Module 4 topic 1 2nd
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This document provides examples for determining the quadrant of coordinates, finding the midpoint between two points, identifying if a relation is a function, and determining the domain and range of a relation. It also gives an example of evaluating a function for a given input value.
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Here are the key steps to factoring when a ≠ 1:
1. Check for GCF. If there is a GCF, factor it out.
2. If no GCF, find two numbers whose product is ac and sum is b. These will be the coefficients of the factors.
3. Split the middle term into two terms using the factors.
4. Factor using the box method, finding the GCF of each row and column.
Some examples:
2x2 + 11x + 5
ac = 10, factors are 2, 5
(2x + 1)(x + 5)
3x2 - 5x - 2
ac = -
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This document defines and provides examples of relations and functions. It defines a relation as a set of ordered pairs, and gives an example relation. It explains that the domain is the set of all x values in the relation, and the range is the set of all y values. For the given example relation, it lists the domain and range. It provides a mapping diagram and explains that a relation assigns x's to y's.
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It provides an example of students in a class as the
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2. Functions are defined as relations where each input is mapped to only one output. Several examples are given to demonstrate determining if a relation qualifies as a function using techniques like the vertical line test.
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1. Check for GCF. If there is a GCF, factor it out.
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1. The document discusses relations and functions, including identifying their domain and range. It provides examples of plotting points on a Cartesian plane and using graphs to represent equations.
2. Functions are defined as relations where each input is mapped to only one output. Several examples are given to demonstrate determining if a relation qualifies as a function using techniques like the vertical line test.
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- Examples of identifying ordered pairs from tables and graphing the points on a coordinate plane. The examples show a direct relationship between variables.
- A warm up on identifying points in quadrants.
- An example application showing pounds of food purchased and its cost, with the relationship that cost is pounds multiplied by a rate.
- A lesson quiz testing the skills of writing ordered pairs from a table and graphing the points.
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- The highest power of the variable x is 2, so the equation is of degree 2.
- Quadratic functions have a parabolic shape when graphed.
- They can model many real-world phenomena like projectile motion, profit, area of shapes, etc.
- Quadratic functions have properties like axis of symmetry, vertex, intercepts, etc. that can be used to analyze and solve problems.
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1h. Pedagogy of Mathematics (Part II) - Set language Activities and exercise
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Pedagogy of Mathematics (Part II) - Set language Activities and exercise, Set Language, Maths, IX std Maths, Samacheerkalvi maths, II year B.Ed., PedagogyOpen sentences.ppt
The document defines key terms related to expressions, equations, and inequalities. It provides examples of each and has students identify whether examples given are expressions, equations, or inequalities. It also covers solving equations by applying order of operations and defining variables, replacement sets, and solution sets when solving equations and inequalities.
(7) Lesson 5.7 - Subtract Linear Expressions
To subtract two linear expressions:
1. Arrange the like terms in column form with the subtrahend expression underneath the minuend.
2. Take the additive inverse of the subtrahend expression by changing the sign of each term.
3. Add the subtrahend expression with its inverse to the minuend expression.
4. Simplify the resulting expression.
Module 4 topic 3 similar problems
This document provides examples of writing equations of linear functions in slope-intercept form. It demonstrates finding the slope from two points, using the slope and a point to write the equation in point-slope form, writing equations for horizontal and vertical lines, writing equations from tables of values, and solving linear equations for y.
Math 235 - Summer 2015Homework 2Due Monday June 8 in cla.docx
Math 235 - Summer 2015
Homework 2
Due Monday June 8 in class
Remember: In this course, you must always show reasoning for your answers. You can use any result we have
proved in class, in textbook reading, or in a previous homework.
Problem 1 For each of the following problems, you must justify your answer by finding the general solution
to the corresponding system of linear equations, or by showing that no solution exists.
(a) In the vector space P3(R), can −2x3 − 11x2 + 3x+ 2 be written as a linear combination of vectors in
{x3 − 2x2 + 3x− 1, 2x3 + x2 + 3x− 2}?
(b) In the vector space M2×2(R), can
(
1 0
0 1
)
be written as a linear combination of vectors
in
{(
1 0
−1 0
)
,
(
0 1
0 1
)
,
(
1 1
0 0
)}
?
Problem 2 Show that a subset W of a vector space V (over a field F ) is a subspace of V if and only if
span(W ) = W .
