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Pre-Calculus Objectives
- Trigonometry
- Algebraic Fundamentals (Functions and Relations)
- Determine whether a given relation is a function and identify the domain and range.
- Perform operations with functions, find composite functions, and find the inverse of a given function, if it exists.
- Use transformations to graph simple functions (e.g., y = Af(Bx + C) + D)
- Investigate and identify the characteristics of polynomial, radical, rational, and special functions (absolute value, step, and piecewise) in order to graph these functions, solve equations, and solve real-world problems. Calculators may be used to find zeros and intersections.
- Triangular and Circular Functions
- Define the six triangular trigonometric functions of an angle in a right triangle.
- Define the six circular trigonometric functions of an angle in standard position.
- Make the connection between the triangular and circular trigonometric functions.
- Recognize and draw an angle in standard position.
- Show how a point on the terminal side of an angle determines a reference triangle.
- Given one trigonometric function value, find the other five trigonometric function values.
- Develop the unit circle, using both degrees and radians.
- Change from radian to degree measure and vice versa, find angles that are coterminal with a given angle, and find the reference angle for a given angle.
- Solve problems, using the circular function definitions and the properties of the unit circle.
- Recognize the connections between the coordinates of points on a unit circle and
- coordinate geometry;
- cosine and sine values; and
- lengths of sides of special right triangles (30o-60o-90o and 45o-45o-90o).
- Find trigonometric function values of special angles and their related angles in both degrees and radians.
- Apply the properties of the unit circle without using a calculator.
- Use a conversion factor to convert from radians to degrees and vice versa without using a calculator.
- Find the length of an arc, linear speed, and angular velocities.
- Solving Triangles
- Write a practical problem involving triangles.
- Solve practical problems involving triangles.
- Use the trigonometric functions, Pythagorean Theorem, Law of Sines, and Law of Cosines to solve practical problems.
- Identify a solution technique that could be used with a given problem.
- Find the area of triangles using Hero's formula and the sine formula.
- Trigonometric Identities, Equations and Inverses
- Use trigonometric identities to make algebraic substitutions to simplify and verify trigonometric identities.
The basic trigonometric identities include:
- reciprocal identities;
- Pythagorean identities;
- quotient identities;
- sum and difference identities;
- double-angle identities; and
- half-angle identities.
- Solve trigonometric equations with restricted domains algebraically and by using a graphing utility.
- Solve trigonometric equations with infinite solutions algebraically and by using a graphing utility.
- Check for reasonableness of results, and verify algebraic solutions, using a graphing utility.
- Use a calculator to find the trigonometric function values of any angle in either degrees or radians.
- Define inverse trigonometric functions.
- Find angle measures by using the inverse trigonometric functions when the trigonometric function values are given.
- Find the domain and range of the inverse trigonometric functions.
- Use the restrictions on the domains of the inverse trigonometric functions in finding the values of the inverse trigonometric functions.
- Use the half-angle identities to verify other trigonometric identities and solve equations.
- Graphs of Trigonometric Functions and Inverses
- Determine the domain, range, amplitude, period, phase shift, and vertical shift of a trigonometric function from the equation of the function and from the graph of the function.
- Describe the effect of changing A, B, C, or D in the standard form of a trigonometric equation {e.g., y = A sin (Bx + C) + D or y = A cos [B(x + C)] + D}.
- State the domain and the range of a function written in standard form {e.g., y = A sin (Bx + C) + D or y = A cos [B(x + C)] + D}.
- Sketch the graph of a function written in standard form {e.g., y = A sin (Bx + C) + D or y = A cos [B(x + C)] + D} by using transformations for at least one period or one cycle.
- Identify the graphs of the inverse trigonometric functions.
- Write equations of trigonometric functions given the amplitude, period, phase shift, and vertical shift.
- Graph inverse trigonometric functions and identify their domain and range.
- Graph compound trigonometric functions using addition of ordinates.
- Polars and Vectors
- Find equal, opposite, and parallel vectors and add and subtract them geometrically.
- Find ordered pairs that represent vectors and add, subtract, multiply, and find the magnitude of vectors algebraically.
- Solve real-world problems using vectors.
- Investigate and identify the characteristics of the graphs of polar equations using a graphing calculator. Change points and equations from polar to rectangular form and vice versa, classify polar equations, predict the effects of changes in the constants, and determine the points of intersection of polar graphs.
