MATH 1465 - Accelerated Precalculus

Credit Hours: 5.00

Objectives: The student will:

1. Determine domain, range, intercepts, and graph a function by plotting points.
2. Determine if a relation is a function.
3. Demonstrate a working knowledge of function notation and terminology.
4. Perform operations on functions including the difference quotient.
5. Demonstrate a working knowledge of the features of the graph of a function.
6. Graph functions using transformations. The functions used should include: x2, x3, |x|, 1/x, 1/x2, square root of x, cube root of x. Use asymptotes to help graph f(x) = 1/x, g(x) = 1/x2, and their transformations.
7. Graph piecewise-defined functions.

Outcome 2: Upon completion of this course, students will be able to demonstrate a working knowledge of polynomial and rational functions.

Objectives: The student will:

1. Recognize polynomial and rational functions and determine domain and range of these functions.
2. Graph quadratic functions using its vertex, intercepts, and axis of symmetry.
3. Graph power functions using transformations.
4. Graph and analyze higher order polynomial functions (use end behavior, zeros, and sign tests).
5. Find all zeros using Rational Roots Theorem and division of polynomials.
6. Use quadratic and polynomial functions in various applications.
7. Graph rational functions using asymptotes, zeros, and sign tests.

Outcome 3: Upon completion of this course, students will be able to demonstrate a working knowledge of exponential and logarithmic functions.

Objectives: The student will:

1. Demonstrate a working knowledge of the definition of an exponential function.
2. Demonstrate a working knowledge of the definition of a logarithm, including the fact that logarithmic and exponential functions are inverses.
3. Use the laws of exponents and properties of logarithms to simplify or evaluate expressions.
4. Graph exponential and logarithmic functions using transformations.
5. Determine domain and range of exponential and logarithmic functions.
6. Use the numbers 10 and e as bases for exponential and logarithmic functions.
7. Use properties of logarithms to expand and condense logarithmic expressions.
8. Solve exponential and logarithmic equations.
9. Solve applications such as exponential growth/decay and other applications in physical science.

Outcome 4: Upon completion of this course, students will be able to demonstrate a working knowledge of the equations and graphs of conic sections.

Objectives: The student will:

1. Find the center and radius of a circle from its equation and graph.
2. Find the vertex, focus, and directrix of a parabola from its equation and graph.
3. Identify key features of the ellipse and hyperbola including the center, foci, vertices, and asymptotes from their equations and graph.
4. Convert equations of conic sections from general form to standard form.
5. Identify a conic section from its equation.
6. Find an equation of a conic section from its graph.

Outcome 5: Upon completion of this course, students will be able to demonstrate a working knowledge of angles and their measure.

Objectives: The student will:

1. Draw angles in degrees and radians.
2. Convert angles between radians and degrees.
3. Find the arc length of a circle.
4. Find the area of a sector of a circle.
5. Find coterminal and reference angles.

Outcome 6: Upon completion of this course, students will be able to use and apply the trigonometry of right triangles.

Objectives: The student will:

1. Use the sine, cosine, tangent, cotangent, secant, and cosecant ratios to find exact values of trigonometric functions of acute angles.
2. Use the Reciprocal, Quotient, and Pythagorean Identities along with Complementary Angle Theorem to find exact trigonometric values of acute angles.
3. Solve right triangles and right triangle applications.
4. Find the exact values of the trigonometric functions of 30°-60°-90°.
5. Find the exact values of the trigonometric functions of 45°-45°-90°.

Outcome 7: Upon completion of this course, students will be able to evaluate and graph trigonometric functions.

Objectives: The student will:

1. Find the exact values of the six trigonometric functions of any angle using a point on the terminal side of the angle.
2. Determine the signs of the trigonometric functions of an angle in a given quadrant.
3. Use the reference angle to find the exact value of a trigonometric function.
4. Find the exact values of the six trigonometric functions of an angle using its corresponding point on the Unit Circle.
5. Know the domain and range of the six trigonometric functions.
6. Use coterminal angles, periodic properties, and even/odd properties to find exact values of the trigonometric functions.
7. Graph the six trigonometric functions f in the form y=Af(Bx-C)+D.
8. Find an equation of a trigonometric function from its graph.
9. Use the Law of Sines and Law of Cosines to solve non-right triangles and applications.
10. Find the area of a triangle.

Outcome 8: Upon completion of this course, students will be able to demonstrate knowledge of analytic trigonometry.

Objectives: The student will:

1. Know the domain and range of the six inverse trigonometric functions.
2. Find the exact values of inverse trigonometric functions.
3. Find the exact values of composite trigonometric functions involving inverses.
4. Solve trigonometric equations.
5. Establish trigonometric identities using fundamental identities and conjugates.
6. Use the sum, difference, double-angle, and half-angle formulas to find exact values and establish identities.

Outcome 9: Upon completion of this course, students will be able to demonstrate knowledge of and use the polar coordinate system.

Objectives: The student will:

1. Plot points using the polar coordinate system.
2. Convert between polar coordinates and rectangular coordinates.
3. Transform equations between polar and rectangular forms.
4. Graph polar equations by plotting points.

Outcome 10: Upon completion of this course, students will be able to demonstrate knowledge of and use vectors.

Objectives: The student will:

1. Graph vectors.
2. Find a position vector.
3. Add and subtract vectors geometrically and algebraically.
4. Find a scalar multiple of a vector.
5. Find the magnitude of a vector.
6. Find a unit vector.
7. Find the dot product of two vectors.

