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AP Precalculus Video Notes: Functions, Rates of Change, Polynomials, and End Behavior (Flashcards)

1.1 Change in Tandem

  • Function: a mathematical relation that maps a set of inputs to a set of outputs with the property that each input value corresponds to exactly one output value.

    • Input values = Domain = Independent variable (x).

    • Output values = Range = Dependent variable (y).

  • Function notation and recognition:

    • Using different labels helps recognition (e.g., if we throw a football and measure height over time, use variables for height h(t) and time t).

    • Function notation immediately identifies inputs vs outputs (e.g., w(x) is a function where w = amount of water in a pool in gallons and x = length of the pool).

    • Four ways to express a function rule:

    • Verbal

    • Analytic (algebraic)

    • Numerical

    • Graphical

  • Examples:

    • w(x): water in a pool (gallons) as a function of pool length (x).

    • Height of a football as a function of time: h(t).

  • Increasing function (as input increases, output increases):

    • Verbal: as input values increase, outputs always increase.

    • Analytic: for all a < b in the interval, f(a) < f(b) (strict increase). Non-decreasing version uses ≤.

    • Graphical: secant slopes are positive over the interval.

  • Decreasing function (as input increases, output decreases):

    • Verbal: as input values increase, the outputs decrease.

    • Analytic: for all a < b in the interval, f(a) > f(b) (strict decrease). Non-increasing uses ≥.

    • Graphical: secant slopes are negative over the interval.

  • Basic graph terminology:

    • Zero (x-intercept): the graph intersects the x-axis where the output value is zero, i.e., f(x) = 0.

    • y-intercept: the graph intersects the y-axis where the input value is zero, i.e., at x = 0, y = f(0).

    • Concavity:

    • Concave Up: bowl facing up, f''(x) > 0; the graph curves upward.

    • Concave Down: bowl facing down, f''(x) < 0; the graph curves downward.

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

    • Straight lines have no concavity.

  • Example exercise prompts (referencing typical graphs):

    • a) When is the graph concave up? (intervals where f''(x) > 0)

    • b) When is the graph concave down? (intervals where f''(x) < 0)

    • c) Find the zero(s) of the function (solve f(x) = 0).

    • d) Find the y-intercept(s) (evaluate f(0)).

    • e) When is the graph increasing? (identify intervals where f'(x) > 0).

    • f) When is the graph decreasing? (identify intervals where f'(x) < 0).

  • Note: These ideas are illustrated in the Algebros from FlippedMath.com.

  • Practical takeaways:

    • A function relates inputs to outputs with a single output per input; this underpins rate-of-change and graph behavior.

1.2 Rates of Change

  • Average rate of change (ARC):

    • Definition: the change in output over the change in input across an interval.

    • Formula: ext{ARC} = rac{f(b)-f(a)}{b-a}

    • Interpretation: slope of the secant line joining the points frac{(a,f(a))}{(b,f(b))} on the graph.

  • Worked example structure (typical):

    • Given a function such as f(x) = -x^{2} + 3x - 1, compute ARC between two points, e.g., a = 1 and b = 3.

    • Practice with a data table: time vs. distance, to compute ARC on a chosen interval.

  • Example with time-distance data:

    • Time (seconds): 10, 13, 19, 25

    • Distance (meters): 80, 76, 43, 30

    • Find ARC on [13,25]:

    • f(13) = 76, f(25) = 30, so

    • ext{ARC}_{[13,25]} = rac{30 - 76}{25 - 13} = rac{-46}{12} = - rac{23}{6} \,( ext{m/s}) \, ext{(approximately } -3.83 ext{ m/s)}.

    • The word “per” signals a rate (e.g., meters per second, miles per gallon).

  • Rate of change at a point (derivative intuition):

    • Goal: approximate the instantaneous rate of change dy/dx at a point by using ARC over small intervals containing the point.

    • Example approach (conceptual): estimate slope of the tangent by computing ARC over [x0, x0 + h] and/or [x0 - h, x0].