- Operate with vectors in the coordinate plane, including addition, subtraction, scalar multiplication, inner (dot) product, normal of a vector, unit vector, components, and graphing properties, and solve practical problems using vectors.
- Apply DeMoivre's theorem to convert expressions from complex form to polar and vice versa as well as to find powers and roots of complex numbers.
- Perform operations with vectors in space, including the cross product, and solve practical problems.
- Apply the techniques of rotation of axes in the coordinate plane to graph functions and conic sections.
- Mathematical Analysis
- Polynomial and Rational Functions
- Identify a polynomial function, given an equation or graph.
- Identify rational functions, given an equation or graph.
- Sketch the graph of a polynomial function.
- Sketch the graph of a rational function.
- Investigate and verify characteristics of a polynomial or rational function, using a graphing calculator.
- Find the composition of functions.
- Find the inverse of a function algebraically and graphically.
- Determine the domain and range of the composite functions.
- Determine the domain and range of the inverse of a function.
- Identify absolute value, step, and piece-wise-defined functions.
- Use transformations to sketch absolute value, step, and rational functions.
- Verify the accuracy of sketches of functions, using a graphing utility.
- Identify all possible rational roots and determine all roots of a polynomial equation.
- Solve equations and inequalities both algebraically and graphically.
- Solve systems of equations including linear and nonlinear functions.
- Decompose a rational expression into partial fractions.
- Expand binomials having positive integral exponents.
- Use the Binomial Theorem, the formula for combinations, and Pascal's Triangle to expand binomials.
- Exponential and Logarithmic Functions
- Identify exponential functions from an equation or a graph.
- Identify logarithmic functions from an equation or a graph.
- Define e, and know its approximate value.
- Write logarithmic equations in exponential form and vice versa.
- Identify common and natural logarithms.
- Use laws of exponents and logarithms to solve equations and simplify expressions.
- Model practical problems, using exponential and logarithmic functions.
- Graph exponential and logarithmic functions, using a graphing utility, and identify asymptotes, intercepts, domain, and range.
- Preparation for Calculus
- Solve problems involving infinite arithmetic and geometric sequences and series, including finding the sum of finite and infinite series that will lead to an intuitive understanding of the concept of limit.
- Verify intuitive reasoning about the limit of a function, using a graphing utility.
- Find the limit of a function algebraically, and verify with a graphing utility.
- Find the limit of a function numerically, and verify with a graphing utility.
- Investigate and describe the end behavior and continuity of functions (including piecewise, absolute value, and step functions) with graphing calculators using the formal definition of continuity.
- Investigate the continuity of absolute value, step, rational, and piece-wise-defined functions.
- Identify zeros, upper and lower bounds, y-intercepts, symmetry, asymptotes, intervals for which the function is increasing or decreasing, points of discontinuity, end behavior, and maximum and minimum points, given a graph of a function.
- Explain and illustrate the derivative of a polynomial function as the slope of the line tangent to the graph of a function at a given point.
- Find the derivative of polynomial, rational, and radical functions using the definition of derivative and apply the derivative to problems concerning the equation of the tangent line at a point, increasing or decreasing intervals of a function, and maximum or minimum values of a function.
- Find the derivatives of polynomial, rational, and radical functions using the power, product, quotient, and chain rules.
- Find the second derivative of a polynomial function and use it to determine the intervals of concavity and inflection points of the function.
- Parametric Equations
- Graph parametric equations, using a graphing utility.
- Use parametric equations to model motion over time.
- Determine solutions to parametric equations, using a graphing utility.
- Compare and contrast traditional solution methods with parametric methods.
- Investigate and identify the characteristics of the graphs of polar equations in parametric form. Predict the effects of changes in the parameters, find the maximum r-values, and determine the points of intersection of polar graphs.
- Eliminate the parameters in a given equation, sketch a curve defined by parametric equations, and use parametric equations to model and solve real-world problems.
- Induction, Best Fit, and Recursion
- Apply the method of mathematical induction to prove formulas and statements.
- Express a discrete data relation implicitly and recursively and apply this method to real-world data.
- Find a best-fit curve for real-world discrete data using linear, quadratic, polynomial, exponential, power, and logarithmic functions.
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