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High School Math Review

This site provides a comprehensive summary of the Ontario Grade 12
Advanced Functions (MHF4U) and Calculus and Vectors (MCV4U) courses.

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The course tab headings will show you the number of matching topics.

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Advanced Functions
Functions
Definition of a Function
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function:definition; relation; vertical line test; independent variable; dependent variable; function notation; mapping; domain; range;
479:10:49:17
110:27:43:17
Even and Odd Functions
x
even functions; odd functions
AAGQabc
lookup phrase
Absolute Value
x
absolute value
Interval Notation
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interval notation
Domain and Range
x
domain of a function; range of a function
Intervals of Increase/Decrease and Turning Points
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intervals of increase and decrease; turning points
AACQab
Order of operations for transformations of functions
x
order of operations for transformations; stretch/compress/reflect/translate
AABQabcde
Image of a point after transformation
x
image of a point after transformation; transformations
Transformation Equations (given two graphs)
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transformation equations; graphs
AACQcd
Inverse of a Function
x
inverse of a function
AAHQa,AAAQa
Piecewise Functions
x
piecewise functions
Solving linear inequalities
x
solving linear inequalities; solving double inequalities
Average and Instantaneous Rates of Change
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average rate of change; instantaneous rate of change; difference quotient
Polynomials
Finite Differences for Polynomial Functions
x
finite differences; successive finite differences; find value of leading coefficient from table of values
AAEQa
411:18:114:17
Polynomial Characteristics
Degree
x
degree of a polynomial; even/odd degree polynomial
AADQb
Leading Coefficient
x
leading coefficient
AADQc
Roots
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number of real roots; number of roots; degree of a polynomial; order of a root; even order roots; odd order roots; double order roots; triple order roots; complex roots; fundamental theorem of algebra
AADQa
Turning Points
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turning points; number of turning points; maximum; minimum
End Behaviour
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end behaviour; even/odd degree polynomial
Families of Polynomials
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families of polynomials; same set of roots
AAKQa,AALQa
Factoring polynomials of degree three or greater
Factoring Theorems
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remainder theorem; factor theorem; rational zero test; factoring
ZAAQab
Polynomial Division (Long and Synthetic)
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long division; synthetic division; factoring
AAJQabc,ZAAQc
Sum and Difference of Cubes
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sum of cubes; difference of cubes; factoring
Solving Polynomial Equations and Inequalities
Solving Polynomial Equations
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solving polynomial equations
Solving factorable polynomial inequalities
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solving factorable polynomial inequalities
Solving non factorable polynomial inequalities
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solving non factorable polynomial inequalities
Working with Polynomials
Finding the equation of polynomial given roots and a point
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finding the equation of a polynomial given roots and a point
AALQa
Finding the equation of a polynomial given a set of data points
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finding the equation of a polynomial given a set of data points;find value of leading coefficient from table of values
AAEQa
Transformations of Polynomial Functions
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transformations of polynomial functions
Graphing Polynomial Functions
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Average and instantaneous rates of change
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average rate of change; instantaneous rate of change; difference quotient
Rational Functions
Graphs of reciprocal functions
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graphs of reciprocal functions; vertical asymptotes
Discontinuities - vertical asymptotes and holes
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discontinuities; asymptotes; holes
End Behaviour - Horizontal, oblique and curved asymptotes
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end behaviour; horizontal asymptotes; oblique asymptotes; curved asymptotes
Finding equations of rational expressions given the graph
Roots
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finding the equation of a rational expression given a graph - using root information
Vertical Asymptotes
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finding the equation of a rational expression given a graph - using vertical asymptote information
Holes
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finding the equation of a rational expression given a graph - using hole infomration
Horizontal Asymptotes
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finding the equation of a rational expression given a graph - using horizontal asymptote information
Oblique Asymptotes
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finding the equation of a rational expression given a graph - using oblique asymptote information
Solving Rational Equations and Inequalities
Solving Rational Equations
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solving rational equations
Solving Rational Inequalities Graphically
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solving rational inequalities graphically
Solving Rational Inequalities Algebraically
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solving rational inequalities algebraically
Average and instantaneous rates of change