    • Sample illustration (generic): for f(x) = x^{2} + 3x, estimate dy/dx at x = 1 using nearby points (e.g., x = 1 and x = 2) to approximate; as h → 0, the ARC approaches f'(1).

  • Positive vs. negative rates of change:

    • Positive rate of change: as the input increases, the output increases.

    • Negative rate of change: as the input increases, the output decreases.

    • Real-world intuition: population growth with time is generally positive; a negative rate indicates inverse behavior (e.g., as price increases, demand decreases).

1.3 Rates of Change in Linear and Quadratic Functions

  • Average rate of change (ARC) for linear functions:

    • A linear function has a constant ARC equal to its slope, independent of the interval length.

    • Examples:

    • y = x → ARC = 1 for any interval.

    • y = 2x + 3 → ARC = 2 for any interval.

    • y = -3 → ARC = 0 for any interval (horizontal line).

  • Practice: Find the ARC for several linear functions; observe the constancy.

  • Rate of change of the ARC for linear functions: constant and equal to the slope of the line.

  • Average rate of change for quadratic functions:

    • For f(x) = x^{2} - 2x, the ARC on [a,b] is:
      ext{ARC} = rac{f(b) - f(a)}{b - a} = rac{(b^{2} - 2b) - (a^{2} - 2a)}{b - a} = rac{(b^{2}-a^{2}) - 2(b-a)}{b-a} = rac{(b-a)(b+a-2)}{b-a} = b + a - 2.

    • If the interval has length h (i.e., b = a + h), then
      ext{ARC} = (a+h) + a - 2 = 2a + h - 2.
      As a increases by h, the ARC changes by 2h, so the rate of change of ARC with respect to a is 2. Thus, the rate of change of ARC for this quadratic, over equal-length subintervals, is constant and equals 2.

  • Worked quadratics example structure (from the notes):

    • For a quadratic like f(x) = -2x^{2} + 10, compute ARC over: [-2,-1], [-1,0], [0,1], etc., to observe how ARC changes with the interval and to illustrate the constant second difference idea embedded in quadratics.

  • Key takeaway:

    • For linear functions, ARC is constant (the slope).

    • For quadratic functions, the ARC over equal-length intervals changes linearly with the left endpoint, with a constant rate of change equal to the second difference (often 2 for the standard x^{2} form, after appropriate shifts).

  • Quick checks with simple quadratics and the idea of averaging across subintervals help build intuition for the derivative concept.

1.4 Polynomial Functions and Rates of Change

  • Polynomial definition:

    • A nonconstant polynomial has the standard form

    • p(x) = a{n}x^{n} + a{n-1}x^{n-1} + \, \cdots \, + a{1}x + a{0},

    • where n is a positive integer and each a_i is a real number.

  • Leading term, degree, and leading coefficient:

    • Leading term: the term with the highest exponent, e.g., in p(x) = 7x^{5} - 3x^{2} + 6x - 2, the leading term is 7x^{5}.

    • Degree: the highest exponent (here, 5).

    • Leading coefficient: the coefficient of the leading term (here, 7).

  • Nonstandard forms:

    • If a polynomial is not written in standard form, identify the largest exponent to find the degree and its corresponding coefficient for the leading term.

  • Extrema (local and global):

    • Local (relative) extrema occur where a function switches from increasing to decreasing (local max) or from decreasing to increasing (local min).

    • An included endpoint of a restricted-domain polynomial may also be a local extremum.

    • Global (absolute) extrema: the largest/smallest values over the entire domain; if none exist, report NONE.

    • A general note: If a polynomial has two zeros, there must be at least one extremum between them (intuition from intermediate values and derivative behavior; ties to Rolle’s theorem).

  • End behavior for even vs. odd degree polynomials:

    • Even degree:

    • Both ends go in the same direction.

    • If leading coefficient is positive, both ends go to +∞; if negative, both go to -∞.

    • There is at least one global extremum (absolute max or min) depending on the sign of the leading coefficient.

