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Master Graphing Reciprocals of Linear Functions | Easy Guide

Demonstrate an understanding of the absolute value of real numbers

Absolute value functions

Expressing an Absolute Value Linear Function as a Piecewise Function

Expressing an Absolute Value Quadratic Function as a Piecewise Function

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Solving absolute value equations

Solving absolute value inequalities

Solving Absolute Value Inequalities Involving "less than"

Solving Absolute Value Inequalities Involving "greater than"

Recognizing Absolute Value Inequalities with "No Solution" or "All Real Numbers"

Solve problems involving operations on radicals and radical expressions with numerical and variable radicands

Conversion between entire radicals and mixed radicals

Adding and subtracting radicals

Combining Like Radicals

Adding and Subtracting Radicals

Multiplying radicals

Multiplying Radicals

Operations with radicals

Operations with Radicals

Square and square roots

Square Root of a Perfect Square

Cubic and cube roots

Cube root of a perfect cube

Solve problems involving radical equations limited to square roots

Solving radical equations

Solving Radical Equations Involving One Square Root Term

Solving Radical Equations Involving More than One Square Root Term

Determine equivalent forms of rational expressions with numerators and denominators that are monomials, binomials or trinomials

Partial fraction decomposition

Simplifying rational expressions and restrictions

Simplifying complex fractions

Perform operations on rational expressions with numerators and denominators that are monomials, binomials or trinomials

Adding and subtracting rational expressions

Adding and Subtracting Rational Expressions

Multiplying rational expressions

Multiplying Rational Expressions

Multiplying Rational Expressions in Factored Form

Dividing rational expressions

Dividing Rational Expressions

Dividing Rational Expressions in Factored Form

Solve problems involving rational equations with numerators and denominators that are monomials, binomials or trinomials

Solving rational equations

Applications of rational equations

Factor polynomial expressions of various forms

Factoring difference of cubes

Factoring sum of cubes

Factor by taking out the greatest common factor

Factor by taking out GCF

Factor by grouping

Factoring difference of squares: \(x^2 - y^2\)

Factoring difference of squares: <katex>x^2 - y^2</katex>

Factoring difference of squares: x^2 - y^2

Factoring trinomials

Factoring Trinomials

Analyze geometric sequences and series to solve problems

Geometric sequences

Geometric series

Geometric Series

Infinite geometric series

Graph and analyze reciprocal functions limited to the reciprocal of linear and quadratic functions

Graphing reciprocals of linear functions

Graphing reciprocals of quadratic functions

Analyze quadratic functions of the form y = a(x - p)² + q and determine characteristics of the corresponding graph

Characteristics of quadratic functions

Transformations of quadratic functions

Converting from general to vertex form by completing the square

Shortcut: Vertex formula

Converting general form into vertex form by applying the vertex formula

Vertex Formula Application

Quadratic function in vertex form: y = a(x-p)^2 + q

Completing the square

Completing the Square

Analyze quadratic functions of the form y = ax² + bx + c to identify characteristics of the corresponding graph

Introduction to quadratic functions

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

Graphing quadratic functions: General form VS. Vertex form

Finding the quadratic functions for given parabolas

Applications of quadratic functions

Solve problems involving quadratic equations

Solving polynomial equations by iteration

Solving quadratic equations by factoring

Solving quadratic equations by completing the square

Using quadratic formula to solve quadratic equations

Solving a quadratic equation with TWO REAL SOLUTIONS

Solving a quadratic equation with TWO COMPLEX SOLUTIONS

Nature of roots of quadratic equations: The discriminant

Applications of quadratic equations

Solve algebraically and graphically problems involving systems of linear-quadratic and quadratic-quadratic equations

Solving 3 variable systems of equations by substitution

Solving 3 variable systems of equations by elimination

System of linear equations

System of linear-quadratic equations

System of quadratic-quadratic equations

Solve problems involving linear and quadratic inequalities in two variables

Graphing linear inequalities in two variables

Graphing quadratic inequalities in two variables

Graphing systems of quadratic inequalities

Graphing systems of linear inequalities

Applications of inequalities

Solve problems involving quadratic inequalities in one variable

Solving quadratic inequalities

Analyze arithmetic sequences and series to solve problems

Arithmetic sequences

Arithmetic series

Arithmetic series formula

Demonstrate an understanding of angles in standard position from 0° to 360°

Angle in standard position

Coterminal angles

Coterminal Angles

Reference angle

Solve problems using the three primary trigonometric ratios for angles from 0° to 360° in standard position

Unit circle

Find the exact value of trigonometric ratios

ASTC rule in trigonometry (All Students Take Calculus)

ASTC rule in trigonometry

Solve problems using the cosine law and sine law, including the ambiguous case

Law of sines

Ambiguous case: SSA triangles

Law of cosines

Applications of the sine law and cosine law

Grade 11 Pre-Calculus Mathematics (30S)

Develop algebraic reasoning and number sense

Develop algebraic and graphical reasoning through the study of relations

Develop trigonometric reasoning

math

Intro

Intro to reciprocal functions: properties, invariant points, and asymptotes

Ex.1

Graphing the reciprocal of f(x) = 4x using five steps

Ex.2

Graphing the reciprocal of f(x) = x + 5 using the 5-step method

Ex.3

Graphing 1/(2x-1) using the five-step reciprocal method

Ex.4

Graphing the reciprocal of y = 1/2 - 5x using five steps

Ex.5

Graphing the reciprocal of y = 1/3 - x/9 using five steps

We know that taking the reciprocal of a very large number will grant us a very small number. Conversely, if we take the reciprocal of a very small number, we will obtain a very small number. What will happen if we take the reciprocal of a linear function? In this section, we will learn about the concept behind the reciprocal of a linear function, as well as how to graph it, while following the "5-steps Approach" noted below.

How to graph reciprocals of linear functions

Graphing Reciprocals of Linear Functions: A Comprehensive Guide

What You'll Learn

Understand how reciprocal transformations affect the behavior of linear functions

Identify invariant points where y-values remain unchanged after taking reciprocals

Locate vertical asymptotes by finding x-values that make the original function equal zero

Recognize that reciprocals of large values yield small values and vice versa

Apply the 5-step approach to systematically graph reciprocal functions

What You'll Practice

Finding vertical and horizontal asymptotes algebraically and graphically

Calculating invariant points by setting the original function equal to ±1

Determining y-intercepts of reciprocal functions

Sketching reciprocal graphs using the 5-step method

Why This Matters

This Unit Includes

6 Video lessons

Practice exercises

Learning resources

Skills

Reciprocal Functions

Vertical Asymptotes

Horizontal Asymptotes

Invariant Points

Function Transformations

Graphing

Rational Functions