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Graphs of Rational Functions: Asymptotes & Holes

Grade 12 Math: Master Advanced Concepts & Skills

12AF.A1.1

What is a logarithm?

Logarithms and Exponential Growth

Understanding Logarithms through Exponential Growth

Understanding Logarithms

Converting from logarithmic form to exponential form

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Evaluating logarithms without a calculator

12AF.A1.2

Common logarithms

Natural log: ln

Evaluating logarithms using change-of-base formula

12AF.A1.3

Converting from exponential form to logarithmic form

Solving exponential equations with logarithms

12AF.A1.4

Product rule of logarithms

Quotient rule of logarithms

Combining product rule and quotient rule in logarithms

Power rule of logarithms

Evaluating logarithms using logarithm rules

12AF.A2.1

Graphing logarithmic functions

Graphing transformations of logarithmic functions

12AF.A2.2

Inverse functions

Graphing exponential functions

Graphing transformations of exponential functions

12AF.A2.3

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

12AF.A2.4

Finding an exponential function given its graph

Finding a logarithmic function given its graph

12AF.A3.1

Solving exponential equations using exponent rules

12AF.A3.3

Solving logarithmic equations

12AF.A3.4

Exponential growth and decay by a factor

Exponential decay: Half-life

Exponential growth and decay by percentage

Finance: Compound interest

Finance: Compound Interest

Continuous growth and decay

Logarithmic scale: Richter scale (earthquake)

Logarithmic scale: pH scale

Logarithmic scale: dB scale

Finance: Future value and present value

12AF.B1.1

Converting between degrees and radians

12AF.B1.2

Trigonometric ratios of angles in radians

12AF.B1.3

Radian measure and arc length

12AF.B1.4

Find the exact value of trigonometric ratios

12AF.B2.1

Sine graph: y = sin x

Cosine graph: y = cos x

12AF.B2.2

Tangent graph: y = tan x

12AF.B2.3

Cotangent graph: y = cot x

Secant graph: y = sec x

Cosecant graph: y = csc x

12AF.B2.4

Graphing transformations of trigonometric functions

12AF.B2.6

Determining trigonometric functions given their graphs

12AF.B2.7

Ferris wheel trig problems

Tides and water depth trig problems

Spring (simple harmonic motion) trig problems

12AF.B3.1

Quotient identities and reciprocal identities

Pythagorean identities

Sum and difference identities

Sum and Difference Identities

Cofunction identities

12AF.B3.2

Double-angle identities

12AF.B3.3

Determining non-permissible values for trig expressions

12AF.B3.4

Solving first degree trigonometric equations

Solving second degree trigonometric equations

Solving trigonometric equations involving multiple angles

Solving trigonometric equations using pythagorean identities

Solving trigonometric equations using Pythagorean identities

Solving trigonometric equations using sum and difference identities

Solving trigonometric equations using double-angle identities

12AF.C1.1

Polynomial functions

Recognizing a Polynomial Function

What is a polynomial?

Classifying Polynomials

Polynomial components

12AF.C1.2

Multiplying monomial by monomial

Multiply a Monomial by a Monomial

Multiplying monomial by binomial

Multiplying binomial by binomial

Multiplying polynomial by polynomial

12AF.C1.3

Characteristics of polynomial graphs

Applications of polynomials

12AF.C1.4

Common factors of polynomials

12AF.C1.5

Find the difference of squares: (a - b)(a + b) = (a^2 - b^2)

Find the difference of squares

Evaluating polynomials

Using algebra tiles to factor polynomials

Factor theorem

Using the Factor Theorem to Test For Factors of a Polynomial

12AF.C1.6

Solving polynomial equations

Word problems of polynomials

12AF.C1.7

Determining the equation of a polynomial function

Factoring polynomials by grouping

Solving polynomials with unknown coefficients

Solving polynomials with unknown constant terms

12AF.C1.8

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

Factoring polynomials: x^2 + bx + c

Applications of polynomials: x^2 + bx + c

12AF.C1.9

Even and odd functions

12AF.C2.1

Graphing reciprocals of linear functions

Graphing reciprocals of quadratic functions

12AF.C2.2

What is a rational function?

Graphs of rational functions

12AF.C2.3

Simplifying rational expressions and restrictions

12AF.C3.1

Factoring difference of cubes

Factoring sum of cubes

Remainder theorem

Finding the Remainder Using Synthetic Division and the Remainder Theorem

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

12AF.C3.3

We&apos;ve got you covered with our complete help for Grade 12 math, whether it&apos;s Ontario curriculum, <a href="https://www.alberta.ca/writing-diploma-exams.aspx">Diploma exam</a> (Alberta), <a href="http://www.edu.gov.mb.ca/k12/assess/tests/index.html">Provincial test</a> (Manitoba) or <a href="https://www.saskatchewan.ca/residents/education-and-learning/departmental-exams">Departmental exam</a> (Saskatchewan).

Aligned with your class or textbook, you will get grade 12 math help on topics like Trigonometric identities, Vectors, Permutations and Combinations, Matrices, Polynomial functions and so many more. Learn the concepts with our video tutorials that show you step-by-step solutions to even the hardest grade 12 math questions. Then, strengthen your understanding with tons of grade 12 math practice. 

All our lessons are taught by experienced Grade 12 math teachers. Let&apos;s finish your homework in no time, and ACE that final.

Grade 12 Advanced Functions, University Preparation (MHF4U)

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Overview

Graphs of Rational Functions

What You'll Learn

A rational function graph is shaped by its vertical asymptotes, horizontal or slant asymptotes, and any holes left over from cancelled factors.

Vertical asymptotes come from denominator zeros that do not cancel, while holes come from factors that cancel completely.

Comparing the degrees of the numerator and denominator tells you whether the graph has a horizontal asymptote, a slant asymptote, or neither.

Plotting a few intercepts alongside the asymptotes is usually enough to sketch an accurate graph by hand.

The basic reciprocal function y equals 1 over x is the simplest rational graph and a good starting model for the rest.

What You'll Practice

Finding vertical and horizontal asymptotes for various rational functions

Factoring polynomials to identify points of discontinuity

Using synthetic and long division to determine slant asymptote equations

Plotting rational functions by combining asymptotes with intercepts and test points

Why This Matters

This Unit Includes

14 Video lessons

Practice exercises

Learning resources

Skills

Rational Functions

Vertical Asymptotes

Horizontal Asymptotes

Slant Asymptotes

Discontinuity

Factoring

Graphing

Polynomial Division