Problem 3 You are given a subset S of a vector space V . Determine whether S is linearly dependent or
linearly independent using exclusively methods developed in this course, and justify your answers.
(a) V = R3 and S = {(1, 2,−1), (2,−3, 1), (2, 3,−5)}.
(b) V = P3(R) and S = {1, 1 + 2t+ t2, 1− 2t+ t3, t2 + t3}.
(c) V = F(R,R) and S = {t, et, sin(t)}.
Problem 4 Prove that a subset S of a vector space V is linearly dependent if and only if there exists a
proper subset S′ ( S with the same span as S.
Problem 5 Exercise 1.6.13 from the textbook.
Problem 6 You are given a subspace S of M2×2(F ), the vector space of 2 × 2 matrices with entries in a
field F . You are required to find a basis for this subspace, and to find the dimension of this subspace.
For each problem, you DO NOT need to prove that S is a subspace, but you DO need to prove that your
conjectured basis is, in fact, a basis (that is, you need to show it is a linearly independent generating set for
S).
(a) S is the subspace of all diagonal 2× 2 matrices with entries in F .
(b) S is the subspace of all symmetric 2× 2 matrices with entries in F .
(c) S is the subspace of all skew-symmetric 2× 2 matrices with entries in F .
Problem 7 Let W1 and W2 be subspaces of a finite-dimensional vector space V . Prove that dim(W1∩W2) ≤
min{dim(W1),dim(W2)} and dim(W1 +W2) ≥ max{dim(W1),dim(W2)}.
Problem 8 Each of the maps below goes from one vector space to another (where both vectors spaces are
over the same field). For each map: prove that it is linear, determine whether it is one-to-one or not (prove
your answer), and determine whether it is onto or not (prove your answer).
(a) T : P3(R)→M2×2(R) defined by T (p) =
(
p(0) p′(0)
p′′(0) p′′′(0)
)
.
(b) T : M2×2(F ) → F defined by T (A) = tr(A), where F is a field. (Recall that for an n × n matrix,
tr(A) =
∑n
i=1Aii.)
1
(c) T : R2 → R3 defined by T ((a, b)) = (a, b, a+ b).
(Hint: You may find an analysis of rank and nullity useful here.)
Problem 9 Suppose that T : R2 → R2 is linear and that T ((1, 2)) = (3, 4) and T ((1, 3)) = (0, 1). Find
T ((1, 0)). Is T one-to-one? Justify your answer.
Problem 10 Let ...
Algebraic Simplification and evaluation
Lessons on Algebraic simplification and evaluation for students preparing for WASSCE, NECO and JAMB exams.
5.1 Linear Functions And Graphs
The document defines domain, range, and functions. It provides examples of determining the domain and range of relations and whether relations are functions. It also gives exercises on writing equations and ordered pairs for linear functions based on tables of values.
6 4 Absolute Value And Graphing
This document reviews absolute value concepts including:
- Graphing absolute value equations and finding their domains and ranges
- Procedures for solving absolute value equations like |x-2|=7 and |x-10|=4.5
- Graphing the solutions to inequality equations like |x - (-5)| ≤ 8 and |x - 6| > 5
Section 41-42.ppt
The document discusses the rectangular coordinate system and graphing linear equations. It defines key terms like coordinates, quadrants, and linear vs. nonlinear relationships. It provides examples of how to determine if an ordered pair satisfies an equation, find solutions to equations with two variables, and graph linear equations by plotting solutions on a coordinate plane and connecting points with a straight line.
rectangular coordinate system in electromagnetic fields
The document discusses the rectangular coordinate system and graphing linear equations. It defines key terms like coordinates, quadrants, and linear vs. nonlinear graphs. It provides examples of how to determine if an ordered pair satisfies an equation, find solutions to equations with two variables, and graph linear equations by plotting solutions on a coordinate plane and connecting the points with a straight line.
Quadratic Equation
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.
Rbsc class-9-maths-chapter-4
This document provides solutions to exercises from NCERT Class 9 Maths Chapter 4 on linear equations in two variables. It includes:
1) Solving linear equations representing word problems and expressing equations in the form ax + by + c = 0.
2) Finding solutions that satisfy given linear equations and determining the value of k if a given point is a solution.
3) Drawing graphs of various linear equations by plotting points that satisfy each equation.
4) Giving two equations of lines passing through a point and noting there are infinitely many such lines.
