MATH 1415 - Precalculus I: College Algebra

Credit Hours: 4.00

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.

1. Review of Fundamentals of Algebra
   1. Linear and nonlinear equations and inequalities
   2. Absolute value equations and inequalities
   3. Radical equations
   4. Equations and graphs of lines
2. 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
3. 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
4. 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
5. 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