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average rate of change; instantaneous rate of change;difference quotient
Trigonometry
Radian Measure
Definition of radian
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radian; arch length
Converting between degrees and radians
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converting between degrees and radians; radians to degrees; degrees to radians
Angular velocity
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angular velocity
Finding Equations of Trig Functions given graph or data
Period of sine, cosine, tangent and their reciprocals
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finding equations of trig functions; trig functions - period; sine; cosine; tangent; secant; cosecant; cotangent; sin; cos; tan; sec; csc; cot
Stretch/Compression Factor of a Periodic Function
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finding equations of trig functions; sine and cosine - period; sine and cosine - stretch/compression; sine; cosine; sin; cos
Phase Shift of a Periodic Function
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finding equations of trig functions; phase shift; sine; cosine; sin; cos
Amplitude of a Sine or Cosine Curve
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finding equations of trig functions; sine and cosine - amplitude; sine; cosine; sin; cos
Axis of a Sine or Cosine Curve
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finding equations of trig functions; sine and cosine - axis; sine; cosine; sin; cos
Solving Equations involving mixed trig functions
Cofunction Identities
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cofunction identities
Identities involving x
x
cofunction identities
Working with angles
Primary trig ratios
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primary trig ratios; sohcahtoa; right angle triangles; sine; cosine; tangent; sin; cos; tan
Trig Ratios for Points on the Cartesian Plane
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trig ratios for points on the cartesian plane; (x,y) angles; sine; cosine; tangent; sin; cos; tan
Unit Circle
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unit circle; sine; cosine; sin; cos
Finding all the solutions in a given domain
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finding all solutions to trig equations
The domain of inverse trig functions on your calculator
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inverse trig functions on a calculator; domain of inverse trig functions on a calculator
Working with compound angles
Compound Angle Formulas
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compound angle formulas; sin(a+b); sin(a-b); cos(a+b);cos(a-b); sin(x+y); sin(x-y); cos(x+y); cos(x-y)
Double Angle Formulas
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double angle formula; sin(2x); cos(2x)
Problems involving multiples of x
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finding exact values of angles involving pi/12
Trigonometric Identities and Equations
Proving Trigonometric Identities
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proving trig identities; pythagorean identities; quotient identities
Solving Linear Trigonometric Equations
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solving linear trig equations
Solving Quadratic Trigonometric Equations
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solving quadratic trig equations
Exponential and Logarithmic Functions
Finite Differences for Exponential and Logarithmic Functions
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finite differences for exponential and logarithmic functions
Definition of a Logarithm
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definition of a logarithm
Properties of Logarithms
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properties of logarithms
Laws of Logarithms
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laws of logarithms; rules of logarithms
Graphs of Exponential and Logarithmic Functions
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graphs of exponential and logarithmic functions; graphing exponential and logarithmic functions
Transformations of Exponential and Logarithmic Functions
y = a(bk(x - d)) + c, y = a(logb(k(x - d))) + c
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transformations of exponential and logarithmic functions
Vertical stretch can be expressed as a horizontal translation
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transformations - vertical stretch can be expressed as a horizontal translation
Horizontal translation of exponential same as vertical translation of log
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transformations - horizontal translation of exponential same as vertical translation of log
Vertical translation of exponential same as horizontal translation of log
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transformations - vertical translation of exponential same as horizontal translation of log
Solving Exponential Equations
Solving by changing the base and equating exponents
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solving exponential equations by changing the base and equating exponents; solve
Solving by rewriting with a common exponent
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solving exponential equations by rewriting with a common exponent; solve
Solving by taking logarithms of both sides
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solving exponential equations by taking logarithms of both sides; solve
Solving quadratic exponential equations
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solving exponential equations by defining a substitution variable; solving quadratic exponential equations; solve
Solving Logarithmic Equations
Solve by Using Definition of a Logarithm
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solving logarithmic equations by using the definition of a logarithm; solve
Solve by taking logarithms of both sides
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solving logarithmic equations by taking logarithms of both sides; solve; logarithm
Solve by Equating Logarithmic Arguments
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solving logarithmic equations by equating logarithmic arguments; solve; logarithm
