About Us

Why Choose StudyPug

How it works

Pricing

Testimonials

Contact Us

Blog

Basic Math

Algebra

Geometry

Trigonometry

Calculus

Linear Algebra

Differential Equations

Statistics

Chemistry

Organic Chemistry

Physics

Microeconomics

For Students

For Parents

For Home Schoolers

For Teachers

Australia

Canada

Ireland

New Zealand

Singapore

United Kingdom

United States

Sitemap

Terms of Service

Privacy Policy

Learn to represent patterns in linear relations

TX.AR.2.A

auto-gen

Understand relations between x- and y-intercepts

Domain and range of a function

Identifying functions

Function notation

Rate of change

Applications of linear relations

Arithmetic sequences

Geometric sequences

TX.AR.2.B

Relationship between two variables

Representing patterns in linear relations

Represent Patterns in Linear Relations Equation Decimal

Represent Patterns in Linear Relations Equation Integer Start From 0

Represent Patterns in Linear Relations Equation Integer Start From 1

Represent Patterns in Linear Relations Equation Negative Decimal

Identify the Linear Equation

Reading linear relation graphs

Solving linear equations by graphing

Understanding graphs of linear relationships

Linear relations

Understanding tables of values of linear relationships

Linear relation

Applications of linear relationships

Characteristics of quadratic functions

TX.AR.2.C

Graphing linear functions using table of values

Graphing linear functions using x- and y-intercepts

Graphing from slope-intercept form y=mx+b

Graphing linear functions using a single point and slope

Graphing quadratic functions: General form VS. Vertex form

TX.AR.2.D

Word problems of graphing linear functions

Money related questions in linear equations

Unknown number related questions in linear equations

Distance and time related questions in linear equations

Rectangular shape related questions in linear equations

Applications of quadratic functions

TX.AR.3.A

Slope intercept form:  y = mx + b

Slope intercept form: y = mx + b

Quadratic function in general form: y = ax^2 + bx + c

Transformations of functions: Horizontal translations

Transformations of functions: Vertical translations

Reflection across the y-axis: y = f(-x)

Reflection across the x-axis: y = -f(x)

Transformations of functions: Horizontal stretches

Transformations of functions: Vertical stretches

Graphing exponential functions

TX.AR.3.B

Combining transformations of functions

Inverse functions

Even and odd functions

TX.AR.3.C

Adding functions

Subtracting functions

Multiplying functions

Dividing functions

Operations of Functions – In a Nutshell

TX.AR.3.D

Operations with functions

TX.AR.3.E

One to one functions

Function notation (advanced)

Composite functions

TX.AR.3.F

Difference quotient: applications of functions

TX.AR.4.A

Characteristics of polynomials

What are polynomials

Equivalent expressions of polynomials

Adding and subtracting polynomials

Multiplying and dividing monomials

Divide a monomial by a monomial

Multiply a monomial by a monomial

Multiplying polynomials by monomials

Multiply a polynomial by a monomial

Dividing polynomials by monomials

Multiplying polynomial by polynomial

TX.AR.4.B

Inequalities of combined functions

Inequalities of Combined Functions

TX.AR.4.C

Polynomial long division

Polynomial synthetic division

TX.AR.4.D

Solving polynomials with the unknown 

Solving polynomials with the unknown "b" from ax^2 + bx + c

Factoring polynomials: ax^2 + bx + c

Factoring Polynomials: ax^2 + bx + c

Factoring perfect square trinomials: (a + b)^2 = a^2 + 2ab + b^2 or (a - b)^2 = a^2 - 2ab + b^2

Factoring perfect square trinomials

Rational zero theorem

Rational Zero Theorem

Common factors of polynomials

Factoring polynomials by grouping

Solving polynomials with unknown coefficients

Solving polynomials with unknown constant terms

Factoring polynomials: x^2 + bx + c

Applications of polynomials: x^2 + bx + c

TX.AR.5.A

Adding and subtracting matrices

TX.AR.5.B

Matrix multiplication

TX.AR.5.C

Scalar multiplication of matrices

TX.AR.5.D

System of linear equations

Representing a linear system as a matrix

Solving a linear system with matrices using Gaussian elimination

TX.AR.5.E

Solving 3 variable systems of equations by substitution

Solving 3 variable systems of equations by elimination

The determinant of a 2 x 2 matrix

The determinant of a 3 x 3 matrix (General & Shortcut Method)

