, 4)
B. (∞, -12)
C. (-∞, 2)
D. (-∞, -3)
7) An artifact originally had 16 grams of carbon-14 present. The decay model A = 16e -0.000121t describes the amount of carbon-14 present after t years. How many grams of carbon-14 will be present in 5715 years?
A. Approximately 7 grams
B. Approximately 8 grams
C. Approximately 23 grams
D. Approximately 4 grams
8) Use properties of logarithms to expand the following logarithmic expression as much as possible.
logb (x2 y) / z2
A. 2 logb x + logb y - 2 logb z
B. 4 logb x - logb y - 2 logb z
C. 2 logb x + 2 logb y + 2 logb z
D. logb x - logb y + 2 logb z
9) The exponential function f with base b is defined by f(x) = __________, b > 0 and b ≠ 1. Using interval notation, the domain of this function is __________ and the range is __________.
A. bx; (∞, -∞); (1, ∞)
B. bx; (-∞, -∞); (2, ∞)
C. bx; (-∞, ∞); (0, ∞)
D. bx; (-∞, -∞); (-1, ∞)
10) Approximate the following using a calculator; round your answer to three decimal places.
3√5
A. .765
B. 14297
C. 11.494
D. 11.665
11) Write the following equation in its equivalent exponential form.
4 = log2 16
A. 2 log4 = 16
B. 22 = 4
C. 44 = 256
D. 24 = 16
12) Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.
31-x = 1/27
A. {2}
B. {-7}
C. {4}
D. {3}
13) Use properties of logarithms to expand the following logarithmic expression as much as possible.
logb (x2y)
A. 2 logy x + logx y
B. 2 logb x + logb y
C. logx - logb y
D. logb x – ...
This document discusses quadratic equations and their properties. It defines quadratic equations as equations of the form y=ax^2 +bx + c, where the highest power is 2. It explains that quadratic equations can be solved using the quadratic formula, x = -b ± √(b^2 - 4ac) / 2a. The number of solutions depends on the discriminant, b^2 - 4ac. If it is greater than 0, there are two solutions, if equal to 0 there is one solution, and if less than 0 there are no solutions. Examples are provided to demonstrate solving quadratic equations.
This document provides notes and formulas for mathematics topics covered in Form 1 through Form 4 in Malaysian secondary schools. It includes formulas and explanations for topics like solid geometry, circle theorems, polygons, factorisation, expansion of algebraic expressions, indices, algebraic fractions, linear equations, simultaneous equations, quadratic expressions, sets, statistics, trigonometry, angles of elevation and depression, and lines and planes. The document is intended to serve as a single reference for key mathematics concepts and formulas for secondary school students.
The document provides information and practice questions about linear functions and slope:
- It instructs the reader to prepare for an upcoming final exam and coordinate test.
- Several practice questions are provided about slope, equations of lines, graphs, and using linear functions to model real-world scenarios.
- The questions cover topics like determining the slope between two points, writing equations in point-slope form, interpreting slope in context, and using linear models to solve word problems.
Lecture 15 section 5.4 graph of sin & cosnjit-ronbrown
This document provides examples and procedures for graphing sine and cosine functions of the form y=a*sin(b(x-c))+d and y=a*cos(b(x-c))+d. It explains that the parameters a, b, c, and d control the amplitude, period, phase shift, and vertical shift of the graphs. Several examples are worked through step-by-step to demonstrate how to graph functions with different parameters. The key steps are to identify the function parameters, determine the period and phase shift, find the x-coordinates of important points, and connect those points to sketch the graph over one full cycle.
Increasing and decreasing functions ap calc sec 3.3Ron Eick
The document discusses increasing and decreasing functions and the first derivative test. It defines that a function is increasing if the derivative is positive, decreasing if the derivative is negative, and constant if the derivative is zero. It provides examples of finding the intervals where a function is increasing or decreasing by identifying critical numbers and testing points in each interval. The document also summarizes the first derivative test, stating that a critical point is an extremum if the derivative changes sign there, and whether it is a maximum or minimum depends on if the derivative changes from negative to positive or positive to negative.
This learner's module discusses or talks about the topic of Quadratic Functions. It also discusses what is Quadratic Functions. It also shows how to transform or rewrite the equation f(x)=ax2 + bx + c to f(x)= a(x-h)2 + k. It will also show the different characteristics of Quadratic Functions.
