## Grade 12 Math

## CONTENTS

In the last year of school, students in Grade 12 always battle with mixed emotions of nostalgia and excitement. By this year, they also have clarity on what courses and colleges they want to pursue and what career path aligns with their passion. A student’s interest and ability in Mathematics also plays a critical role in determining their professional goals in this stage.

## FUNCTIONS

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Identify functions

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Find the slope of a linear function

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Graph a linear function

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Write the equation of a linear function

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Linear functions over unit intervals

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Evaluate functions

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Find values using function graphs

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Complete a table for a function graph

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Add, subtract, multiply and divide functions

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Composition of functions

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Identify inverse functions

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Find values of inverse functions from tables

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Find values of inverse functions from graphs

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Find inverse functions and relations

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Identify graphs of continuous functions

## FAMILIES OF FUNCTIONS

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Reflections of functions

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Dilations of functions

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Transformations of functions

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Function transformation rules

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Describe function transformations

## QUADRATIC RELATIONS

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Find the maximum or minimum value of a quadratic function

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Graph a quadratic function

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Match quadratic functions and graphs

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Solve a quadratic equation using square roots

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Solve a quadratic equation by factoring

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Solve a quadratic equation by completing the square

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Solve a quadratic equation using the quadratic formula

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Using the discriminant

## POLYNOMIALS

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Write a polynomial from its roots

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Find the roots of factored polynomials

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Rational root theorem

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Complex conjugate theorem

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Conjugate root theorems

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Descartes’ Rule of Signs

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Fundamental Theorem of Algebra

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Match polynomials and graphs

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Factor sums and differences of cubes

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Solve equations with sums and differences of cubes

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Factor using a quadratic pattern

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Solve equations using a quadratic pattern

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Pascal’s triangle

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Pascal’s triangle and the Binomial Theorem

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Binomial Theorem I

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Binomial Theorem II

## RATIONAL FUNCTIONS

**A function that be written as a polynomial divided by a polynomial is a rational function. It is rational because one is divided by the other, like a ratio. Note that every polynomial function is a rational function with denominator of 1. A function that cannot be written in the form of a polynomial such as f(x)=Sin(x), is not a rational function. **

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Solve rational equations

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Check whether two rational functions are inverses

## LOGARITHMS

**The logarithm base b of a number x is the power to which b must be raised in order to equal x. In other words, A logarithm answers the question of how many of one number do we multiply to get another number. Logarithm is written like ** **. For example, ** ** so we say the logarithm of 8 with base 2 is equal to 3 since ** **.**

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Evaluate logarithms

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Change of base formula

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Identify properties of logarithms

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Product property of logarithms

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Quotient property of logarithms

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Power property of logarithms

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Properties of logarithms: mixed review

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Evaluate logarithms: mixed review

## EXPONENTIAL AND LOGARITHMIC FUNCTIONS

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Evaluate exponential functions

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Match exponential functions and graphs

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Solve exponential equations by factoring

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Solve exponential equations using common logarithms

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Solve logarithmic equations I

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Solve logarithmic equations II

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Identify linear and exponential functions

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Exponential functions over unit intervals

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Describe linear and exponential growth and decay

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Exponential growth and decay: word problems

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Compound interest: word problems

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Continuously compounded interest: word problems

## RADICAL FUNCTIONS

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Solve radical equations

## ROOTS AND RATIONAL EXPONENTS

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Roots of rational numbers

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Find roots using a calculator

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Evaluate rational exponents

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Operations with rational exponents

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Nth roots

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Simplify radical expressions with variables

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Simplify expressions involving rational exponents

## Systems of equations

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Solve a system of equations by graphing: word problems

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Classify a system of equations

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Solve a system of equations using substitution

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Solve a system of equations using substitution: word problems

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Solve a system of equations using elimination

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Solve a system of equations using elimination: word problems

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Solve a system of equations in three variables using substitution

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Solve a system of equations in three variables using elimination

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Determine the number of solutions to a system of equations in three variables

## Systems of inequalities

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Find the vertices of a solution set

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Linear programming

## Nonlinear inequalities

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Solve quadratic inequalities

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Graph solutions to higher-degree inequalities

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Solve higher-degree inequalities

## Matrices

**A matrix is an array of numbers, a rectangular (or square) array of numbers. Matrices can be written using brackets or parentheses. For example, ** ** is a 2 × 3 matrix, because there are two rows and three columns. Matrices are important in mathematics because they help us work with liner equations faster and easier. They are also used for plotting graphs, statistics and a lot more application in science and real life problems.**

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Matrix operation rules

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Add and subtract matrices

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Multiply a matrix by a scalar

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Linear combinations of matrices

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Multiply two matrices

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Simplify matrix expressions

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Solve matrix equations

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Determinant of a matrix

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Is a matrix invertible?