Elements of a sequence
The document provides 5 examples of defining and calculating sequences based on functions. The first two examples show defining sequences by functions f(x)=2x-7 and f(x)=2x^3-4, and calculating the first 5 terms. The third example defines the sequence by f(x)=5x^2-3x+7 and lists the first 3 terms. The fourth example provides 3 infinite sequences defined by patterns in the terms. The fifth example lists 3 finite sequences.
05 tat math class xi_r&f-1_for website
The document defines key concepts related to relations and functions:
1. Ordered pairs, Cartesian products, and domains and ranges of relations are introduced. An ordered pair lists two objects in a particular order, while a Cartesian product is the set of all ordered pairs with the first element from one set and the second from another.
2. The domain of a relation is the set of first elements in its ordered pairs, while the range is the set of second elements. Inverse relations are also defined.
3. Several examples illustrate calculating Cartesian products of sets, determining the domains and ranges of given relations, and finding inverse relations.
System of linear equations and their solution
1. The document presents examples of solving addition and subtraction problems with positive and negative numbers. It also gives examples of finding two numbers whose sum and difference satisfy given values.
2. It defines a system of linear equations as two equations in two variables of the form Ax + By = C and Dx + Ey = F. It explains that an ordered pair (x,y) is a solution if it satisfies both equations.
3. Examples are given of determining if an ordered pair is a solution to a given system of linear equations by substituting the values into each equation and checking if both equations are satisfied. An activity asks students to identify if ordered pairs are solutions to several systems.
L1 functions, domain & range
The document defines functions and discusses:
- Functions are sets of ordered pairs with each first element paired to a unique second element.
- Functions can be one-to-one or many-to-one.
- Functions are represented in set notation, tabular form, as equations, and as graphs.
- The domain of a function is the set of first elements and the range is the set of second elements.
Chapter 8 mini test review
This document contains an 8 question review for Chapter 8 in math that covers solving equations, graphing linear equations, finding slope between two points on a line, and identifying the domain and range and whether relations are functions. The questions include solving equations for specific solutions, graphing linear equations, finding the slope between two given points, writing the domain and range of relations, and determining if relations are functions.
Similar to Module 4 topic 1 2nd (20)
1h. Pedagogy of Mathematics (Part II) - Set language Activities and exercise
1h. Pedagogy of Mathematics (Part II) - Set language Activities and exercise
Math 235 - Summer 2015Homework 2Due Monday June 8 in cla.docx
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Module 4 topic 1 2nd
- 1. More – Coordinate Plane, Midpoint, and Relations and Functions Module 4 Topic 1
- 2. Example 1 Remember that the quadrants are I, II, III, or IV. Quadrant I = (positive x-value, positive y-value) Quadrant II = (negative x-value, positive y-value) Quadrant III = (negative x-value, negative y-value) Quadrant IV = (positive x-value, negative y-value)
- 3. Example 2 Similar Problem: Find the coordinates of the midpoint of M(-3, 0) and N(-7, -5). Midpoint is the average of the x-values and the average of the y-values. Midpoint = (x2 + x1) ÷ 2 , (y2 + y1) ÷ 2 Midpoint = (-7 + -3) ÷ 2 , (-5 + 0) ÷ 2 Midpoint = -10 ÷ 2 , -5 ÷ 2 Midpoint = -5, -2.5 so we write it as an ordered pair (-5, -2.5)
- 4. Example 3 Similar Problem: Use the following relation to answer questions 7 - 9. {(2, 4), (7, 8), (-5, 4), (-2, -8), (0, 0)} Is the relation a function? Yes - because the x-values do not repeat!
- 5. Example 4 Similar Problem: Use the following relation to answer questions 7 - 9. {(2, 4), (7, 8), (-5, 4), (-2, -8), (0, 0)} State the domain of the relation. Domain = all x-values of the relation 2, 7, -5, -2, 0 but let's put these in order so we have -5, -2, 0, 2, 7
- 6. Example 5 Similar Problem: Use the following relation to answer questions 7 - 9. {(2, 4), (7, 8), (-5, 4), (-2, -8), (0, 0)} State the range of the relation. Range = all y-values of the relation 4, 7, 4, -8, 0 but let's put these in order so we have -8, 0, 4, 4, 7 and let's not repeat any so we have -8, 0 , 4, 7
- 7. Example 6 Similar Problem: If f(x) = 3x + 6, find f(-4). Plug in -4 for x in the function 3x + 6 3(-4) + 6 -12 + 6 -6