Piecewise Functions

Graphing Piecewise Functions

Piecewise functions are defined by different expressions for different intervals of the domain. Understanding how to graph them is essential for visualizing their behavior.

• Definition: A piecewise function is a function composed of two or more functions, each defined on a specific interval.

• Graphing Steps:

  1. Identify the intervals and corresponding expressions.

  2. Plot each expression on its interval.

  3. Use open or closed circles to indicate whether endpoints are included.

• Example:

Evaluating Piecewise Functions

To evaluate a piecewise function, determine which interval the input belongs to and use the corresponding expression.

• Key Point: Always check the domain restrictions before substituting the value.

• Example: For above, , .

Polynomial Functions

Zeros, Factors, and Multiplicity

Polynomial functions are central to precalculus, with their zeros, factors, and multiplicity determining their graph's shape and behavior.

• Zero: A value where .

• Factor: If is a zero, then is a factor.

• Multiplicity: The number of times a zero is repeated. If is a factor, is a zero of multiplicity .

• Touch or Cross: If multiplicity is odd, the graph crosses the x-axis at the zero; if even, it touches and turns around.

• Example: : Zero at (multiplicity 2, touches), (multiplicity 1, crosses).

End Behavior

The end behavior of a polynomial function depends on its degree and leading coefficient.

• Degree: The highest power of .

• Leading Coefficient: The coefficient of the term with the highest degree.

• End Behavior Rules:

  • Even degree, positive leading coefficient: both ends up.

  • Even degree, negative leading coefficient: both ends down.

  • Odd degree, positive leading coefficient: left down, right up.

  • Odd degree, negative leading coefficient: left up, right down.

Synthetic and Long Division

These methods are used to divide polynomials and find zeros or factors.

• Synthetic Division: A shortcut for dividing by linear factors .

• Long Division: Used for dividing by higher-degree polynomials.

• Example: Divide by using synthetic division.

Rational Zeros Theorem

This theorem helps identify possible rational zeros of a polynomial.

• Theorem: If has integer coefficients, any rational zero is of the form , where divides the constant term and divides the leading coefficient.

• Example: For , possible zeros are .

Descartes’ Rule of Signs

Descartes’ Rule of Signs estimates the number of positive and negative real zeros.

• Rule: The number of positive real zeros equals the number of sign changes in or less by an even integer.

• Negative Zeros: Apply the rule to .

Intermediate Value Theorem

This theorem guarantees the existence of a zero in an interval where the function changes sign.

• Theorem: If and have opposite signs, there is at least one zero between and .

Turning Points (Min/Max)

Turning points are where the graph changes direction, corresponding to local minima or maxima.

• Maximum Number: A polynomial of degree has at most turning points.

Sketching the Graph by Hand

To sketch a polynomial graph, use zeros, multiplicity, end behavior, and turning points.

• Steps:

  1. Find zeros and their multiplicities.

  2. Determine end behavior.

  3. Plot turning points.

  4. Connect with smooth curves.

Finding the Function from Given Zeros

Construct a polynomial by multiplying factors corresponding to the given zeros.

• Example: Zeros at (multiplicity 2): .

Finding Remaining Zeros

Use division or factoring to find all zeros of a polynomial.

• Example: If is a zero, divide by to find remaining zeros.

Imaginary Numbers

Polynomials may have complex zeros. Imaginary numbers are used when the discriminant is negative.

• Definition:

• Example:

Rational Functions

Domain

The domain of a rational function excludes values that make the denominator zero.

• Definition: A rational function is where .

• Example: , domain: .

Asymptotes

Rational functions may have vertical, horizontal, or oblique (slant) asymptotes.

• Vertical Asymptotes: Occur where and .

• Horizontal Asymptotes: Determined by degrees of and .

  • If degree :

  • If degree :

  • If degree : No horizontal asymptote; may have slant asymptote.

• Oblique (Slant) Asymptotes: Occur when degree ; found by division.

Holes

Holes occur where both numerator and denominator have a common factor.

• Example: has a hole at .

Graphing Rational Functions by Hand

To graph rational functions, identify domain, intercepts, asymptotes, and holes.

• Steps:

  1. Find domain.

  2. Locate vertical/horizontal/slant asymptotes.

  3. Find holes.

  4. Plot intercepts.

  5. Sketch behavior near asymptotes.

Miscellaneous Topics

Intercepts

Intercepts are points where the graph crosses the axes.

• x-intercept: Set and solve for .

• y-intercept: Set and solve for .

Quadratic Formula

The quadratic formula solves .

• Formula:

• Discriminant: determines the nature of the roots.

Factoring

Factoring is used to solve equations and simplify expressions.

• Methods: Common factor, difference of squares, trinomials, grouping.

• Example:

Calculator Usage

Graphing calculators can be used for graphing, evaluating, finding intersections, and min/max points.

• Allowed: TI-84 or lower; no cell phone calculators.

• Functions: Graph, evaluate, find intersection, min/max.

Additional info:

• Bring scrap paper, graph paper, pencil, and calculator.

• No scantron or formula sheet will be provided.

• Leaving during the test is not allowed.