Functions and Their Properties

Concavity and Inflection Points

Understanding the concavity of a function and identifying inflection points are essential for analyzing the shape and behavior of graphs.

• Concave Upward: A function is concave upward on an interval if its graph lies above its tangent lines on that interval. The second derivative is positive: .

• Concave Downward: A function is concave downward on an interval if its graph lies below its tangent lines on that interval. The second derivative is negative: .

• Inflection Point: A point where the function changes concavity (from up to down or vice versa). At an inflection point, and the sign of changes.

• Example: For a given graph, determine intervals of concavity and locate inflection points by observing where the curve changes from 'cup' to 'cap' shape or vice versa.

Limits and Continuity

Evaluating Limits from Graphs

Limits describe the behavior of a function as the input approaches a particular value.

• Left-Hand Limit: is the value as approaches from the left.

• Right-Hand Limit: is the value as approaches from the right.

• Existence of Limit: The limit exists at if both left and right limits are equal.

• Function Value: is the actual value of the function at (may differ from the limit if there is a hole or jump).

• Example: For a piecewise or discontinuous graph, compare the left and right limits and the function value at the point of interest.

Types of Functions

Polynomial and Rational Functions

Classifying functions helps in predicting their behavior and properties.

• Linear Polynomial: A polynomial of degree 1, e.g., .

• Non-linear Polynomial: A polynomial of degree 2 or higher, e.g., .

• Rational Function: A function of the form , where and are polynomials and .

• Example: is a rational function.

Zeros and Multiplicity

Finding Zeros and Their Behavior

The zeros of a function are the values of for which . The multiplicity of a zero affects how the graph behaves at that point.

• Zero of Multiplicity 1: The graph crosses the x-axis at this zero.

• Zero of Even Multiplicity: The graph touches the x-axis and turns around at this zero.

• Zero of Odd Multiplicity (greater than 1): The graph crosses the x-axis but flattens out at the zero.

• Example: For , is a zero of multiplicity 2 (touches and turns), is a zero of multiplicity 1 (crosses).

Quadratic Functions

Vertex and Maximum/Minimum Values

The vertex of a parabola defined by a quadratic function is the point where the function reaches its maximum or minimum value.

• Vertex Formula: ,

• Maximum/Minimum: If , the parabola opens upward and the vertex is a minimum. If , it opens downward and the vertex is a maximum.

• Example: For , the vertex is at , .

End Behavior of Polynomials

Leading Coefficient Test

The end behavior of a polynomial function is determined by its leading term .

• Even Degree, Positive Leading Coefficient: Rises to the left and right.

• Even Degree, Negative Leading Coefficient: Falls to the left and right.

• Odd Degree, Positive Leading Coefficient: Falls to the left, rises to the right.

• Odd Degree, Negative Leading Coefficient: Rises to the left, falls to the right.

• Example: (odd degree, negative leading coefficient): rises left, falls right.

Inequalities and Critical Points

Solving Polynomial Inequalities

Critical points are values where the expression equals zero or is undefined. They divide the number line into intervals for testing the inequality.

• Steps:

  1. Set the inequality to zero.

  2. Find the zeros (critical points).

  3. Test intervals between critical points.

• Example: Solve ; critical points are and .

Rational Functions: Asymptotes and Behavior

Vertical and Horizontal Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches.

• Vertical Asymptote: Occurs where the denominator is zero and the numerator is nonzero. Set .

• Horizontal Asymptote: Determined by comparing degrees of numerator and denominator:

  • If degree numerator < degree denominator:

  • If degrees are equal: (ratio of leading coefficients)

  • If degree numerator > degree denominator: No horizontal asymptote (may have slant/oblique asymptote)

• Slant (Oblique) Asymptote: Occurs when the degree of the numerator is exactly one more than the denominator. Found by polynomial long division.

• Example: has a slant asymptote.

Table: Types of Asymptotes in Rational Functions

Finding Zeros of Polynomials

Solving Polynomial Equations

To find the zeros, set the polynomial equal to zero and solve for .

• Factoring: Factor the polynomial and set each factor equal to zero.

• Quadratic Formula: For degree 2:

• Example: factors to , so zeros are .

Applications: Optimization

Maximizing Area with Constraints

Optimization problems involve finding the maximum or minimum value of a function given certain constraints.

• Example: A developer has 285 feet of fencing to enclose a rectangular grassy lot along a street (one side not fenced). Let be the width and the length parallel to the street. The constraint is . The area is maximized by expressing in terms of and finding the vertex of the resulting quadratic.

Polynomial Long Division

Dividing Polynomials

Long division is used to divide polynomials, especially when finding slant asymptotes or simplifying rational expressions.

• Steps:

  1. Divide the leading term of the numerator by the leading term of the denominator.

  2. Multiply the entire divisor by this result and subtract from the numerator.

  3. Repeat with the new polynomial until the degree of the remainder is less than the divisor.

• Example: Divide by .

Summary Table: Key Concepts