Honors Pre-Calc 220 Chapter Objectives

HONORS PRE-CALC 220 CHAPTER OBJECTIVES

Upon successful completion of this course, a student should be able to successfully work with the following mathematical concepts which are listed by the corresponding textbook chapter. While Chapters 1-3 are not formally included in this course, a student should, nevertheless, still be able to successfully work with the listed concepts due to the successful completion of previous algebra and geometry courses.

Chapter 1 Objectives:

Use the slope, midpoint and distance formulas with points on the Cartesian coordinate plane.
Graph equations by hand and/or with a graphing calculator.
Determine the symmetry of an equation with respect to the x-axis, y-axis, and the origin.
Determine and write the equation of a line in General Form and in Slope-Intercept form.
Define parallel and perpendicular lines.
Fine the equations of lines parallel and/or perpendicular to an existing line.
Distinguish between linear and nonlinear relations.
Use a graphing calculator to fine the line of best fit for a set of data.
Use the line of best fit to make predictions for the data set.
Solve linear and quadratic equations.
Know the quadratic formula and its applications.
Set up and solve various application problems.
Solve linear inequalities algebraically and with a graphing calculator.

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Chapter 2 Objectives:

Determine whether a relation represents a function.
Identify the graph of a function, determine its value at a particular point, and determine its domain and range.
Find the average rate of change of a function.
Determine where a graph is increasing or decreasing and be able to locate local maxima and minima points.
Determine whether a function is odd or even.
Be able to graph piece-wise, absolute value, and greatest interger functions.
Graph functions involving a variety of transformations (i.e. horizontal/vertical shifts, compressions/stretches, and reflections about the x-axis and y-axis.
Form the sum, difference, product, and quotient of two functions.
Form a composite function and find its domain.
Construct and solve a variety of functions.

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Chapter 3 Objectives:

Graph a quadratic function by hand and/or with a graphing calculator.
Identify the vertex and axis of symmetry of a quadratic function.
Use the discriminant of the quadratic formula to identify the number of x-intercepts of a quadratic function.
Graph and analyze power functions of degree n.
Identify polynomials and their degree.
Identify the zeros of a polynomial and their multiplicity.
Analyze the graph of a polynomial.
Analyze rational functions to determine their domain, and occurance of any vertical, horizontal, or oblique asymptotes.
Graph a rational function by hand and/or with a graphing calculator.
Find the real zeroes of a polynomial function.
Use synthetic division and other techniques to solve polynomial functions.
Work with complex numbers and their conjugates.
Solve polynomial and rational inequalities graphically and algebraically.

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Chapter 4 Objectives:

Determine whether a function is one-to-one.
Find the inverse function of a one-to-one function.
Be able to graph the function and its inverse.
Be able to apply the horizontal and vertical line tests to functions.
Define and use exponential functions.
Solve and graph exponential equations.
Define and use logarithmic functions.
Solve and graph logarithmic functions.
Convert exponential expressions to logarithmic expressions
Covert logarithmic expressions to exponential expressions
State and apply the properties of logarithms.
Apply the Change-of-Base formula to logarithms whose base is neither 10 or e.
Solve problems involving compound interest, growth and decay, and others involving the application of exponential and logarithmic functions.

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Chapter 5 Objectives:

Convert between angle measures in Degrees, Minutes, Seconds (DMS) and Decimal Degrees (DD).
Measure angles in radians and in degrees.
Measure arcs in degrees and radians.
Convert angle measures from degrees to radians and vice versa.
Solve problems involving circular motion ( i.e. angular and linear velocity).
Define the six trigonometric functions relative to the unit circle.
Determine the values of the six trigonometric functions relative to the unit circle.
Define the six trigonometric functions for a right triangle.
Determine the values of the six trigonometric functions in a right triangle.
In the unit circle, determine the reference and coterminal angles for any given angle.
Find the measure of an angle given the trigonometric ratio.
Determine the domain, range, and period of the six trigonometric functions.
Identify and use the fundamental trigonometric identities.
Solve application problems involving the trigonometric functions.
Graph the six trigonometric functions with various amplitudes, periods, phase shifts and vertical shifts.
Develop the inverse trigonometric functions.
Evaluate expressions using the inverse trigonometric functions.