The Inverse of a 2 x 2 matrix

The inverse of 3 x 3 matrices with matrix row operations

The inverse of 3 x 3 matrix with determinants and adjugate

2 x 2 invertible matrix

Solving linear systems using Cramer's Rule

Solving linear systems using Cramer"s Rule

Cramer"s Rule with 2 x 2 matrices

Solving linear systems using 2 x 2 inverse matrices

TX.AR.6.B

Introduction to linear equations

Solving quadratic equations by factoring

TX.AR.6.C

Solving logarithmic equations

Solving radical equations

Solving Radical Equations Involving One Square Root Term

Solving Radical Equations Involving More than One Square Root Term

Solving exponential equations using exponent rules

TX.AR.7.A

Set notation

Understanding How to Use Set Notation

Set builder notation

Intersection and union of 2 sets

Intersection and Union of 2 Sets

Intersection and union of 3 sets

Intersection and Union of 3 Sets

TX.AR.7.C

Regression analysis

TX.AR.7.D

Equation of the best fit line

Algebraic Reasoning 

Modeling from data

Number and algebraic methods

Patterns and structure

math

Ex. 1:

Ex. 2:

Ex. 3:

<p>- Let's take a look. We're going to come up with an equation to represent the relationship between the number of sides we have, sorry the number of octagons that we have versus the number of sides that we have right here. This is the input and this is the output. The purpose of this chapter here is to figure out the relationship between the two variables and the relationship is always called the linear. They are all linear functions in this chapter. It's a good thing. Why?<br />
Because if you're doing linear functions, guess what? You have one equation. It's y equals to mx plus a b. That's the equation you are trying to get. This is the output, which is the y. This is the x, which is the input. Basically, one times something and add another thing should give you that eight over here. It should work for all the points here. Are we going to take a guess or are we going to calculate? Personal suggestion, we should calculate. Because guessing might take forever. What we're going to do is this. It's very simple. We're going to calculate the m and we're calculate the b. To calculate m right now, we're just going to give you an equation. It's very straightforward. It's y2 minus y1 divided by x2 minus the x1. What are the twos? What are the ones? It's just representing the second point versus the first point. And they don't have to be the second versus the first. They can be any two points from your data sets right here. So for example, I can pick the third one versus the second one. But try to pick the later one subtracting the preceding ones. Let's say I pick this set here and let's pick another set right here. What happened here we're just following the equation. It's says y two, so let's say this is my y two, which is right here, the 20 because this is the y and that's the x. This is the second point and this is the first point that we're going to pick. 20 minus 14. Be careful you're subtracting the two y's together. Then x two minus the x one. We've got three minus one. We've got a six over here on the top and a two down here. We've got ourselves a three for our m. You see right here, we've got m right here now which is three. What about b? For b, it's a little bit easier. What happened is, if you write down that equation one more time and this time you can pick any points that you wish. Just pick one. Let's say you pick this one right here. Let's say you like the five better. You put 32 into your y position. The m that you just calculated is a three. The x with the 32, make sure it is coming from the same pair. You're going to put the five right there and there's your b that's missing. Let's do a little bit of math right here for you. We've got 32 equals to 15 plus the b. If you subtract the 15 over to the other side, your b turns out to be 17. That's your b right there. Once you have your b, and once you have your m, guess what happens? Your final answer for this question here is actually just going to be-- therefore, three triangles we're going to go y equals the m, 3x plus the 17 and this is your final answer for this question. Without any guessing, the process may seem a little messy but you just have to remember apply this m calculation right here. Then solve for b by writing down the equation one more time but plug in the m you just calculated which is a three and you can pick any points you like. Just plug it in and you will find the b. Rewrite them one more time, leave out the y, leave out the x. Why is that? Because we're going to put different numbers for x in here to calculate the y for future questions. That's it for this part. Let's take a look at what's happening for the next part.</p>


In this section, we will learn how to represent patterns in linear relations. In other words, these patterns will form a straight line if they are plotted on a graph paper. We can start with creating a table of values which has the independent variable as X, and <a href='[[glossary.html|{"keyword":"Dependent Variable","term":"Dependent Variable"}]]'>dependent variable</a> as Y. One way to represent patterns in linear relations is to use the slope-intercept form.