The document is the marking scheme for a mathematics exam consisting of 26 questions divided into 3 sections. Section A has 6 one-mark questions, Section B has 13 four-mark questions, and Section C has 7 six-mark questions. For questions involving calculus, the marking scheme provides the full working and steps to arrive at the solution. For other questions it states the final answer or shows a short reasoning to justify the answer. The marking scheme also sometimes explains the concepts involved in the question to help examiners understand the approach and marking.
This module introduces polynomial functions of degree greater than 2. It covers identifying polynomial functions from relations, determining the degree of a polynomial, finding quotients of polynomials using division algorithm and synthetic division, and applying the remainder and factor theorems. The document provides examples and practice problems for each topic. It aims to teach students how to work with higher degree polynomial functions.
The student reflects on completing a math project for their calculus course as a way to study for an upcoming exam. They acknowledge that they procrastinated significantly but were able to cover a broad range of calculus concepts through multi-step word problems selected from different units. While the assignment did not dramatically increase their knowledge, it helped reinforce some details and connections between topics. The student resolves to select deadlines more wisely and stop procrastinating for future projects.
Similar to Trigonometric Functions and Graphs Assignment-Advance Function - VHS - MHF4U (20)
Critical Essay - Virtual High School (VHS) - HZT4UMichael Taylor
Critical Essay ‐ Political Philosophy
For this unit’s culminating activity you are asked to write an essay on one of the four major political theories discussed in this unit (i.e., either liberalism, conservatism, Marxism or libertarianism). Most importantly, this essay must be critical meaning that your main aim should be to present
both the strengths and weaknesses of your particular chosen theory so as to represent the theory objectively. You may conclude the essay by offering your opinion, however, this is not necessary.
In your essay you must consult at least 3 secondary sources and cite them both within and at the end of your essay in a proper referencing format.
The assignment should be 34 pages in length, double spaced and in Times New Roman font. You will be graded on your ability to recognise and explain both the strengths and weaknesses of the theory, write in a clear, purposeful and organised fashion and properly cite all references.
Qualitative Analysis of Functional Groups Assignment - Virtual High School (V...Michael Taylor
Procedure
1. Create a data table with the headings and test column
similar to the one on the website.
2. Do the 'Review Tests' first. Click on the test tube icon to
determine what indicates a positive or negative result and
the difference between water soluble and insoluble
compounds. Record your observations in your data table.
3. Click each test tube for the 5 unknowns you've been
assigned and record your observations in your data table.
Qualitative Analysis of Functional
Groups Assignment
Communicate an analysis of your results in a lab report. Note: there is only ONE
functional group per unknown. Refer to the "Writing a Formal Lab Report"
page of the Scientific Skills and Formatting module in the introduction unit to
ensure that you follow the proper format.
Note on Writing In Science
All information and relevant data are to be included in a logically sequenced manner. In your writing, it is important to use appropriate writing style, tone, and scientific terminology. Conduct your research using reliable, peer Reviewed and industry sources and ensure that sources listed in your reference list are directly related to information presented in your paper. Use the appropriate referencing style for science to cite your sources. For more information, see the page titled “References” in the Scientific Skills and Formatting module of the Introduction unit in your course.
Oxidizing and Reducing Agents Lab Assignment - Virtual High School (VHS) - ...Michael Taylor
The document describes an experiment to investigate how different solutions (acidic, neutral, basic) affect the reduction of manganese. Students are instructed to add a potassium permanganate solution to beakers containing acidic, neutral, and basic solutions of sodium hydrosulfite and observe the resulting reactions. They are asked to record initial and final pH measurements, concentrations of aqueous species, colors, and any solid materials produced. The results show that an acidic solution produces a pink color and manganese(II) ions, a neutral solution remains colorless and produces manganese dioxide, and a basic solution turns green and produces less manganese dioxide. Students are then asked to write a formal lab report following the proper scientific format and style to communicate
Issues Related to Energy Changes Assignment - Virtual High School (VHS) - SCH4UMichael Taylor
The document provides instructions for an assignment on issues related to energy changes. Students are asked to write an opinion paper on the issue after reviewing peers' posts and teacher feedback. The paper must thoroughly research and reflect on the issue, include all relevant information and data, and be written in an appropriate scientific style with reliable sources cited using the appropriate referencing style. The document also introduces deep lake water cooling as a new technology that uses large bodies of cold water as a heat sink. It describes how the process works and provides an example of how it could be implemented in Toronto to cool large downtown buildings.
Equilibrium Assignment - Virtual High School (VHS) - SCH4UMichael Taylor
1. The equilibrium constant for the reaction below is 0.18 at 25oC. PCl3(g) + Cl2(g) = PCl5(g)
The following concentrations were measured from the reaction vessel:
[PCl3(g)] = 0.0420
[Cl2(g)] = 0.0240
[PCl5(g)] = 0.00500
a. Is this system in equilibrium? Explain
b. If not, in which direction will the system shift?
