Precalculus Study Guide: Piecewise, Polynomial, and Rational Functions

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: $f(x) = \begin{cases} x^2 & x < 0 \\ 2x + 1 & x \geq 0 \end{cases}$

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 $f(x)$ above, $f(-2) = (-2)^2 = 4$, $f(1) = 2 \times 1 + 1 = 3$.

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 $x$ where $f(x) = 0$.

• Factor: If $x = a$ is a zero, then $(x - a)$ is a factor.

• Multiplicity: The number of times a zero is repeated. If $(x - a)^k$ is a factor, $a$ is a zero of multiplicity $k$.

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

• Example: $f(x) = (x - 2)^2(x + 1)$: Zero at $x = 2$ (multiplicity 2, touches), $x = -1$ (multiplicity 1, crosses).

End Behavior

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

• Degree: The highest power of $x$.

• 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 $(x - a)$.

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

• Example: Divide $x^3 - 6x^2 + 11x - 6$ by $x - 1$ using synthetic division.

Rational Zeros Theorem

This theorem helps identify possible rational zeros of a polynomial.

• Theorem: If $f(x)$ has integer coefficients, any rational zero is of the form $\frac{p}{q}$, where $p$ divides the constant term and $q$ divides the leading coefficient.

• Example: For $f(x) = 2x^3 + 3x^2 - 8x + 3$, possible zeros are $\pm1, \pm3, \pm\frac{1}{2}, \pm\frac{3}{2}$.

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 $f(x)$ or less by an even integer.

• Negative Zeros: Apply the rule to $f(-x)$.

Intermediate Value Theorem

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

• Theorem: If $f(a)$ and $f(b)$ have opposite signs, there is at least one zero between $a$ and $b$.

Turning Points (Min/Max)

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

• Maximum Number: A polynomial of degree $n$ has at most $n-1$ 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 $x = 1, x = -2$ (multiplicity 2): $f(x) = (x - 1)(x + 2)^2$.

Finding Remaining Zeros

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

• Example: If $x = 2$ is a zero, divide by $(x - 2)$ to find remaining zeros.

Imaginary Numbers

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

• Definition: $i = \sqrt{-1}$

• Example: $x^2 + 4 = 0 \Rightarrow x = \pm 2i$

Rational Functions

Domain

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

• Definition: A rational function is $f(x) = \frac{P(x)}{Q(x)}$ where $Q(x) \neq 0$.

• Example: $f(x) = \frac{1}{x - 3}$, domain: $x \neq 3$.

Asymptotes

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

• Vertical Asymptotes: Occur where $Q(x) = 0$ and $P(x) \neq 0$.

• Horizontal Asymptotes: Determined by degrees of $P(x)$ and $Q(x)$.

  • If degree $P < Q$: $y = 0$

  • If degree $P = Q$: $y = \frac{\text{leading coefficient of } P}{\text{leading coefficient of } Q}$

  • If degree $P > Q$: No horizontal asymptote; may have slant asymptote.

• Oblique (Slant) Asymptotes: Occur when degree $P = Q + 1$; found by division.

Holes

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

• Example: $f(x) = \frac{(x - 2)(x + 1)}{(x - 2)}$ has a hole at $x = 2$.

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 $y = 0$ and solve for $x$.

• y-intercept: Set $x = 0$ and solve for $y$.

Quadratic Formula

The quadratic formula solves $ax^2 + bx + c = 0$.

• Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Discriminant: $b^2 - 4ac$ 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: $x^2 - 9 = (x - 3)(x + 3)$

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.