Comparing Intensities/Concentrations
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compare intensities or concentrations for richter, ph or decibel (db) scale; earthquake intensities; acid/base intensities
Finding Exponential and Logarithmic Equations
Establish base from two points and asymptote
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finding exponential and logarithmic equations - establish base from two points and asymptote
Establish base from three equally spaced points
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finding exponential and logarithmic equations - establish base from three equally spaced points
Problem Solving Tools
Finding Doubling Time
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finding doubling time
Change of Base Formula
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change of base formula
Four ways to write loga x
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four ways to write loga x; four ways to write loga(x)
Combination Functions
Sum, Difference, Product and Quotient Function
Domain
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domain of a sum, difference, product or quotient function
Range
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range of a sum, difference, product or quotient function
Even/Odd Nature of Combinations
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even, odd nature of of sum, difference, product or quotient functions
AAGQdef
Graphing Sum and Difference Functions
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graphing or sketching sum and difference functions
Graphing Product Functions
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graphing or sketching product functions
Graphing Quotient Functions
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graphing or sketching quotient functions
Composite Functions
Definition
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definition of a composite functions
Equation of a Composite Function
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equation of a composite function
Domain of x
x
domain of a composite functions
Range of x
x
range of a composite functions
Even/Odd Nature of Composite Functions
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even odd nature of composite functions
Calculus and Vectors
Limits and Continuity
Requirements for Existence of a Limit at a Point
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requirements for existence of a limit at a point; limit from the left; limit from the right; left limit; right limit
Requirements for Continuity at a Point
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requirements for continuity at a point; limit from the left; limit from the right; left limit; right limit; continuity; piecewise functions
Existence of Limits as x
x
existence of limits as x approaches infinity; horizontal asymptote
Types of Discontinuities
Holes
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continuity; discontinuity; removable discontinuity; holes
Jumps
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continuity; discontinuity; jump discontinuity
Vertical Asymptotes
x
continuity; discontinuity; vertical asymptotes
Methods of Finding Limits as x
Direct Substitution
x
finding limits; finding limits - direct substitution
Factoring
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finding limits; finding limits - factoring; limits as x approaches a
Multiply by conjugate (for square root limits)
x
finding limits; finding limits - multiply by conjugate; square root limits; limits as x approaches a
Difference of Cubes or Cube Roots (u substitutions)
x
finding limits; finding limits - difference of cubes; cube root limits; limits as x approaches a
Methods of Finding Limits as x
Degree of NumeratorxDegree of Denominator
x
finding limits; finding limits - horizontal asymptotes; limits as x approaches infinity
Degree of Numerator > Degree of Denominator
x
finding limits; finding limits - oblique asymptotes or curved asymptotes; limits as x approaches infinity
Derivatives
Definition of Derivative
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definition of derivative; instantaneous rate of change; slope of the tangent to the curve
Finding Derivatives from First Principles
For a specific point
x
finding the derivative from first principles - for a specific point
For any point
x
finding the derivative from first principles - for any point
Requirements for Differentiability
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requirements for differentiability; differentiable
Differentiation Rules
Derivative of a Constant
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differentiation rules; differentiation rules - derivative of a constant
Power Rule
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differentiation rules; differentiation rules - power rule
Constant Multiple Rule
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differentiation rules; differentiation rules - constant multiple rule
Sum and Difference Rule
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differentiation rules; differentiation rules - sum and difference rule; sum rule; difference rule
Product Rule
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differentiation rules; differentiation rules - product rule
Quotient Rule
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differentiation rules; differentiation rules - quotient rule
Chain Rule
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differentiation rules; differentiation rules - chain rule
Exponential and Logarithmic Formulas
Exponential Formula
x
exponential derivative formulas; e^x; a^x
Logarithmic Formula
x
logarithmic derivative formulas; ln; ln(x); log; log(x)
Trigonometric Formulas
Sine, Cosine, Tangent
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trig derivative formulas; sine; cosine; tangent; sin; cos; tan
Secant, Cosecant, Cotangent
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trig derivative formulas; secant; cosecant; cotangent; sec; csc; cot
Implicit Differentiation
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implicit differentiation
Curve Sketching
Find Critical Points
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find critical points
Find Vertical Asymptotes and Holes
