MAT-152 Precalculus (Survey of Functions II)
This course is a continuation of the study of families of functions from those included in MAT 145, Survey of Functions I. Exponential, logarithmic, trigonometric/sinusoidal, and rational functions are analyzed in depth. Embedded within the study of each of these families are composition, decomposition, and the creation of inverse functions. An introduction to limit notation is used to describe both long and short run behavior. The use of realistic applications and modeling with these families of functions is an essential element of this course. Emphasis on multiple methods of solving equations (algebraic, graphic, and numeric) is included as are multiple representations (algebraic, graphic, numeric, and verbal) of mathematical information. This course carries SUNY General Education Mathematics (and Quantitative Reasoning) credit.
Prerequisite
MAT-145 with a C- or Higher or Placement into Math Level 3 or Higher
For more detailed course information view the
Course Outline
Course Description
This course is a continuation of the study of families of functions from those included in MAT 145, Survey of Functions I. Exponential, logarithmic, trigonometric/sinusoidal, and rational functions are analyzed in depth. Embedded within the study of each of these families are composition, decomposition, and the creation of inverse functions. An introduction to limit notation is used to describe both long and short run behavior. The use of realistic applications and modeling with these families of functions is an essential element of this course. Emphasis on multiple methods of solving equations (algebraic, graphic, and numeric) is included as are multiple representations (algebraic, graphic, numeric, and verbal) of mathematical information. This course carries SUNY General Education Mathematics (and Quantitative Reasoning) credit.
Credit Hours
3Semester(s) Course Will Be Offered
Fall, Spring, Summer
Prerequisites
MAT-145 with a C- or Higher or Placement into Math Level 3 or Higher
SUNY General Education Course
- Mathematics and Quantitative Reasoning
Course Learning Outcomes
- Model realistic scenarios using exponential, logarithmic, sinusoidal, and rational functions.
- Use algebraic skills to combine, compose, decompose and invert functions.
- Use limit notation to explain long range behavior, asymptotes, and removable discontinuities of functions.
- Solve equations algebraically, graphically, and numerically (via tables) and evaluate the result for reasonableness.
Topic Outline
- General outline of topics covered:
- 1. Common to all function families below (embed throughout the course)
- a. Understanding and using function notation
- b. Function evaluation
- c. Characteristics of their graphs (increasing, decreasing, concavity, asymptotes, holes, etc.)
- d. Choosing bounds to graph functions in an appropriate window using a graphing utility
- e. Finding zeros and the vertical intercept algebraically and graphically
- f. Solving for the input of a function given an output algebraically and graphically
- g. Solving inequalities related to functions graphically
- h. Interpreting the realistic meaning of the inputs and outputs, zeros, and vertical intercept
- i. Stating domain and range, both abstract and realistic or relevant
- j. Effects of transformations (graphical, algebraic, and verbal)
- k. Defining a formula for a function from a given graph, table, or verbal description
- l. Using limit notation to describe the long-run behavior of functions
- m. Calculating and interpreting average rate of change (AROC)
- 2. Functions (embed throughout the course)
- a. Evaluating functions with expressions (e.g., the difference quotient)
- b. Composition of functions
- i. Evaluation
- ii. Finding formulas algebraically (for all families of functions in the course)
- iii. Domain and range
- iv. Decomposition
- c. Inverse functions
- i. Definition of a function (review)
- ii. Determining when a function is invertible over its entire domain
- (1) Restricting the domain to make functions invertible
- iii. Notation x = 𝑓⁻¹(y) and interpretation
- iv. Constructing the formula of an inverse of a function
- v. Determining if two functions are inverses
- (1) Graphically
- (2) Algebraically as individual functions
- (3) Algebraically through composition
- 3. Exponential functions
- a. General forms and properties of exponential functions: y = a(1 + r)ᵗ, y = abᵗ, y = aeᵏᵗ, and y = P(1 + R/n)ⁿᵗ
- b. Modeling with exponential functions
- c. Converting growth factors between time scales (e.g., monthly to annual percent rate of change and vice versa)
- d. Development of the continuous growth form from the compound interest formula
- e. Converting between the different forms in 3a
- f. Limits at positive and negative infinity
- g. Applications (e.g., time value of money, population growth, doubling time, and half-life)
- 4. Logarithmic functions
- a. Common, natural, and other-base logarithms
- b. Converting between exponential and log equations (e.g., 2ᵗ = 8 to log₂(8) = t)
- c. Using single-sided limit notation to describe the vertical asymptote
- d. Properties of logarithms
- i. Relationship to exponent rules
- ii. Using them to simplify and expand algebraic expressions
- e. Using logarithms to solve exponential equations
- f. Solving logarithmic equations
- g. Applications (e.g., comparing orders of magnitude, graphing using log scales, decibels, and the Richter and pH scales)
- 5. Trigonometric functions
- a. Sine, cosine, and tangent of an angle (review)
- b. Center-radius form of a circle
- c. Determining (x, y) coordinates on a circle with a given radius and angle measure
- d. Pythagorean identity: sin²θ + cos²θ = 1
- e. Defining radian measure through arc length
- f. Converting between radian and degree measure
- g. Inverse trigonometric functions (using both sin⁻¹(x) and arcsin(x) notation)
- h. Solving equations with sine, cosine, and tangent using radian measure over a restricted domain
- i. Sinusoidal functions
- i. Finding period, amplitude, frequency, and shift (vertical and horizontal) algebraically and graphically
- ii. Interpreting period, amplitude, frequency, and shift (vertical and horizontal) algebraically and graphically
- iii. Creating sinusoidal models from graphs, tables, or verbal descriptions
- j. Applications (e.g., Ferris wheels, daylight hours, and pendulums)
- 6. Rational functions
- a. Polynomial functions (review)
- b. Algebraic manipulation into r(x) = p(x)/q(x) form (including manipulation of complex fractions)
- c. Long-run behavior from the ratio of leading terms
- d. Using limit notation to describe asymptotic behavior (vertical, horizontal, and optionally slant), long-run behavior, and holes
- e. Short-run behavior: identifying (algebraically and graphically) the y-intercept, zeros, undefined values, and their connection to vertical asymptotes
- f. Applications (e.g., average cost and concentration)