Integrated Math II

Building on the foundations laid in Integrated Math I students will extend their understanding of functions to encompass function arithmetic (including function composition), domain, range, local extrema, periodicity, and even and odd functions. Students will conduct detailed explorations of function invertibility and investigate how to perform multi-step graph transformations. They will learn about formal sequence notation and explore recursive sequences.

As part of this course, students will master techniques for solving quadratic equations and explore quadratic functions in-depth, including how they're used to model real-world problems. Students will also deepen their understanding of exponential functions, explore the relationship between exponentials and logarithms, and the laws of logarithms.

This course aims to broaden students' knowledge of geometry, including concepts such as congruence, similarity, circle theorems, and solid geometry. Students will lay robust foundations in trigonometry, including exploring the unit circle, the Pythagorean identity, and how trigonometry is used to solve practical problems.

For the first time, the concept of a rigorous mathematical proof is introduced. Students begin by exploring proof fundamentals, such as postulates, definitions, theorems, and the properties of congruence. Once mastered, they will use these ideas to prove important geometric theorems. Opportunities to demonstrate and develop their proof-writing skills are ongoing throughout this course.

Finally, students will develop foundations in probability and combinatorics and explore essential tools for analyzing bivariate data.

Quadratics

• 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.
• Identifying elementary quadratic functions undergoing shifts and reflections.
• Calculating roots of quadratic functions.
• 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.
• Finding intersection points of lines and quadratic functions.
• Using quadratic functions to model real-world applications involving vertical motion, revenue, cost, and profit.

Functions & Sequences

• Function arithmetic, including function composition.
• Defining and determining 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 and quadratic functions
• Calculating the average rate of change of a function over an interval.
• Graphing transformations of functions, including composite and absolute value transformations.
• Calculating a sequence's terms explicitly or recursively and working with Fibonacci sequences.

Exponentials & Logarithms

• Applying composite transformations to graphs of exponential functions and determining their properties.
• Converting expressions between exponential and logarithmic form, including those containing common and natural logarithms.
• Applying the laws of logarithms to simplify expressions.

Geometry

• Proving the vertical angles theorem.
• Proving the alternate and consecutive interior and exterior angle theorems and their converses.
• Proving the linear pair perpendicular theorem, the perpendicular bisector theorem, and their converses.
• Proving the sum of the interior angles of a triangle equals 180 degrees.
• Proving the exterior angle theorem.
• Understanding and applying the ASA, AAS, SAS, SSS, and HL congruence criteria.
• Understanding and applying the reflexive, symmetric, and transitive properties of congruence.
• Proving that two triangles are congruent and applying CPCTC.
• Understanding and applying the concept of similarity and solving problems involving similar polygons.
• The AA, SSS, and SAS similarity criteria and the midpoint and proportionality theorems.
• Showing that two figures are similar using similarity transformations.
• Proving that two triangles are similar, and applying CPSTP and corresponding angles in similar triangles.
• Understanding and applying the midpoint and triangle proportionality theorems.
• Proving the midpoint theorem.
• 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.
• Applying Euler's formula for polyhedra.
• Describing the five Platonic solids.
• Calculating surface areas and volumes of similar solids.
• Calculating lengths within rectangular solids using the distance formulas.
• Calculating surface areas and volumes of polyhedra and non-polyhedra,
• Describing solids of revolution and computing their volumes.

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.

Probability & Statistics

• Essential concepts from probability, including sample spaces, event complements, compound events, Venn diagrams, unions, and intersections of events, and using these ideas to solve real-world and mathematical problems.
• Carrying out elementary analyses on bivariate data, including interpreting scatter plots, making predictions using trend lines, interpreting residues, interpolation vs. extrapolation, and linear correlation
• Distinguishing between correlation and causation.