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Chapter 6 Objectives:

Establish and prove trigonometric identities.
Use known trigonometric identities to prove other trigonometric identities.
Use known trigonometric identities to write equivalent trigonometric expressions.
Use the fundamental trigonometric identities to determine the exact trigonometric value of an angle.
Use trigonometric identities such as the sum/difference, double-angle, half-angle, product-to-sum, and sum-to-product identities to determine the exact trigonometric value of an angle.
Find exact and/or approximate solutions to trigonometric equations.

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Chapter 7 Objectives:

Solve right triangles.
Use the Law of Sines to solve oblique triangles where one side and two angles are known (SAA or ASA) or when two sides and the angle opposite one of them are known (SSA), commonly referred to as the ambiguous case.
Use the Law of Cosines to solve oblique triangles where two sides and the included angle are known (SAS) or when all three sides are known (SSS).
Find the area of a triangle by a variety of means such as Heron's formula, etc.
Determine the parameters (amplitude, period, phase shift, and vertical shift) of a trigonometric function from its graph.
Apply graphing to problem situations.
Solve problems involving simple harmonic motion.

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Chapter 8 Objectives:

Define polar coordinates and polar equations.
Convert measures from rectangular coordinates to polar coordinates and vice versa.
Graph polar equations by hand and/or by using a graphing calculator.
Classify and name polar equations.
Convert complex numbers from rectangular form to polar form.
Plot points on the complex plane.
Write a complex number in polar form.
Find the products and quotients of complex numbers in polar form.
Use Demoivre's Theorem to raise a complex number to any positive integral power.
Find a complex root to a complex number when written in polar form.
Add and subtract vectors, and multiply vectors by scalor quantities.
Find the magnitude of a vector and determine its unit vector.
Find the Dot Product of two vectors.
Find the measure of the direction angle between two vectors and the magnitude of the resultant vector.
Decompose a vector into two orthogonal vectors.
Find the distance between two points in 3 dimensional space and/or the magnitude of a vector in 3 dimensional space.
Find the direction angles of vectors in 3 dimensional space.

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Chapter 9 Objectives:

Know the names of the various conics and how they are derived from a double napped cone.
Be able to determine the equation of and the key parameters of the various conic functions (i.e. parabola, ellipse, and hyperbola).
Be able to graph by hand and with a graphing calculator the various conic functions (i.e. parabola, ellipse, and hyperbola).
Be able to apply conic functions to real world applications.

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Chapter 10 Objectives:

Solve systems of linear equations through substitution and/or linear combination.
Solve systems of linear equations through the applications of matrices.
Be able to add, subtract, and multiply matrices by hand and by using an appropriate calculator.
Be able to calculate the determinant of a matrix by hand and by using an appropriate calculator.
Perform the partial fraction decomposition of a rational expression.
Solve and graph systems of nonlinear equations and inequalities.
Set up, graph, and solve applications of linear programming.

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Chapter 11 Objectives:

Define arithmetic and geometric sequences and series.
Find terms of arithmetic and geometric sequences by the application of appropriate formulas.
Find partial sums of arithmetic and geometric series by the application of appropriate formulas
Use the concept of Mathematical Induction to prove statements involving natural numbers are true for all natural numbers.
Use the Binomial Theorem to expand and simplify binomials raised to an integral power.
Use the fundamental counting principle.
Find the number of permutations of n elements taken r at a time.
Find the number of combinations of n elements taken r at a time.
Specify the sample space for a random event.
Calculate the probability of a given event.
Calculate the odds of a given event.
Calculate the probability of mutually exclusive and independent events.

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Chapter 12 Objectives:

Determine the limit of various functions as the independent variable approaches a given value.
Use synthetic division to determine the factors of a given function.
Determine whether a given function is continuous over a specified domain or at a specific point.
Find the equation for the Tangent Line to the graph of a function at particular point on the curve.
Find and evaluate the first and second derivative of a given function.
Determine the relative maximum and minimum values for a given function.

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