Precalculus Study Guide: Functions, Polynomials, Rational Functions, and Graph Analysis

Functions and Their Properties

Limits and Function Values

Understanding limits and function values is essential for analyzing the behavior of functions, especially at points of discontinuity or endpoints.

• Limit of a function as x approaches a value describes the behavior of the function near that point, not necessarily the value at the point.

• Function value at a point is the actual output of the function for that input.

• Discontinuities occur when the limit does not equal the function value or the function is not defined at that point.

Example:

• For a piecewise graph, $\\lim_{x \to 2} f(x)$ may differ from $f(2)$ if there is a jump or hole at $x=2$.

Polynomial Functions

Types of Polynomial Functions

Polynomials are classified based on their degree and structure.

• Linear Polynomial: Degree 1, e.g., $f(x) = ax + b$

• Non-linear Polynomial: Degree 2 or higher, e.g., $f(x) = ax^2 + bx + c$

• Rational Function: Ratio of two polynomials, e.g., $f(x) = \frac{p(x)}{q(x)}$

Zeros of Polynomial Functions

Zeros (roots) are values of $x$ where $f(x) = 0$.

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

• For $f(x) = x^3 - x^2 - 2x + 12$, factor or use synthetic division to find $x$-intercepts.

Example:

• $f(x) = x^3 - x^2 - 2x + 12$ has zeros at $x = -2, x = 2, x = 3$.

Vertex of a Quadratic Function

The vertex of a parabola defined by $f(x) = ax^2 + bx + c$ is the point where the function reaches its maximum or minimum.

• Vertex formula: $x = -\frac{b}{2a}$

• Substitute $x$ into $f(x)$ to find the $y$-coordinate.

Example:

• For $f(x) = x^2 - 8x + 8$, vertex at $(4, -8)$.

End Behavior of Polynomials

The end behavior describes how the function behaves as $x \to \infty$ or $x \to -\infty$.

• Determined by the leading coefficient and degree.

• Leading Coefficient Test:

  • If degree is even and leading coefficient is positive, $f(x) \to \infty$ as $x \to \pm\infty$.

  • If degree is odd and leading coefficient is positive, $f(x) \to \infty$ as $x \to \infty$, $f(x) \to -\infty$ as $x \to -\infty$.

Example:

• $f(x) = -3x^3 - 5x^2 - 2x + 4$ falls to the left and rises to the right.

Concavity and Inflection Points

Concave Upward and Downward

Concavity describes the direction of curvature of a function's graph.

• Concave upward: Graph opens upwards, $f''(x) > 0$.

• Concave downward: Graph opens downwards, $f''(x) < 0$.

Inflection Points

Inflection points are where the graph changes concavity.

• Occurs where $f''(x) = 0$ and the sign of $f''(x)$ changes.

Example:

• From a graph, inflection points can be estimated visually where the curvature changes.

Rational Functions

Vertical and Horizontal Asymptotes

Asymptotes are lines that the graph approaches but never touches.

• Vertical asymptote: Occurs at $x = a$ where the denominator is zero and the numerator is not zero.

• Horizontal asymptote: Determined by the degrees of numerator and denominator.

  • If degrees are equal, $y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$.

  • If degree of numerator < denominator, $y = 0$.

  • If degree of numerator > denominator, no horizontal asymptote (may be slant/oblique).

Example:

• $f(x) = \frac{-3x + 5}{5x - 7}$ has horizontal asymptote $y = -\frac{3}{5}$.

Oblique (Slant) Asymptotes

Occurs when the degree of the numerator is exactly one more than the degree of the denominator.

• Found by polynomial long division.

Example:

• $f(x) = \frac{x^2 + 4x + 2}{x + 2}$ has a slant asymptote.

Finding Asymptotes and Holes

• Vertical asymptotes: Set denominator equal to zero and solve for $x$.

• Holes: Occur when a factor cancels in numerator and denominator.

Example:

• $f(x) = \frac{x^2 - 16}{x^2 - 16}$ has a hole at $x = 4$ and $x = -4$.

Graph Analysis

Multiplicity of Zeros

Multiplicity refers to the number of times a zero occurs.

• If a zero has even multiplicity, the graph touches and turns around at the $x$-axis.

• If a zero has odd multiplicity, the graph crosses the $x$-axis.

Critical Points in Inequalities

Critical points are values where the function changes sign, important for solving inequalities.

• Set each factor equal to zero to find critical points.

Example:

• For $(x - 1)(x - 3) < 0$, critical points are $x = 1$ and $x = 3$.

Optimization Problems

Maximizing Area with Constraints

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

• Express the quantity to be maximized/minimized as a function of one variable.

• Use calculus or algebraic methods to find the maximum/minimum.

Example:

• Maximize area of a rectangle with fixed perimeter using $A = lw$ and constraint $2l + w = P$.

Polynomial Long Division

Dividing Polynomials

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

• Divide the highest degree term of the numerator by the highest degree term of the denominator.

• Multiply, subtract, and repeat until the remainder is of lower degree than the divisor.

Example:

• Divide $9x^3 - 18x^2 + 17x - 3$ by $3x + 4$.

Summary Table: Asymptotes and Holes in Rational Functions

Function	Vertical Asymptote	Horizontal Asymptote	Hole	Oblique Asymptote
$\frac{-3x + 5}{5x - 7}$	$x = \frac{7}{5}$	$y = -\frac{3}{5}$	None	None
$\frac{x^2 + 4x + 2}{x + 2}$	$x = -2$	None	None	Slant
$\frac{x^2 - 16}{x^2 - 16}$	None	None	$x = 4, x = -4$	None

Additional info:

• Some questions involve graphical analysis, requiring students to interpret graphs for limits, function values, and asymptotes.

• Optimization and critical point problems are foundational for calculus and further mathematical study.