2. Write the Solubility Product Constant (Ksp) expression for the following solution at 275 oC.
3. Given the solubility of MgF2(s) is 8.4 x 10-7 M, calculate the value of Ksp for the reaction shown in question #2 at 275 oC.
Corrosion Assignment - Virtual High School (VHS) - SCH4UMichael Taylor
Corrosion Assignment
After reviewing the issues and opinions posted in the discussion forum by your
peers and receiving feedback from your teacher, write an opinion paper that will
show you have thoroughly researched and reflected on the issue you chose to
explore.
Be sure to include in your paper a discussion of how these substances are used
by industry to prevent corrosion. Provide examples wherever possible. Note on Writing In Science
All information and relevant data are to be included in a logically sequenced manner. In your writing, it is important to use appropriate writing style, tone, and scientific terminology. Conduct your research using reliable, peer reviewed and industry sources and ensure that sources listed in your reference list are directly related to information presented in your paper. Use the appropriate referencing style for science to cite your sources. For more information, see the page titled “References” in the Scientific Skills and Formatting module of the Introduction unit in your course.
Normal Distribution Assignment - Virtual High School (VHS) - MDM4UMichael Taylor
2. In many situations, the normal distribution can be used to approximate the binomial distribution.
a. Explain the conditions in which this can be done, and explain why we might want to take advantage of this property.
b. Give an example of a situation in which we could do this.
c. Give an example of a situation in which we would not be able to make this approximation and explain why. 3. A species of alien has a mean height of 23 cm and a standard deviation of 3.6 cm. What is the probability that an alien chosen at random has a height of more than 20cm? 4. Researchers have observed that regular smokers have an average lifespan that is normally distributed and is 68 years with a standard deviation of 10 years. What percent of smokers will live beyond age 76? 5. The life span of a particular species of turtle are normally distributed with a mean of 180 years and a standard deviation of 40 years. What is the probability that one of these turtles will live more than a century? 6. A second species of alien has a mean height of 71 cm and a standard deviation of 5.3 cm. An alientologist discovers that 30% of them bump their heads getting into their spaceship. What is the height of the spaceship door? 7. In Bayfield, 65% of residents read the Bayfield Breeze, a local online blog. Dennis wants to know what people think of the blog, so he stops 40 people on the street to ask them if they read it. a. Verify that the normal distribution can be used to approximate this situation. b. What is the mean and standard deviation of the number of people he finds that read the Breeze? c. What is the probability that at least 25 of the people he asks read the blog? 8. Yuen Zhi is running a ring toss event at a school fair.
There is a 15% chance that each attempt wins a prize. She has 45 prizes and believes 250 people attempt the event. She is worried she won't have enough prizes. Can you reassure her she will probably be OK ? 9. We have been using the normal distribution to approximate situations that are in fact binomial
events. a. Demonstrate how accurate the approximation is by using both approaches to find the probability of the same event. b. Describe the conditions under which the normal would give a less accurate approximation. c. Explain a situation in which the criteria for using the approximation would be met, ie. np ≥ 5 and n(1 − p) ≥ 5, and yet you would decide not to use the normal distribution.
trp Operon - Virtual High School (VHS) - SBI4UMichael Taylor
trp Operon
Recall
Cells respond to their environments by modifying gene expression.
Prokaryotic cells can individually adapt to their changing environments by
regulating the expression of their genes.
Bacteria can adjust to their environments using a special sequence of DNA
known as an operon, which controls the synthesis of protein in response to the environment. An operon has three key areas on the DNA strand: the promoter, the operator, and the transcription unit. The bacteria Escherichia coli need the amino acid tryptophan to survive. Amazingly, if tryptophan is not readily available in its environment, E. coli will actually make its own tryptophan from another compound by activating a metabolic pathway called the trp operon (trp for tryptophan). This metabolic process “shuts off’ as soon as tryptophan is readily available again. How the trp Operon Works Tryptophan is made using a sequence of five different enzymatic reactions. The genes that code for these five enzymes are clustered together on the same chromosome. A single promoter transcribes the entire cluster at once. In this way, the cell can turn “off” or “on” the entire series of functionally associated genes simultaneously. Let’s look at the molecules that cause these switches to be “on” or “off”. 1. How does gene regulation involving the trp repressor protein differ from what you observed with the lac repressor in the previous lesson? (2 points) 2. Based on your understanding of gene regulation in the cell and the function of the trp operon, describe how this process would be affected if there was a mutation in the operator region so that the operator could not carry out its function. (4 points) 3. What would be the result in this process of a genetic mutation that altered the shape of the trp repressor protein? (4 points)
Issues Related to Gene Therapy Assignment - Virtual High School (VHS) - SBI4UMichael Taylor
After reviewing the issues and opinions posted by your peers and receiving
feedback from your teacher, write an opinion paper that will show you have
thoroughly researched and reflected on this issue. Be sure to choose a particular
disorder/disease to use as an example.