    • Odd degree:

    • Ends go in opposite directions (one up, one down).

    • Example patterns follow the sign of the leading coefficient:

      • If leading coefficient positive, as x → ∞, P(x) → ∞ and as x → −∞, P(x) → −∞.

      • If leading coefficient negative, as x → ∞, P(x) → −∞ and as x → −∞, P(x) → ∞.

  • 1.4 Practice and examples cover identifying degree, leading coefficient, extrema, end behavior, and how these relate to the shape of the graph.

1.5A Polynomial Functions and Complex Zeros

  • Zeros (real zeros) and x-intercepts:

    • If p(a) = 0, then a is a zero (root) of p, and the x-intercept is at (a, 0).

    • If a is real, then (x − a) is a real linear factor of p.

    • Factoring example:

    • For g(x) = x^{3} − 2x^{2} − 8x, factor: g(x) = x(x^{2} − 2x − 8) = x(x − 4)(x + 2).

    • Zeros at x = 0, 4, and −2.

  • Intervals where a quadratic is nonnegative (example):

    • For h(x) = x^{2} − 2x − 3, factor: h(x) = (x − 3)(x + 1).

    • Sign chart gives intervals where h(x) ≥ 0: (−∞, −1] ∪ [3, ∞).

  • Multiplicity and graph behavior:

    • If a linear factor (x − a) is repeated n times, the corresponding zero a has multiplicity n.

    • If a real zero has even multiplicity, the graph bounces off the x-axis at x = a.

  • Complex zeros and the Fundamental Theorem of Algebra:

    • A polynomial of degree n has exactly n complex zeros when counted with multiplicities.

    • Non-real zeros come in conjugate pairs: if a + bi is a non-real zero, then a − bi is also a zero.

    • For real polynomials, the number of real zeros can be between 0 and n, but non-real zeros occur in pairs.

  • Real vs. non-real zero counts (illustrative table concept):

    • Quadratic (degree 2): number of real zeros ∈ {0,1,2}; non-real zeros ∈ {0,2} depending on discriminant.

    • Cubic (degree 3): number of real zeros ∈ {1,3}; non-real zeros ∈ {0,2}.

    • Quartic (degree 4): number of real zeros ∈ {0,2,4}; non-real zeros ∈ {0,2,4} depending on polynomial.

  • Complex zeros and conjugates with examples:

    • If one non-real zero is 3 + 4i, then 3 − 4i is also a zero.

  • Zeros, complex zeros, and real factors bridge to factoring and graphing polynomials.

  • 1.5A also covers techniques for identifying zeros from graphs, intervals, and factoring, and introduces the concept of multiplicity and its geometric impact on the graph.

  • 1.5A: Zeros, Complex Zeros, and Degree:

    • A polynomial of degree n has exactly n complex zeros counting multiplicities.

    • If there are any non-real zeros, they occur in conjugate pairs.

    • The discussion includes quick reference to how to determine the number of real vs. non-real zeros for common degrees (2, 3, 4) using discriminants and sign patterns.

  • 1.5A: Degree by Differences (a method for data):

    • If you have a table of input-output values for a polynomial, you can determine the degree by looking at differences.

    • Compute successive differences: if the nth differences are constant (for equal-spaced inputs), the degree is n.

    • Example outline:

    • Input: 0,1,2,3,4,5,6,7

    • Output: 35, 49, 38, 5, -62, -175, -346, -587

    • The first, second, or higher-order differences become constant at the degree.

  • 1.5B Even and Odd Polynomials (symmetry and tests)

    • Even function: symmetric about the y-axis; f(x) = f(-x).

    • Odd function: symmetric about the origin; f(-x) = -f(x).

    • How to test analytically:

    • Replace x with −x and compare to f(x).

    • Quick shortcuts:

    • If a polynomial contains only even powers (x^{2}, x^{4}, …), it is even.

    • If it contains only odd powers (x^{1}, x^{3}, …), it is odd.

