Pre-Calculus

Course description:

Precalculus is an advanced mathematics course that uses meaningful problems and appropriate technologies to build upon previously learned mathematical concepts to develop the underpinnings of calculus.

Standard 1.0:  Models for Real-World Phenomena

Students will model and analyze real-world phenomena using techniques from algebra and data analysis.

Learning Expectations:

The student will:

• 1.1     select and use appropriate algebraic functions to model real-world situations;
• 1.2     select and use appropriate techniques from data analysis to model real-world phenomena.

Student Performance Indicators:

• model real-world phenomena using techniques of data analysis;
• recognize and apply mathematical models of linear, quadratic, exponential, logarithmic, and trigonometric functions;
• use scatterplot residuals, and/or correlation coefficients to determine whether a model is appropriate;
• apply equations and graphs of conic sections to model real-world phenomena.
• use models when appropriate to draw conclusions or make predictions.

Standard 2.0: Algebraic Functions

Students will extend the concepts of function from earlier courses to a wider variety of functions and their graphs and real-world applications.

Learning Expectations:

The student will: 

• 2.1     represent a variety of functions graphically;
• 2.2     use a variety of methods to analyze and interpret functions;
• 2.3     determine the slope and equations of lines tangent to curves;
• 2.4     apply functions in problem situations.

Student Performance Indicators:

• sketch the graphs of the basic functions (linear, quadratic, cubic, square root, absolute value, reciprocal, trigonometric, exponential, logarithmic, and greatest integer);
• graph transformations and combinations of transformations for all basic functions;
• analyze functions, such as by decomposing into simpler functions;
• determine if a function is even, odd, or neither;
• use an appropriate technology to solve inequalities;
• demonstrate an understanding of  the concept of the limit of a function;
• apply the limit of a function to find the slope of a line tangent to a curve;
• write equations of tangents and normals to conic sections;
• apply limits to develop the concept of continuity and identify intervals of increase and decrease;
• locate critical points on the graphs of polynomial functions and determine if each critical point is a minimum, a maximum, or a point of inflection;
• determine an equation of a rational function from a written description.
• define and use the logarithmic function as the inverse of the exponential function;
• sketch the graphs of exponential and logarithmic functions;
• solve exponential and logarithmic equations modeling real-world problems (e.g. growth and decay).

Standard 3.0: Trigonometric Functions

The student will

• 3.1     apply trigonometry concepts and applications to model and solve problems;
• 3.2     use trigonometric concepts to represent, apply, and operate with complex numbers;
• 3.3     solve trigonometric equations and inequalities algebraically or graphically;
• 3.4     interpret transformations of trigonometric functions.

Student Performance Indicators:

• define six circular functions;
• sketch graphs of the six trigonometric functions involving period change, amplitude change, phase shift, and/or vertical shift;
• use trigonometric functions to model periodic phenomena;
• use graphs to develop and verify trigonometric identities;
• find values of inverse trigonometric functions, applying appropriate domain and range restrictions;
• solve trigonometric equations and inequalities either algebraically or using graphing technology.
• derive the Law of Sines and the Law of Cosines and apply them to solve problems involving triangles and vectors;
• derive and apply the formulas for the area of a triangle and the sector of a circle;
• understand the relationship between measurements in radians and degrees;
• apply radian measures in problems related to linear and angular velocity;
• understand and apply vectors to solve real world problems;
• represent complex numbers in both rectangular and polar form;
• apply the trigonometric form of complex number in calculations; 
• prove and apply DeMoivre's Theorem to find roots and powers of complex numbers.

Standard 4.0:  Sequences and Series

Students will develop the concept of limit  by examining infinite sequences and series.

Learning Expectations: 

The student will:

• 4.1     represent sequences and series;
• 4.2     determine, when possible, the sums of infinite series.

Student Performance Indicators:

• demonstrate an understanding of sequences by representing them recursively and explicitly;
• use sigma notation to represent a series;
• determine whether a given series converges or diverges;
• find the sum of an infinite series that converge;
• find the sum of an infinite geometric series;
• use the Binomial Theorem to expand binomials.