Below are a couple of links to sites that may help you in organizing your opinion
paper. If you have any questions, please contact your teacher.
Resources:
Writing a Science Paper
Writing an Opinion Essay
Note on Writing In Science
All information and relevant data are to be included in a logically sequenced
manner. In your writing, it is important to use appropriate writing style, tone, and
scientific terminology. Conduct your research using reliable, peerreviewed and
industry sources and ensure that sources listed in your reference list are directly related to information presented in your paper. Use the appropriate referencing
style for science to cite your sources. For more information, see the page titled
“References” in the Scientific Skills and Formatting module of the Introduction unit in your course.
Intersections Unit Assignment - Virtual High School (VHS) - MCV4UMichael Taylor
MCV4Ud3—Intersections Assignment
Answer all questions with full solutions. Make sure your work is legible, even after you have scanned iT, and submit it as 0 single file.
1. The equation of a line can be determined using two points on the line.
a. Find the vector, parametricand symmetric equations of the line through the points
(-2,6,1)and(2,1,3)
b. Explain the features of the equations ofa line that is parallel to the xy plane, but does not lie on the plane, and is not parallel to any of the axes. include a Lan Graph of your line.
2. Two given lines are either parallel, skew or intersecting.
e
a. Determine, ifthere is one, the point ofintersection of the lines given by the equations (x-5)/1=(y-1)/(-2)=(z+1)/(-4) and (x-6)/3=(y-7)/2=(z-2)/(-5)
b. Give the equations of two lines that meet at the point (3,2,-4) and which meet at right angles, but do not use that point in either of the equations. Explain your reasoning and include a LanGraph of your line.
3. The equation of a plane can be determined using three points on the plane.
a. Find the vector, parametricand general equations of the plane through the points
(3,1,-2) , (-2,4,3) and (5,-1,4)
b. Give the equation ofa plane that crosses the axes at points equidistant from the origin. Explain your reasoning and include a Lan Graph of your plane.
4. A Line can either lie on a plane, lie parallel to it or intersect it.
a. Determine, ifthere is one, the point ofintersection between:
the line given by the equation (x-3)/3=(y+1)/(-2)=(z-10)/4
and the plane given by the equation [x,y,z]=[-6, 3, 6 ] +s[1, 2, 3 ] +T[2,-1,2] b. Determine the angle between the line and the plane.
c. Give the equation ofa plane and three lines, one of which is parallel to the plane, one of which lies on the plane, and one of which intersects the plane. Explain your reasoning and include a Lan Graph.
5. The angle between two planes can also be determined
Economics Assignment 2 4th year advanced micro universityMichael Taylor
This document contains 30 questions related to producer theory and the theory of the firm. The questions cover various topics including production functions, cost minimization, profit maximization, elasticities, and comparative statics. Students are asked to analyze production functions, solve optimization problems, examine how input demands and costs change in response to price changes, and prove various relationships between theoretical concepts.
Biology 204 Principles of Biology I Assignment 2CMichael Taylor
Biology 204 Principles of Biology I Assignment 2C
For students with first names starting with the letters O to Z.
This assignment is graded out of 110 points, and is worth 10% of your final mark. Please submit this assignment after you have completed Chapter 16 and before you write the final exam
Biology 204 Principles of Biology I Assignment 1CMichael Taylor
Biology 204 Principles of Biology I Assignment 1C
For students with first names starting with the letters O to Z.
This assignment is graded out of 110 points, and is worth 10% of your final mark. Please submit this assignment after you have completed Chapter 7 and before you write the midterm exam.
Qualitative Analysis of Functional Groups Assignment - SCH4U h5, Chemistry, 1...Michael Taylor
SCH4U h5+, Chemistry, 12, University Virtual High School
Unit Assignment: Qualitative Analysis of Functional Groups Assignment
Request the complete assignment now.