    • Examples from notes:

    • f(x) = x^{6} − 4x^{2} is even.

    • f(x) = -2x^{3} + 5x is odd.

    • f(x) = 6x^{4} − 2x is neither even nor odd.

  • 1.5B Quick check exercises verify whether given polynomials are even, odd, or neither.

  • 1.6 Polynomial Functions and End Behavior

    • End behavior describes what the function does as x → ±∞, summarized by the leading term (the term with the largest exponent).

    • Concept: for large |x|, the leading term dominates all others, so the end behavior is determined by the leading term.

    • End behavior rules (based on leading term and degree):

    • Let p(x) have leading term an x^n with an ≠ 0.

    • If n is even and a_n > 0, then as x → ±∞, p(x) → ∞ (both ends up).

    • If n is even and a_n < 0, then as x → ±∞, p(x) → −∞ (both ends down).

    • If n is odd and a_n > 0, then as x → −∞, p(x) → −∞ and as x → ∞, p(x) → ∞ (ends go in opposite directions).

    • If n is odd and a_n < 0, then as x → −∞, p(x) → ∞ and as x → ∞, p(x) → −∞ (ends go in opposite directions).

    • Notational practice: describe end behavior with limits, e.g.,

    • ext{Left end: }\lim_{x o - ofty} p(x) ext{ is } - ext{∞ or ∞ depending on leading term.}

    • ext{Right end: }\lim_{x o o ty} p(x) ext{ is } - ext{∞ or ∞ depending on leading term.}

    • Example prompts in notes: given a polynomial, determine end behavior by inspecting the leading term and applying the rule above.

    • Key takeaway: end behavior is dictated by the leading term, which dominates behavior for large |x|.

  • Overall connections and practical implications:

    • Understanding function behavior (increasing/decreasing, concavity, end behavior) helps predict long-run trends without computing every point.

    • Polynomial functions serve as a flexible model for many real-world relationships; their zeros, extrema, and end behavior provide essential qualitative understanding for approximation and graphing.

    • Complex zeros and multiplicities influence the shape and factorization of polynomials, informing both graph behavior and algebraic factorings.

  • Notation recap (LaTeX):

    • Average Rate of Change: ext{ARC} = rac{f(b)-f(a)}{b-a}

    • End behavior cases (examples):

    • If degree n is even and leading coefficient an > 0: \lim{x o -0} p(x) = , \lim_{x o 0} p(x) = .

    • If degree n is odd and leading coefficient an > 0: \lim{x o -0} p(x) = -0, \lim_{x o 0} p(x) = 0.

    • Zeros and factors: if p(a) = 0 with a real, then (x − a) is a factor; multiplicity n implies a repeated root of order n; even multiplicity yields a bounce on the x-axis.

  • Connections to prior material:

    • 1.1 establishes the language of inputs/outputs and function behavior; 1.2 and 1.3 develop rates of change and how slopes behave for linear vs. quadratic functions; 1.4/1.5 expand toward polynomials, zeros, and factoring; 1.6 unifies the view with end behavior and leading terms. Together, these sections build a framework for analyzing qualitative and quantitative aspects of polynomial and non-polynomial functions, essential for AP Precalculus and subsequent calculus topics.

  • Examples for quick practice (to reinforce the notes):

    • Determine if f is increasing on an interval by testing f'(x) > 0 across that interval (conceptual from 1.1).

    • For f(x) = x^{2} - 4x, find ARC on [1,3]:

    • f(1) = 1 - 4 = -3; f(3) = 9 - 12 = -3; ARC =

    • rac{-3 - (-3)}{3 - 1} = 0.

    • This reflects a quadratic with a turning point between 1 and 3; the graph changes slope there.

  • Tip for exam prep:

    • Be comfortable with translating between representations (verbal, analytic, numeric, graphical).

    • Practice finding and interpreting zeros, multiplicities, and end behavior from a polynomial’s leading term.

    • Remember the key formulas, especially ARC and the end-behavior rules tied to degree and leading coefficient.