Integrated Math II (Honors)

Students will fully master various techniques for solving problems involving quadratic equations and functions. They will explore how quadratic functions are used to model real-world situations and use their knowledge as a stepping stone to begin investigating the fascinating world of complex numbers.

Building on foundations laid out in Integrated Math I (Honors), students will extend their understanding of functions to encompass function arithmetic, composition, periodicity, even and odd functions, and invertible functions. They will carefully investigate the process of performing multi-step graph transformations and learn how to compute inverse functions.

This course will deeply enrich students' knowledge of exponential functions. This includes a deep dive into logarithms, examining their graphs and properties. Students will learn to solve both exponential and logarithmic equations and further develop their aptitude for modeling scenarios with exponential functions, such as calculating compound interest.

Building upon the foundations laid in Integrated Math I, students will strengthen their knowledge of sequences and series, including geometric sequences, arithmetic series, sigma notation, and the binomial theorem.

This course aims to broaden students' knowledge of geometry to include crucial concepts such as similarity, fundamental circle theorems, conic sections, and solid geometry. Students will lay a robust foundation in trigonometry, delving into the unit circle and its pivotal role in extending trigonometric ratios and graphing trigonometric functions.

Finally, students lay solid foundations in probability and combinatorics and build on previous understandings of correlation and regression, further solidifying their knowledge and skills.

Number Systems

• Describing the hierarchical structure of the real number system and classifying numbers accordingly.
• Expressing repeating decimals as fractions.
• Understanding the properties of sums and products of rational and irrational numbers.
• Addition, subtraction, and multiplication of complex numbers,
• Understanding the cyclic property of the imaginary unit and solving related problems.
• Solving quadratic equations with complex roots.

Polynomials

• Finding the least common multiple of two polynomials.
• Describing numerical relationships (such as listing Pythagorean triples) using polynomial identities.
• Solving quadratic equations by factoring, completing the square, and using the quadratic formula.
• Using the discriminant to determine the number of solutions to a quadratic equation.
• Factoring polynomials containing greatest common factors, differences of squares, differences of cubes, and biquadratic expressions.
• Solving polynomial equations.

Quadratic Functions

• Identifying elementary quadratic functions undergoing shifts and reflections.
• Calculating the roots of a quadratic function.
• Computing the axis of symmetry of a parabola by taking the average of the roots, expressing it in vertex form, or using a formula.
• Sketching graphs of general quadratic functions and stating their domain and range.
• Solving systems of equations involving quadratic functions and lines.
• Using quadratic functions to model real-world applications involving vertical motion, revenue, cost, and profit.

Functions

• Function arithmetic, including function composition.
• Defining and determining the local extrema of functions.
• Accurately describing the behavior of a function near a vertical asymptote.
• Classifying functions as periodic, even, odd, one-to-one, and many-to-one, and solving problems using these properties.
• Describing the conditions required for a function to be invertible and how the graph, domain, and range of a function are related to that of its inverse.
• Calculating inverses of linear, quadratic, exponential, and logarithmic functions
• Calculating the average rate of change of a function over an interval.
• Graphing transformations of functions, including composite and absolute value transformations.

Absolute Value Functions

• Plotting graphs of shifted, stretched, and reflected absolute value functions.
• Finding the range and roots of an absolute value function.
• Solving equations involving absolute value functions, including cases with extraneous solutions.

Exponentials & Logarithms

• Converting between exponential and logarithmic form.
• Applying the laws of logarithms to simplify expressions.
• Solving exponential equations using logarithms, including cases with different bases.
• Solving logarithmic equations.
• Graphing exponential and logarithmic functions and describing their properties.
• Calculating inverses of exponential and logarithmic functions.
• Using exponential functions to model real-world situations involving compound interest, including continuous compounding.

Sequences & Series

• Constructing an explicit formula for the nth term of a geometric sequence.
• Converting between explicit and recursive formulas for geometric sequences.
• Solving mathematical problems related to geometric sequences (e.g., finding the common ratio of a geometric sequence given two non-consecutive terms).
• Calculating the sum of an arithmetic series.
• Using arithmetic series to model and solve real-world and mathematical problems.
• Expanding binomials using Pascal's triangle and the binomial theorem.
• Approximating binomials at specific points, including calculating the percentage error of an approximation.

Rationals & Radicals

• Adding, subtracting, multiplying, dividing, and simplifying rational expressions.
• Solving simple and advanced radical equations, including cases with no real solutions.

Geometry

• Understanding and applying the concept of similarity and solving problems involving similar polygons.
• The AA, SSS, and SAS similarity criteria.
• Proving that two triangles are similar, and applying CPSTP and corresponding angles in similar triangles.
• Proving the sum of the interior angles of a triangle equals 180 degrees.
• Understanding and applying the midpoint and triangle proportionality theorems.
• Proving the exterior angle theorem.
• Solving problems using the midpoint and triangle proportionality theorems.
• Proving the midpoint theorem.
• Using similarity transformations to show that two figures are similar.
• Proving properties of parallelograms and theorems relating parallelograms and rectangles.
• Solving circle problems using the inscribed angle theorem, Thales' theorem, and properties of inscribed quadrilaterals.
• Solving problems with tangent lines to circles, tangent-tangent, secant-sectant, and sectant-tangent lines, and applying the tangent-chord theorem.
• Proving that any two circles are similar.
• Finding equations of circles and parabolas in a coordinate plane, describing their mathematical properties, and calculating intersections.
• Finding equations of tangent lines to circles.
• Using the properties of the perpendicular bisectors of diameters and chords to solve problems in the coordinate plane.
• Applying Thales' theorem in the coordinate plane.

Solid Geometry

• Calculating lengths within rectangular solids using the distance formulas.
• Calculating surface areas and volumes of similar solids.
• Calculating surface areas and volumes of polyhedra and non-polyhedra.
• Describing solids of revolution and their volumes.
• Applying Euler's formula for polyhedra.
• Describing the five Platonic solids.

Trigonometry

• Calculating trigonometric and reciprocal trigonometric ratios in right triangles.
• Using trigonometry to calculate angles and sides in right triangles.
• Deriving and applying the special trigonometric ratios for acute angles.
• Understanding the relationship between cofunctions.
• Solving practical problems using trigonometry, including cases with multiple triangles.
• Understand the concept of radian measure and use radians to calculate arc lengths and areas of circular sectors.
• Describing how the unit circle extends the trigonometric ratios to non-acute and negative angles.
• Applying the Pythagorean identity.
• Graphing transformed trigonometric and reciprocal functions and computing properties such as the amplitude, period, zeros, and local extrema of transformed trigonometric functions.

Probability & Statistics

• Essential concepts from probability, including sample spaces, event complements, compound events, and Venn diagrams,
• Calculating probabilities involving unions and intersections of events and using these ideas to solve real-world problems.
• Applying the addition law and understanding mutually exclusive events.
• Conditional probabilities, including the multiplication law and its derivation, the law of total probability, independent events, and tree diagrams.
• Applying the rule of sum and the rule of product.
• Determining the number of ways to order objects and calculating permutations and combinations.
• Calculating probabilities using combinatorics.
• Analyzing data sets by computing their mean, variance, standard deviation, z-scores, and covariance and interpreting these within the context.
• Finding and interpreting the linear correlation coefficient of a given data set, calculating a regression line, and finding and analyzing residuals.