Vectors Unit Assignment- Calculus and Vector - VHS - MCV4UMichael Taylor
This document contains an assignment on vectors for a calculus and vectors course. It includes 8 questions covering topics like identifying vectors and scalars, vector addition and subtraction, converting between Cartesian and direction/magnitude forms, and properties of vector addition like commutativity and associativity. The questions are multiple choice, short answer, and require diagrams. The final question asks students to research a real-world application of vectors such as in engineering, animation, gaming or GPS technology.
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
How to Download & Install Module From the Odoo App Store in Odoo 17Celine George
Custom modules offer the flexibility to extend Odoo's capabilities, address unique requirements, and optimize workflows to align seamlessly with your organization's processes. By leveraging custom modules, businesses can unlock greater efficiency, productivity, and innovation, empowering them to stay competitive in today's dynamic market landscape. In this tutorial, we'll guide you step by step on how to easily download and install modules from the Odoo App Store.
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
A Visual Guide to 1 Samuel | A Tale of Two HeartsSteve Thomason
These slides walk through the story of 1 Samuel. Samuel is the last judge of Israel. The people reject God and want a king. Saul is anointed as the first king, but he is not a good king. David, the shepherd boy is anointed and Saul is envious of him. David shows honor while Saul continues to self destruct.
Gender and Mental Health - Counselling and Family Therapy Applications and In...PsychoTech Services
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
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ملزمة تشريح الجهاز الهيكلي (نظري 3)
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6- تحتوي الملزمة في اول سلايد على خارطة تتضمن جميع تفرُعات معلومات الجهاز الهيكلي المذكورة في هذهِ الملزمة
واخيراً هذهِ الملزمة حلالٌ عليكم وإتمنى منكم إن تدعولي بالخير والصحة والعافية فقط
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Trigonometric Functions and Graphs Assignment-Advance Function - VHS - MHF4U
1. PanHelp
MHF4U k1+, Advanced Functions, 12, University Virtual High School
Unit Assignment: Trigonometry unit and graph assignment
Assignment Help | 100% Plagiarism Free | Success Assured | Email Now to get quote – admin@panhelp.com
2. Assignment Help | 100% Plagiarism Free | Success Assured | Email Now to get quote – admin@panhelp.com
Question 1
For each of the following functions, state
the domain;
the range
the minimum and maximum values
the period
the phase shift
and the amplitude
a. 𝑓 𝑥 = 4 sin
1
3
𝑥 + 2𝜋 −
3
2
A) given,
𝑓 𝑥 = 4 sin ((1/3) x + 2π) −
3
2
Domain {x|x E R}
Range -4 <4*sin
1
3
x + 2π < 4
-4 ≤4 sin
1
3
x + 2π < 4
Now subtract (3/2) on both sides
−4 – (3/2) ≤ 4 sin ((1/3) x + 2π) – (3/2) ≤ 4 −
3
2
−
11
2
≤ 4sin
1
3
x + 2π –
3
2
≤
5
2
Range f x −
11
2
≤ f x ≤
5
2
, f(x) E R}
Minimum value = -11/2
Maximum value= 5/2
Phase Shift = 6 π (left)
Amplitude = 4
Period= 2π/ (1/3)
= 2π / (1/3)
= 6π
3. Assignment Help | 100% Plagiarism Free | Success Assured | Email Now to get quote – admin@panhelp.com
Question 2
Describe the transformations in each graph.
Sketch the graphs of the following functions by
hand, using transformations. Show your work
(show how the transformations map from the
base function to the transformed function) for
the graph in addition to listing the information.
Graph values from 0 ≤ x ≤ 2π.
a. f(x) = −2 cos 3𝑥 −
𝜋
3
+
1
2
b. f(x) =
1
4
𝑠𝑖𝑛
1
2
𝑥 + 𝜋 − 3
Answer 2
a) Phase Shift = π/9 (right)
Period = 2π/3
Amplitude = 2
Vertical Shift = ½
b) Phase Shift = 2π (left)
Period = 2π/ (1/2) = 4π
Amplitude = 1/4
Vertical Shift = -3
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Question 3
For each of the graphs below, create a function
of the form f(x) = a cos(k(x − d)) + c or f(x) = a
sin(k(x − d)) + c. Show all of your calculations
a.
b.
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Question 4
On a particular lake, the difference between high
and low tide is 6 hours. At low tide, 6 a.m., a
student measures a 1.4 m distance from the
dock she is standing on, down to the water. two
hours later, she measures the distance from the
dock down to the water to be 1.29 m. After three
hours, the distance is 1.175 m down to the
water.
a. Determine the function for the distance from
the dock down to the water with respect to
the number of hours since midnight.
b. Determine the distance to the water at 4:30
a.m. and 1:45 p.m.
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