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Inverse of a 2 x 2 matrix

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Inverse of a 3 x 3 matrix

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Identify inverse matrices

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Solve matrix equations using inverses

## Trigonometry

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Radians and arc length

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Quadrants

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Coterminal and reference angles

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Find trigonometric ratios using right triangles

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Find trigonometric ratios using the unit circle

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Find trigonometric ratios using reference angles

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Inverses of trigonometric functions

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Solve trigonometric equations

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Trigonometric ratios: find a side length

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Trigonometric ratios: find an angle measure

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Solve a right triangle

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Law of Sines

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Law of Cosines

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Solve a triangle

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Area of a triangle: sine formula

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Area of a triangle: Heron’s formula

## Trigonometric functions

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Write equations of sine functions from graphs

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Write equations of sine functions using properties

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Graph sine functions

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Find properties of cosine functions

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Write equations of cosine functions from graphs

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Write equations of cosine functions using properties

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Graph cosine functions

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Graph sine and cosine functions

## Trigonometric identities

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Symmetry and periodicity of trigonometric functions

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Trigonometric identities I

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Trigonometric identities II

## Conic sections

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Write equations of parabolas in vertex form

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Graph parabolas

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Find properties of circles

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Write equations of circles in standard form

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Graph circles

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Find properties of ellipses

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Find the eccentricity of an ellipse

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Write equations of ellipses in standard form

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Find properties of hyperbolas

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Find the eccentricity of a hyperbola

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Write equations of hyperbolas in standard form

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Convert equations of conic sections from general to standard form

## Complex numbers

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Add and subtract complex numbers

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Complex conjugates

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Multiply and divide complex numbers

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Add, subtract, multiply and divide complex numbers

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Absolute values of complex numbers

## Complex plane

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Graph complex numbers

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Addition in the complex plane

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Subtraction in the complex plane

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Graph complex conjugates

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Absolute value in the complex plane

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Midpoints in the complex plane

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Distance in the complex plane

## Polar form

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Convert complex numbers from rectangular to polar form

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Convert complex numbers from polar to rectangular form

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Convert complex numbers between rectangular and polar form

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Match polar equations and graphs

## Two-dimensional vectors

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Find the direction angle of a vector

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Find the component form of a vector

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Find the component form of a vector from its magnitude and direction angle

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Find a unit vector

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Add and subtract vectors

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Multiply a vector by a scalar

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Find the magnitude or direction of a vector scalar multiple

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Find the magnitude and direction of a vector sum

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Linear combinations of vectors

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Graph a resultant vector using the triangle method

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Graph a resultant vector using the parallelogram method

## Three-dimensional vectors

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Find the component form of a three-dimensional vector

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Find a three-dimensional unit vector

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Add and subtract three-dimensional vectors

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Scalar multiples of three-dimensional vectors

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Linear combinations of three-dimensional vectors

## Sequences and series

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Find terms of a recursive sequence

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Identify a sequence as explicit or recursive

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Find a recursive formula

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Find recursive and explicit formulas

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Convert a recursive formula to an explicit formula

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Convert an explicit formula to a recursive formula

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Convert between explicit and recursive formulas

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Introduction to sigma notation

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Identify arithmetic and geometric series

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Find the sum of a finite arithmetic or geometric series

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Introduction to partial sums

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Partial sums of arithmetic series

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Partial sums of geometric series

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Partial sums: mixed review

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Convergent and divergent geometric series

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Find the value of an infinite geometric series

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Write a repeating decimal as a fraction

## Logic

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Find the direction angle of a vector

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Find the component form of a vector

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Find the component form of a vector from its magnitude and direction angle

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Find a unit vector

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Add and subtract vectors

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Multiply a vector by a scalar

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Find the magnitude or direction of a vector scalar multiple

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Find the magnitude and direction of a vector sum

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Linear combinations of vectors

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Graph a resultant vector using the triangle method

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Graph a resultant vector using the parallelogram method

## Probability

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Calculate probabilities of events

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Combinations and permutations

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Find probabilities using combinations and permutations

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Find probabilities using two-way frequency tables

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Identify independent events

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Find conditional probabilities

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Independence and conditional probability

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Find conditional probabilities using two-way frequency tables

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Find probabilities using the addition rule

## Probability distributions

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Write a discrete probability distribution

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Graph a discrete probability distribution

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Expected values of random variables

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Variance of random variables

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Standard deviation of random variables

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Write the probability distribution for a game of chance

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Expected values for a game of chance

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Choose the better bet

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Find probabilities using the binomial distribution

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Mean, variance and standard deviation of binomial distributions

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Find probabilities using the normal distribution I

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Find probabilities using the normal distribution II

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Find z-values

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Find values of normal variables

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Distributions of sample means

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The Central Limit Theorem

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Use normal distributions to approximate binomial distributions

## Statistics

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Variance and standard deviation

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Identify an outlier

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Identify an outlier and describe the effect of removing it

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Outliers in scatter plots 6. AA.6 Match correlation coefficients to scatter plots

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Calculate correlation coefficients

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Find the equation of a regression line

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Interpret regression lines

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Analyze a regression line of a data set

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Analyze a regression line using statistics of a data set

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Find confidence intervals for population means

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Find confidence intervals for population proportions

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Interpret confidence intervals for population means