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find vertical asymptotes and holes
Establish End Behaviour
Horizontal Asymptotes
x
end behaviour - horizontal asymptotes
Oblique Asymptotes
x
end behaviour - oblique asymptotes
Use First Derivative Test to Find Turning Points
x
use first derivative test to find turning points
Use Second Derivative Test to Classify Turning Points
x
use second derivative test to classify turning points
Use Second Derivative Test to find POI and Concavity
x
use second derivative test to find poi and concavity; points of inflection; point of inflection
Calculus Applications
Finding Maximum and Minimum of an Interval
x
finding maximum and minimum of an interval; absolute extrema
Equations of Tangents and Normals
Find Equation through a point on the curve
x
find the equation of the tangent to a curve through a point on the curve; find the equation of the tangent to a curve
Find the Equation of a tangent line with a given slope
x
find the equation of the tangent to a curve with a given slope; find the equation of the tangent to a curve
Find Equation through a point not on the curve
x
find the equation of the tangent to a curve through a point NOT on the curve; find the equation of the tangent to a curve
Motion on a Straight Line
Position
x
motion on a straight line - position function
Velocity
x
motion on a straight line - velocity function
Acceleration
x
motion on a straight line - acceleration function
Financial Applications
Revenue, Profit and Cost
x
financial applications - revenue; profit and cost
Demand and Price Functions
x
financial applications - demand function; price function; revenue
Marginal Revenue, Profit and Cost
x
financial applications - marginal revenue; marginal profit; marginal cost

Basic Vector Operations
Add, Subtract, Scalar Multiplication
x
basic vector operations; adding vectors; subtracting vectors; vector addition; vector subtraction
Modelling - Ground Speed, Tension, Friction
Ground Speed and Direction
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modelling with vectors; ground speed; air speed; wind; tension; friction
Tension is Supporting Cables
x
modelling with vectors; tension in supporting cables
Friction Acting on Objects on Ramps
x
modelling with vectors; friction on a ramp
Dot Product (Geometric and Cartesian)
x
dot product - geometric and cartesian/algebraic; vector product; vector product - scalar
Cross Product (Geometric and Cartesian)
x
cross product - geometric and cartesian/algebraic; vector product; vector product - vector; right hand rule
Direction Angles
x
direction angles; coordinate axis/axes angles
Angle Between Two Vectors
x
angle between 2 vectors; dot product
Projection of one Vector onto Another
x
projection of one vector onto another vector; dot product
Working with Lines and Planes in 3 space
Equations of Lines
Vector Equation of a Line
x
vector equation of a line
Parametric Equations for a Line
x
parametric equations for a line; parameter
Symmetric Equations for a Line
x
symmetric equations for a line
Scalar/Cartesian Equation for a Line
x
scalar equation of a line; cartesian equation of a line
Equations of Planes
Vector Equation of a Plane
x
vector equation of a plane
Parametric Equations for a Plane
x
parametric equations for a plane; parameter
Scalar/Cartesian Equation of a Plane
x
scalar equation of a plane; cartesian equation of a plane
Planes Parallel to Coordinate System
Planes Parallel to Coordinate Planes
x
planes parallel to coordinate planes
Planes Parallel to Coordinate Axes
x
planes parallel to coordinate axes
Angles Between Lines and Planes
Angle Between a Line and a Plane
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angle between a line and a plane
Angle Between Two Planes
x
angle between 2 planes
Testing for Collinear and Coplanar Vectors
Test for Collinear Vectors
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test for collinear vectors; scalar multiple
Test for Coplanar Vectors (linear combination)
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test for coplanar vectors; linear combination
Test for Coplanar Vectors (scalar triple product)
x
test for coplanar vectors; scalar triple product
Test for Perpendicular Vectors
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test for perpendicular vectors; dot product
Distance between Points and Lines
Distance Between a Point and a Line
In 2D (using dot product and normal)
x
distance between a point and a line - in 2d; dot product
In 3D (using cross product and direction)
x
distance between a point and a line - in 3d; cross product
In 3D (using dot product and direction)
x
distance between a point and a line - in 3d; dot product
Distance Between a Point and a Plane
x
distance between a point and a plane; dot product
Distance Between Parallel Lines
x
distance between parallel lines; cross product
Distance Between Skew Lines
x
distance between skew lines; dot product
Intersections of Lines and Planes
Intersection of Lines with Lines
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intersection of lines with lines
Intersection of Lines with Planes
x
intersection of lines with planes
Intersection of 3 Planes
Unique Solution (non coplanar normals)
x
intersection of 3 planes; intersection of 3 planes - unique solution; matrix analysis for intersection of 3 planes
Non Unique Solution (coplanar normals)
x
intersection of 3 planes; intersection of 3 planes - non unique solution; coplanar normals; matrix analysis for intersection of 3 planes
No Solution (coplanar normals, 2 or 3 non coincident parallel planes or no parallel planes)
x
intersection of 3 planes; intersection of 3 planes - no solution; coplanar normals; matrix analysis for intersection of 3 planes
1:89:81
a:280:64:55:13
xx x
5:298:200
a:3:19:13:16
b:3:80:13:16
c:3:152:13:16
d:3:209:13:16
e:3:256:13:16
xx xxxxx
4:435:308
a:3:326:13:13
b:3:348:13:15
c:3:373:13:14
d:3:394:13:15
xx xxxx
3:133:227
a:3:62:13:16
b:3:84:13:16
c:3:108:13:16
xx xxx
2:308:262
a:9:234:54:16
b:9:274:54:16
xx xx
6:358:312
a:4:17:11:14
b:4:100:11:14
c:4:149:11:14
d:4:197:11:14
e:4:248:11:14
f:4:295:11:14
xx xxxxxx
1:95:115
a:288:47:60:15
xx x
3:215:165
a:2:10:15:17
b:2:82:15:17
c:2:155:15:17
xx xxx
1:359:127
a:307:335:53:17
xx x
1:332:192
a:308:307:53:15
xx x
3:150:235
a:2:15:15:17
b:2:55:15:17
c:2:93:15:17
xx xxx
leading coefficient
BAP
domain and range
BAE