AP Precalculus Notes

Unit 1: Polynomial and Rational Functions

Average Rate of Change (AROC)

• On an interval [a, b], given points (a, f(a)) and (b, f(b)).

• The AROC represents the slope of the secant line, which provides average information about the function's behavior between these points.

• AROC = \frac{f(b) - f(a)}{b - a}

• Secant Line: A line formed between two points a and b, using the Average Rate of Change formula, illustrating how the function values change across the interval.

Relating Function Behavior, Concavity, and Rate of Change

• Increasing: Positive rate of change indicating that as x increases, f(x) also increases.

• Decreasing: Negative rate of change indicating that as x increases, f(x) decreases.

• Concave Up: The rate of change of the function is increasing, like an upward-opening parabola, indicating an acceleration in the ascent.

• Concave Down: The rate of change of the function is decreasing, like a downward-opening parabola, indicating a deceleration in the ascent or an acceleration in the descent.

• Point of Inflection: A point where the function changes concavity, meaning the rate of change shifts from increasing to decreasing or vice versa.

Graph of f(x)

• Increasing: Represented as a positive slope (+)

• Decreasing: Represented as a negative slope (-)

• Concave Up: Illustrated as \cup indicating an upward curve.

• Concave Down: Illustrated as \cap indicating a downward curve.

Rate of Change (ROC) of f(x)

• This reflects the instantaneous rate of change at a specific point, determined by the derivative of the function.

• Increasing: Indicates that the function itself is rising at that point (+)

• Decreasing: Indicates the function is falling at that point (-)

Function Behavior and Rate of Change

• If f(x) decreases while the rate of change is increasing, the curve approaches a point of inflection, resulting in a concave up curve.

• If f(x) increases while the rate of change is increasing, the curve continues to rise concavely up.

• If f(x) decreases while the rate of change is decreasing, the curve bends concavely down, indicating a push towards the x-axis.

• If f(x) increases while the rate of change is decreasing, the curve rises concavely down towards the x-axis.

Odd/Even Functions

• Odd Functions: Defined by the property f(-x) = -f(x), indicating symmetry through the origin and that the output for a negative input matches the negative output for the positive input.

• Even Functions: Characterized by the property f(-x) = f(x), indicating symmetry about the y-axis, with mirrored outputs for positive and negative inputs.

Important Justification Phrase

• Over equal-length input-value intervals, the behavior of the function can be determined by examining rates of change and concavity.

Zeros, Multiplicity, and Factors

• Zeros of the function indicate where it intersects the x-axis, providing vital information about the function's behavior.

• Multiplicity affects the behavior at the zeros:

  • (x - a)^1: Crosses the x-axis (linearly).

  • (x - a)^2: Bounces off of the x-axis (quadratically).

  • (x - a)^3: Bends at the x-axis (cubically) without crossing it.

Intercepts

• x-intercept(s): Found by setting y = 0 in the function.

• y-intercept: Determined by setting x = 0, giving the output value where the function meets the y-axis.

Complex Conjugate

• For polynomial functions, if (a + bi) is found as a root or factor, (a - bi) must also be a factor due to the Fundamental Theorem of Algebra.

End Behavior

Limit Notation

• Expressing the behavior of the function as x approaches a value can be written as \lim_{x \to c} f(x) = L, defining how the function behaves near the given point.

Degree

• Odd Degree: Functions of odd degree display end-behavior where the left and right ends contrast, typically extending to opposite infinities.

• Even Degree: Functions of even degree share the same end-behavior in both directions, either rising or falling.

Leading Coefficients

• Positive Leading Coefficient: Ensures that the function's end behavior rises on the right side of the graph.

• Negative Leading Coefficient: Indicates that the function's end behavior falls on the right side of the graph.

Transformations of Polynomial Functions

• The general transformation formula is given as F(x) = a \cdot f[b(x - h)] + k, where transformations include:

  • a-value: Vertical dilation by a factor of |a|; reflects across the x-axis if a < 0.

  • b-value: Horizontal dilation by a factor of \frac{1}{b}, impacting the width of the graph.

  • h-value: Represents horizontal translation in the direction of -h; shifts the graph left or right.

  • k-value: Represents vertical translation; moves the graph up +k or down -k.

Analysis of Table Problems

• To analyze polynomial behavior from a set of values:

  • Linear Function: If both input and output values change consistently with a constant first difference.

  • Quadratic Function: If inputs change consistently and the second differences of outputs are constant.

  • Cubic Function: If the inputs change consistently and the third differences of output values are constant.

  • Exponential Function: If input changes consistently while output values exhibit proportional changes.

  • Logarithmic Function: If input values change proportionally while the resulting output changes consistently.

Asymptotes, Holes, and Discontinuities

General Form of Rational Functions

• f(x) = \frac{ax^n + …}{bx^m + …}, providing a foundation for understanding how rational functions behave.

Vertical Asymptotes

• Occur when a factor in the denominator fails to cancel out with the numerator, represented by x = c.

• Specifically, \lim_{x \to c} R(x) = \pm \infty suggests points where the function will break, demonstrating vertical lines on the graph.

Horizontal Asymptotes

• Characterize the behavior at extremes:

  • Case I: If n < m, the horizontal asymptote is y = 0, indicating the function approaches this line as x goes to infinity.

  • Case II: If n = m, the horizontal asymptote is defined as y = \frac{a}{b}.

  • Case III: If n > m, there is no horizontal asymptote, meaning the function becomes infinitely large as x increases or decreases.

Slant/Oblique Asymptotes

• For cases where n > m and the numerator degree exceeds the denominator by one, long division is necessary to deduce the equation of the oblique asymptote, ignoring the remainder.

Holes

• Occur in the graph when a common factor cancels after simplifying equations; the location of the hole can be found by evaluating the simplified expression at the corresponding zero of that factor.

Pascal’s Triangle

Binomial Expansion

• Each row corresponds to increasing powers of a binomial, with specific coefficients that can be represented by the triangle:

  • Row 0: (x + y)^0 = 1

  • Row 1: (x + y)^1 = x + y

  • Row 2: (x + y)^2 = x^2 + 2xy + y^2

  • Row 3: (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3

• The pattern of coefficients follows: 1; 1 1; 1 2 1; 1 3 3 1; 1 4 6 4 1, reaching beyond for higher powers.

Parent Functions

• Defined shapes of fundamental shapes:

  • y = x, a straight line through the origin.

  • y = x^2, a parabola opening upwards.

  • y = x^3, a cubic function demonstrating both increase and bend.

  • y = \frac{1}{x}, a hyperbola which approaches asymptotes.

A polynomial function can be expressed in the standard form as:

f(x) = an x^n + a{n-1}x^{n-1} + … + a1 x + a0

where:

• an, a{n-1}, …, a1, a0 are coefficients, with a_n \neq 0.

• n is a non-negative integer representing the degree of the polynomial, which indicates the highest power of x in the function.

Characteristics of polynomial functions include:

• They are continuous and smooth, having no breaks or sharp turns.

• The end behavior of polynomial functions is determined by the degree and leading coefficient:

  • Even Degree: Both ends go in the same direction (either both up or both down).

  • Odd Degree: Ends go in opposite directions (one end up, the other down).

• Polynomial functions can have multiple roots or zeros, and their behavior at these points can be influenced by the multiplicity of the roots.