1. Functions
   1. Evaluate a function
   2. Sum, difference, product, quotient, and composition of two functions
   3. Difference quotient
   4. The graph of a function
      1. Vertical Line Test
      2. Domain, range, and intercepts
      3. Symmetry
      4. Even and odd
      5. Increasing, decreasing, and constant
      6. Local and absolute extrema
   5. Evaluate and graph piecewise-defined functions
   6. Graph functions using transformations
      1. Vertical and horizontal shifting
      2. Reflection
      3. Stretching and compressing
2. Polynomial and Rational Functions
   1. Quadratic functions
      1. Graph using vertex, axis of symmetry, and intercepts
      2. Maximum and minimum value
      3. Quadratic models and optimization
   2. Graph power functions using transformations
   3. Zeros of a polynomial function
      1. Remainder, factor, and rational root theorems
      2. Division of polynomials including synthetic division
      3. Fundamental Theorem of Algebra
      4. Complex zeros and conjugate pairs
   4. Graph higher-degree polynomial functions
      1. End behavior (Leading Term Test)
      2. Zeros and their multiplicity
      3. Sign tests
   5. Graph rational functions
      1. Domain
      2. Vertical and horizontal asymptotes (optional: slant asymptotes)
      3. Graph y = 1/x and y = 1/x^2 using transformations, intercepts, and asymptotes
      4. Graph other rational functions using intercepts, symmetry, asymptotes, and sign tests
3. Exponential and Logarithmic Functions
   1. One-to-one and inverse functions
   2. Evaluate exponential functions including base 10 and e
   3. Evaluate logarithmic expressions including base 10 and e
   4. Graph exponential and logarithmic functions using domain, intercepts, asymptotes, and transformations
   5. Properties of logarithms including product, quotient, and power rules
   6. Solve exponential and logarithmic equations
   7. Financial and exponential growth and decay models
4. Conic Sections
   1. Graph from standard form of the equation with center/vertex at (h,k)
      1. Circle - use center and radius
      2. Parabola - use vertex, focus, and directrix
      3. Ellipse - use center, vertices, and foci
      4. Hyperbola - use center, vertices, foci, and asymptotes
   2. Find an equation from the graph
   3. Write the standard form of the equation from the general form by completing the square
5. Trigonometric Functions
   1. Angles and their measure
      1. Draw angles in degrees and radians
      2. Convert from degrees to radians
      3. Convert from radians to degrees
      4. Find length of an arc of a circle
      5. Find area of a sector of a circle
      6. Coterminal and reference angles
   2. The Unit Circle
      1. Find the exact values of the six trigonometric functions using a point on the unit circle
      2. Find the exact values of the six trigonometric functions of quadrantal angles
   3. Properties of the trigonometric functions
      1. Find the domain and range of the six trigonometric functions
      2. Find the period of the six trigonometric functions
      3. Find the signs of the six trigonometric functions in a given quadrant
   4. Use transformations to graph y = Asin(Bx-C)+D, y = Acos(Bx-C)+D, y = Atan(Bx-C)+D, y = Acsc(Bx-C)+D, y = Asec(Bx-C)+D, and y= Acot(Bx-C)+D
6. Analytic Trigonometry
   1. Inverse trigonometric functions
      1. Find the domain and range of the 6 inverse trigonometric functions
      2. Find the exact value of all 6 inverse trigonometric functions
      3. Composite functions involving inverses
      4. Write a trigonometric expression as an algebraic expression in u
   2. Establish trigonometric identities
      1. Use fundamental identities
      2. Use a common denominator, factoring, conjugate, or any other algebraic technique
   3. Sum and difference formulas
      1. Find exact values
      2. Establish identities
      3. Find exact values involving inverse trigonometric functions
   4. Double and half-angle formulas
      1. Find exact values
      2. Establish identities
   5. Solving trigonometric equations
      1. Solve equations involving a single trigonometric function
      2. Solve equations involving multiple trigonometric functions
         1. Solve trigonometric equations in quadratic form
         2. Solve trigonometric equations using identities
7. Applications of Trigonometric Functions
   1. Right triangle trigonometric applications
      1. Find the value of trigonometric functions of acute angles
      2. Use the Complementary Angle Theorem
      3. Solve right triangles
      4. Solve applied problems including angles of elevation and depression
   2. Law of Sines
      1. Solve SAA or ASA Triangles
      2. Solve SSA Triangles
      3. Solve Applied Problems
   3. Law of Cosines
      1. Solve SAS Triangles
      2. Solve SSS Triangles
      3. Solve Applied Problems
   4. Area of a triangle
      1. Find the Area of SAS Triangles
      2. Find the Area of SSS Triangles using Heron’s Formula
8. Polar Coordinates
   1. Polar coordinates
      1. Plot points using polar coordinates
      2. Convert from polar coordinates to rectangular coordinates
      3. Convert from rectangular coordinates to polar coordinates
   2. Graph polar equations by plotting points
9. Vectors
   1. Graph Vectors
   2. Find a Position Vector
   3. Add and Subtract Vectors
   4. Find a Scalar Product of a Vector
   5. Find the Magnitude of a Vector
   6. Find a Unit Vector in the direction of the given vector
   7. Find the dot product of two vectors