Standards


 * PRE-CALCULUS STANDARDS (revised 2004) 2007 **
 *  Topics 1 – 8 are appropriate for January contests **
 *  Topics 1 – 11 are appropriate for February contests **
 *  Topics 1 – 14 are appropriate for contests in March and April **

 • given two functions perform the algebra of functions including composition of functions. • determine if a given function is: 1. symmetric (with respect to the axes and/or origin.) 2. periodic 3. monotonic 4. bounded 5. continuous
 * 1. Demonstrate an understanding of the theory of functions.•** **find domains; ranges; an specific values of functions in functional notation.**

• evaluate circular and trigonometric expressions involving any of the six functions and their inverses. • given the equation for a circular (trigonometric) function; identify and/or sketch the graph  and the graph of its inverse relation and state the domain and range of the original  • function and its associated inverse function.  • identify its equation when given a graph of any of the six circular functions.
 * 2**. **Demonstrate an understanding of connection between circular and trigonometric functions and their inverses.**

**3.** **Demonstrate an understanding of the trigonometric identities.**  • prove that an appropriate trigonometric equation is an identity when given the double order formulas for sine; cosine; and tangent. • prove that an appropriate trigonometric equation is an identity when given the half-angle formulas for sine; cosine; and tangent.

• use the appropriate trigonometric function(s) to solve problems involving right or oblique triangles. • <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;">apply the Law of Sines. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • apply the Law of Cosines. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • find the area of an oblique triangle. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • estimate the solution to a problem involving a right or oblique triangle. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • in the SSA case determine whether 0; 1; or 2 triangles exist and determine the triangles (if they exist)
 * 4.** **Demonstrate the ability to apply trigonometry to problem solving situations.**

<span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> **5.** **Demonstrate the ability to solve a variety of trigonometric (circular) equations.** <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • find the general solutions to a trigonometric equation <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • find particular solutions to a trigonometric equation within a given domain <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • solve equations involving inverse of circular/trigonometric functions.

<span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;">**6.** **Demonstrate an understanding of conic sections and loci.** <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • given the description of a locus determine the equation of the locus. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • given equation of an ellipse in standard form; determine the center; foci; and vertices; graph it. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • given the equation of a hyperbola in standard form; determine the foci; vertices; and asymptotes; and graph it. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • determine new equations resulting from translation or rotation of axes. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • express a quadratic equation in general form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 and use B2 - 4AC to distinguish conics. • recognize degenerate and imaginary cases.

• <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;">evaluate expressions involving rational exponents. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • sketch the graphs of exponential functions and logarithmic functions of different bases. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • solve equations involving exponential functions and logarithmic functions. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • solve real-world problems involving exponential functions and logarithmic functions. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • simplify expressions using the relationships between logarithms and exponents. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • express the number e and the expression 'e to the x' as infinite series
 * 7.** **Demonstrate an understanding of the relationship between exponential and logarithmic**
 * functions and their application to problem situations.**

<span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;">**8. Demonstrate the ability to solve problems using concepts from matrix algebra.** <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • apply determinants to solve systems of equations. • invert a square matrix.


 * END OF STANDARDS FOR JANUARY**


 * __Example Tests:__**
 * [|January Regionals 2010]**
 * [|January Regionals 2011]**
 * All January Regionals**


 * [|January Invitational 2009]**
 * [|January Invitational 2010] **
 * (no page with all January Invitationals)**

• <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;">identify perpendicular and parallel vectors. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • determine the measure of the angle between two vectors. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • resolve a vector into component vectors. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • add and subtract vectors and multiply a vector by a scalar. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • find the dot product of two vectors. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • use vectors to solve real world problems.
 * 9. Demonstrate the ability to solve problems using vectors.**

<span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> **10. Demonstrate an understanding of polynomial and rational functions; their parametric** <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> **equations and their graphs.** <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • given a polynomial function determine intercepts and sketch the graph. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • given an equation of rational function determine intercepts and asymptotes and sketch the graph. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • given a set of parametric equations sketch the graph.

<span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;">**11. Demonstrate an understanding of graphs in the polar coordinate system and their relation to** <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> **the Cartesian coordinate system.** <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • graph points in the polar coordinate system. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • convert between polar coordinates and Cartesian coordinates. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • express complex numbers in polar or trigonometric form. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • convert equations in polar form to Cartesian form. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • convert equations in Cartesian form to polar form. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • graph polar equations and identify specific types (roses; limacons; spirals; and conics) • use de Moivre's theorem to find powers and roots of complex numbers.


 * END OF STANDARDS FOR FEBRUARY**


 * [|February Regionals 2009]**
 * [|February Regionals 2011]**
 * All February Regionals**


 * [|February Invitationals 2010]**
 * [|February Invitationals 2011]**



<span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;">**12. Demonstrate understanding of mathematical induction and sequences and series.** <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • given an expression of rule for the nth term find any term of the sequence. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • given a sequence find a formula for the nth term in the sequence. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • find the nth term of a binomial expansion. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • find the sum of an arithmetic series. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • find the sum of a finite or infinite geometric series if it exists. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • define convergent and divergent sequences and series, determine limits if they exist. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • determine whether a sequence is increasing or decreasing. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • find the least upper bound and greatest lower bound of a sequence if they exist. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • express a series in sigma notation. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • use mathematical induction to prove series formulas. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • use mathematical induction to prove inequality formulas. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> **13. Demonstrate the ability to solve problems using probability and statistics.** <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • find probabilities of simple events. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • find probabilities using venn diagrams. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • find probabilities of mutually exclusive events. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • find probabilities of independent events. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • define an event and/or the complement of an event. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • find probabilities of the complement of an event. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • find conditional probabilities. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • find probabilities in binomial distributions. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • determine a standard (z) score in a normal distribution. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> **14. Demonstrate an understanding of the concept of limits and its applications.** <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • geometrically illustrate functions for which x increases without bound and find limits, if they exist. • find when possible for any neighborhood of a number L; a neighborhood of a point //<span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;">a //<span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> such that f(x) is in the neighborhood of L when x is in the neighborhood of //<span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;">a //<span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;">. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • calculate limits of functions using theorems about limits. • <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;">geometrically illustrate functions which are continuous at a point and/or continuous on <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> an interval. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • given a rational function f(x) find the limit if it exists at a point of discontinuity. <span style="font-family: 'Century Gothic',sans-serif; font-size: 10pt; line-height: 115%;"> • using the definition of the derived function of f(x) find the derive function. • determine the equation of tangents to graphs of curves given the slope formula.


 * [|March Regionals 2007]**
 * [|March Regionals 2010]**
 * All March Regionals**


 * [|March Invitationals 2008]**
 * [|March Invitationals 2011]**