# Academic standards list

### Precalculus — Mathematics

### Introduction

Academic standards define the expectations for knowledge and skills that students are to learn in a subject by a certain age or at the end of a school grade level. This page contains a list of standards for a specific content area, grade level, and/or course. The list of standards may be structured using categories and sub-categories.

### Number Expressions

TSS.Math.P.N.NE — Conceptual Category: Number and Quantity

Represent, interpret, compare, and simplify number expressions.

Use the laws of exponents and logarithms to expand or collect terms in expressions; simplify expressions or modify them in order to analyze them or…

Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Classify real numbers and order real numbers that include transcendental expressions, including roots and fractions of Pi and e.

Simplify complex radical and rational expressions; discuss and display understanding that rational numbers are dense in the real numbers and the integers…

### The Complex Number System

TSS.Math.P.N.CN — Conceptual Category: Number and Quantity

Perform complex number arithmetic and understand the representation on the complex plane.

Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and…

Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this…

Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers…

Use complex numbers in polynomial identities and equations.

### Vectors and Matrix Quantities

TSS.Math.P.N.VM — Conceptual Category: Number and Quantity

Represent and model with vector quantities.

Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for…

Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

Understand the graphic representation of vectors and vector arithmetic.

Perform operations on matrices and use matrices in applications.

Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the…

Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The…

Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as…

Work with 2 _ 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

### Sequences and Series

TSS.Math.P.A.S — Conceptual Category: Algebra

Understand and use sequences and series.

Use sigma notation to represent a series; expand and collect expressions in both finite and infinite settings.

Derive and use the formulas for the general term and summation of finite or infinite arithmetic and geometric series, if they exist.

*There are 3 components within this standard*.Understand that series represent the approximation of a number when truncated; estimate truncation error in specific examples.

### Reasoning with Equations and Inequalities

TSS.Math.P.A.REI — Conceptual Category: Algebra

Solve systems of equations and nonlinear inequalities.

Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 _ 3 or greater).

Solve nonlinear inequalities (quadratic, trigonometric, conic, exponential, logarithmic, and rational) by graphing (solutions in interval notation if …

### Parametric Equations

TSS.Math.P.A.PE — Conceptual Category: Algebra

Describe and use parametric equations.

### Conic Sections

TSS.Math.P.A.C — Conceptual Category: Algebra

Understand the properties of conic sections and model real-world phenomena.

Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

From an equation in standard form, graph the appropriate conic section: ellipses, hyperbolas, circles, and parabolas. Demonstrate an understanding of the…

### Building Functions

TSS.Math.P.F.BF — Conceptual Category: Functions

Build new functions from existing functions.

Understand how the algebraic properties of an equation transform the geometric properties of its graph. For example, given a function, describe the…

Develop an understanding of functions as elements that can be operated upon to get new functions: addition, subtraction, multiplication, division, and…

Find inverse functions (including exponential, logarithmic, and trigonometric).

*There are 4 components within this standard*.Explain why the graph of a function and its inverse are reflections of one another over the line y = x.

### Interpreting Functions

TSS.Math.P.F.IF — Conceptual Category: Functions

Analyze functions using different representations.

Analyze qualities of exponential, polynomial, logarithmic, trigonometric, and rational functions and solve real-world problems that can be modeled with…

Identify the real zeros of a function and explain the relationship between the real zeros and the x-intercepts of the graph of a function (exponential,…

Identify characteristics of graphs based on a set of conditions or on a general equation such as y = ax^2 + c.

Visually locate critical points on the graphs of functions and determine if each critical point is a minimum, a maximum, or point of inflection. Describe…

Graph rational functions, identifying zeros, asymptotes (including slant), and holes (when suitable factorizations are available) and showing end-behavior.

### Trigonometric Functions

TSS.Math.P.F.TF — Conceptual Category: Functions

Extend the domain of trigonometric functions using the unit circle.

Use special triangles to determine geometrically the values of sine, cosine, tangent for Pi/3, Pi/4 and Pi/6, and use the unit circle to express the…

Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

### Graphing Trigonometric Functions

TSS.Math.P.F.GT — Conceptual Category: Functions

Model periodic phenomena with trigonometric functions.

Determine the difference made by choice of units for angle measurement when graphing a trigonometric function.

Graph the six trigonometric functions and identify characteristics such as period, amplitude, phase shift, and asymptotes.

Find values of inverse trigonometric expressions (including compositions), applying appropriate domain and range restrictions.

Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

Determine the appropriate domain and corresponding range for each of the inverse trigonometric functions.

### Applied Trigonometry

TSS.Math.P.G.AT — Conceptual Category: Geometry

Use trigonometry to solve problems.

Use the definitions of the six trigonometric ratios as ratios of sides in a right triangle to solve problems about lengths of sides and measures of angles.

Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

### Trigonometric Identities

TSS.Math.P.G.TI — Conceptual Category: Geometry

Apply trigonometric identities to rewrite expressions and solve equations.

Apply trigonometric identities to verify identities and solve equations. Identities include: Pythagorean, reciprocal, quotient, sum/difference,…

Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

### Polar Coordinates

TSS.Math.P.G.PC — Conceptual Category: Geometry

Use polar coordinates.

### Model with Data

TSS.Math.P.S.MD — Conceptual Category: Statistics and Probability

Model data using regressions equations.

Create scatter plots, analyze patterns, and describe relationships for bivariate data (linear, polynomial, trigonometric, or exponential) to model…

Determine a regression equation to model a set of bivariate data. Justify why this equation best fits the data.

♦ Academic Standards marked with a diamond are "

*Supporting Content*" within the grade level or course.

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