AP Precalculus: Change in Tandem (1.1) Notes (Function Basics, Monotonicity, and Graph Analysis)

Function Basics and Representations

• A function is a mathematical relation that maps a set of input values to a set of output values such that each input value is mapped to exactly one output value.

  • Input values = Domain = Independent variable (𝑥)

  • Output values = Range = Dependent variable (𝑦)

• Notation and helper variables

  • To emphasize which quantities are inputs and outputs, we may introduce helper variables. Example: If we throw a football and measure height over time, we can use 𝓉 for time (seconds) and 𝒉(𝓉) for height (feet). This makes the input/output relationship explicit.

  • A simple function example: 𝒘(𝑥) is a function where 𝑤 is the amount of water in a pool (gallons) and 𝑥 is the length of the pool (units of length).

  • Independent variable (input): 𝑥 = length of pool

  • Dependent variable (output): 𝑤 = amount of water in pool (gallons)

• Four common representations of a function (VANG): Verbal, Analytic, Numerical, Graphical

  • Verbal: Describe the rule in words. Example for increasing function: "as the input values increase, the output values always increase."

  • Analytic (Algebraic): express the rule with a functional inequality or equation. Example: for all 𝑎, 𝑏 in the interval, if 𝑎 < 𝑏, then 𝑓(𝑎) < 𝑓(𝑏).

  • Numerical: use data values or a table to show the rule holds for input-output pairs.

  • Graphical: show the behavior on a graph; for increasing functions, the graph rises as 𝑥 increases.

• Increasing function (definition)

  • Verbal: as input values increase, outputs increase.

  • Analytic: ∀ 𝑎, 𝑏 in the domain with 𝑎 < 𝑏, we have 𝑓(𝑎) < 𝑓(𝑏).

  • Graphical: the graph moves upward as you move from left to right.

• Decreasing function (definition)

  • Verbal: as input values increase, outputs decrease.

  • Analytic: ∀ 𝑎, 𝑏 in the interval with 𝑎 < 𝑏, we have 𝑓(𝑎) > 𝑓(𝑏).

  • Graphical: the graph moves downward as you move from left to right.

• Example (numerical): Determine whether a function is increasing or decreasing on a given interval using a table

  • Example data:

  • 𝑥: 4, 6, 7, 10, 20

  • 𝑓(𝑥): 1, 1.01, 1.04, 1.06, 1.29

  • Observation: as 𝑥 increases, 𝑓(𝑥) increases; function is increasing on the interval shown.

Basic elements of a function’s graph

• Zero (x-intercept)

  • The graph intersects the x-axis when the output value (𝑦) is zero.

  • Input values associated with x-intercepts are the x-values in the domain (input values).

• y-intercept (y-intercept on y-axis)

  • The graph intersects the y-axis when the input value (𝑥) is zero.

  • The output value is the y-value; the intercept is the value of the function at 𝑥 = 0, i.e., 𝑓(0).

• Concavity and points of inflection

  • Concave up: the graph curves upward; the second derivative is positive (where applicable) or the slope of the tangent is increasing.

  • Concave down: the graph curves downward; the second derivative is negative (where applicable) or the slope of the tangent is decreasing.

  • Straight lines have no concavity.

  • Point of inflection: a point where concavity changes (from up to down, or from down to up).

• Practice question prompts (referencing a graph of 𝑓 below)

  • a) When is the graph concave up?

  • b) When is the graph concave down?

  • c) Find the zero(s) of the function.

  • d) Find the y-intercept(s) of the function.

  • e) When is the graph increasing?

  • f) When is the graph decreasing?

1.1 Change in Tandem: identifying input vs output

• For each function, identify what the dependent and independent variables represent.
  1) 𝑏(𝑠) is a function where 𝑏 is the number of books in the library and 𝑠 is the number of students in the school.

  • Dependent variable: number of books in the library (𝑏)

  • Independent variable: number of students in the school (𝑠)
    2) 𝑝(𝑡) is a function where 𝑡 is the number of years since kindergarten and 𝑝 is the number of Pokemon cards.

