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## Grade 12 Math

## CONTENTS

## FUNCTIONS

A function related input to an output. It works like a machine that takes something in (input) and at the end gives us something back (output). F(x) is the traditional way of expressing functions. Each function has three parts the Input, the Relationship, and the Output. Understanding functions is important for future math works.

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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

A family of functions is a set of functions whose equations have a similar form. The “parent” of the family is the equation in the family with the simplest form. There are many different kinds of functions, these different types of functions can be grouped together into several different categories called 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

Quadratic relation is a type of relationship, where the second differences within Y coordinates are equal. Quadratic relation is represented by a parabola on a graph. A quadratic relation can be written in the following forms: vertex form y=a〖(x-h)〗^2+k, or standard form y=ax^2+bx+c, or factored form y= a(x-s)(x-r).

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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

The sum or difference of terms which have variables raised to positive integer powers and which have coefficients. A polynomial can have constants, variables and exponents, but never division by variable. Even though the poly- means many the polynomials terms should be finite.

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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

In exponential functions the variable is the power rather than the base. Exponential functions have the form of f(x)=b^x where b>0 and b ≠ 1. Exponential functions are important in real world problems such as modeling populations, computing investments and a lot of other applications. Logarithmic functions are the inverses of exponential 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

A radical function contains a radical expression with the independent variable in the radicand. For example, y= √x is a radical function which is also called a square root function in this case.

Domain and range of radical functions

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Solve radical equations

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Solve radical equations

## ROOTS AND RATIONAL EXPONENTS

Using rational numbers as exponents. A rational exponent represents both an integer exponent and an nth root. The root is found in the denominator and the integer exponent is found in the numerator. In other words, a rational exponent is an exponent that is a fraction. For example, √4= 4^(1/2). Using rational exponents are sometimes easier and quicker to simplify the radical expressions.

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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

Two or more equations containing common variables is called the system of equations. In solving a system of equations, we try to find values for each of the variables that will satisfy every equation in the system. The equations in the system can be linear or non-linear.

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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

Two or more inequalities containing common variables is called the system of inequalities. System of inequalities sometimes include equations as well as inequalities. System of inequalities are used when a problem requires a range of solutions, and there is more than one constraint on those solutions.

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Find the vertices of a solution set

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Linear programming

## Nonlinear inequalities

A nonlinear inequality is an inequality that is not a straight line when it is graphed. In other words, a nonlinear inequality is an inequality containing a nonlinear expression.

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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

Trigonometry is the study of triangles with emphasis on calculations involving the lengths of sides and the angles. Trigonometry is based on six functions, namely Sine, Cosine, Tangent, Cotangent, Secant, and Cosecant. Trigonometry also includes study of the properties of these functions and their graphs. Trigonometry is an important field in mathematics with a lot of applications in real life problems, like calculating the waves and tides in oceans, creating maps, satellite systems and a lot more.

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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

Trigonometry is based on six functions, namely Sine, Cosine, Tangent, Cotangent, Secant, and Cosecant. All the trigonometric functions can be defined in terms of a right triangle. These functions can be graphed, and some, notably the Sine function, produce shapes that frequently occur in nature.

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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

Trigonometric identities are simply ways of writing one of the trigonometric functions using others. For example, secx=1/cosx . This equivalence is called an identity. Understanding the trigonometric identities are important because they help us to simplify and solve the trigonometric problems.

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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

The family of curves including circles, ellipses, parabolas, and hyperbolas are called conic sections. All these geometric figures can be obtained by the intersection a double cone with a plane, hence the name conic section. The general equation that covers all conic sections is Ax^2+Bxy+Cy^2+Dx+Ey+F=0 from this equation we can derive equations for circle, ellipse, parabola and hyperbola. Conic sections have a lot of application in real life such as, studying the path of the planets around the sun, designing car headlights and a lot more applications.

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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

Complex number is a number with two components, namely real part and imaginary part. Complex numbers are written as a+bi where a is the real part and b is the imaginary part where i is the square root of -1. The plus sign in the complex number does not mean the addition, it is only a symbol to separate the two components of the complex number. Complex numbers are important in engineering fields for solving electrical equations.

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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

Complex plane is the coordinate plane used to graph the complex numbers. The x-axis is called the real axis and the y-axis is called the imaginary-axis. The complex number a+bi is graphed as the point (a,b) in the coordinate 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

Polar coordinate is a way to pinpoint where you are on a map or graph by how far away and at what angle the point is. The polar form of a complex number is another way to represent a complex number. In polar form we find the real and imaginary components in terms of r and θ where r is the length of the vector and θ is the angle made with the real axis. The polar form of a complex number a+bi is equal to r(cosθ+i sinθ)

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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

Vectors can be represented graphically using an arrow which points from the vector’s initial point to its terminal point. The length of the arrow-shaft represents the magnitude of the vector. To describe the direction of the vector, we normally use degrees from the horizontal line, in a counter-clockwise direction.

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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

Just like two-dimensional vectors, three-dimensional vectors are quantities with both magnitude and direction, and they are represented by directed line segments (arrows).

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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

A pattern is a series or sequence that repeats. Mathematics patterns are sequences that repeat according to a rule or rules. Numbers can have interesting patterns, like Arithmetic sequences Geometric sequences and so on. Number sequence is a list of numbers in a special order. The sum of infinite terms that follow a rule is called a Series. Sequences and series are important mathematic subject for future and more advanced math problems.

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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

One area of mathematics that has its roots deep in philosophy is the study of logic. Mathematical log helps to detect whether a statement is valid or invalid.

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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

Probability is the chance of something happening or how likely it is that some event will happen. Probability is a number between 0 (not happening) to 1 (certainly happening). Easiest way to understand probability is to toss a coin. There are two outcomes of tossing a coin namely Heads or Tails. So, the probability of the coin landing Heads is 1⁄2 same as the probability of the coin landing tails 1⁄2

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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

A probability distribution is a table or an equation that linked each outcome of a statistical experiment with its probability of occurrence. A probability distribution shows the probabilities of all the possible outcomes of an experiment, instead of just a particular or individual outcome. Probability distributions are important in real life because of most the phenomena happing around us can be described with a probability distribution. For example, heights, blood pressure, measurement errors are all following the normal distribution.

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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

Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of data. Statistics is a strong tool in everyday life to get answers about data and make concrete decisions.

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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