  • Dependent variable: number of Pokemon cards (𝑝)

  • Independent variable: years since kindergarten (𝑡)
    3) 𝑐(𝑡) is a function where 𝑐 is the number of cups of coffee consumed and 𝑡 is the number of teachers at the school.

  • Dependent variable: cups of coffee (𝑐)

  • Independent variable: number of teachers (𝑡)
    4) Dep: Indep: (not filled in the transcript)
    5) Dep: Indep: (not filled in the transcript)

• Let the function 𝑓 be increasing or decreasing, but not both. State whether the function is increasing or decreasing on the given interval and justify.

  • Example data (decreasing):

  • 𝑥: 1, 2, 3, 4, 5

  • 𝑓(𝑥): 95, 90, 75, 50, 10

  • Observation: as 𝑥 increases, 𝑓(𝑥) decreases; the function is decreasing on the interval shown.

Practice data and problem types (representative summaries)

• Basic functions and related areas

  • Example topics listed: area of a circle, radius of a circle, volume of a box, side length of a box

  • Numerical samples accompany x and f(x) values for practice in identifying increasing/decreasing behavior and intercepts.

• Tables with x and f(x) values

  • Multiple problems present data tables (x-values with corresponding f(x) values) to determine monotonicity, zeros, and intercepts.

• Graph-based questions

  • Use a provided graph of 𝑓 to determine:

  • Intervals where the graph is concave up or concave down

  • Intervals where the graph is increasing or decreasing

  • Zeros of the function and the y-intercept

• Specific numeric examples (from the transcript)

  • A set of items labeled 8–13 and 12–13 indicate tasks such as identifying concavity intervals, increasing/decreasing intervals, zeros, and intercepts from given data or graphs.

  • Some items show pairs of x-values with f(x) values that illustrate monotonic behavior and intercepts; others present abstract prompts asking for the interval notation of concavity or monotonicity.

Interpreting graphs and intervals (AP Precalculus Practice)

• Questions 12–13 (and similar): use a graph of 𝑓 to answer:

  • a. On what interval(s) is the graph concave up?

  • b. On what interval(s) is the graph concave down?

  • c. On what interval(s) is the graph increasing?

  • d. On what interval(s) is the graph decreasing?

  • e. Find the zeros of the function.

  • f. Find the y-intercept of the function.

• Additional data-driven prompts show multiple instances of the same type of questions, reinforcing the need to read off intervals for concavity and monotonicity from a graph.

Graph analysis with a provided f-graph (questions 17–18)

• Question 17: The figure shows the graph of a function 𝑓 with x-coordinates labeled a, b, c, d, e at points on the graph. Which point is a point of inflection?

  • Options: (A) point a, (B) point b, (C) point c, (D) point d, (E) point e

• Question 18: On which interval is 𝑓 decreasing and the graph of 𝑓 concave down?

  • Options: (A) a to b, (B) b to c, (C) c to d, (D) d to e, (E) None of the above

• These questions require identifying inflection points and the combination of decreasing behavior with concavity on specific intervals from the graph.

Quick reference: key definitions and notational reminders

• Function f: D → R with domain D (input values) and range R (output values)

• Increasing on an interval I: ∀ a,b ∈ I with a < b, f(a) < f(b)

• Decreasing on an interval I: ∀ a,b ∈ I with a < b, f(a) > f(b)

• Zeros (x-intercepts): solutions to f(x) = 0

• y-intercept: f(0) (value of the function when x = 0)

• Concave up: graph bends upward (second derivative > 0 where applicable)

• Concave down: graph bends downward (second derivative < 0 where applicable)

• Point of inflection: where concavity changes

(Note: The transcript contains several incomplete items (blanks such as 4., 5., and some data entries). The notes above fill in the standard interpretations where the context is clear and indicate where information is missing. When studying, focus on the definitions, how to identify input vs. output, and how to determine increasing/decreasing and concavity from graphs or data.)