Quadratic, Polynomial, and Rational Functions: Precalculus Study Guide

Quadratic Functions

Definition and General Form

Quadratic functions are polynomial functions of degree 2, typically written as:

• General Form: $f(x) = ax^2 + bx + c$, where $a \neq 0$

Key Features of Quadratic Functions

• Vertex: The point $(h, k)$ where the parabola changes direction. Found using $h = -\frac{b}{2a}$ and $k = f(h)$.

• Axis of Symmetry: The vertical line $x = h$.

• Y-intercept: The value $f(0) = c$.

• Direction of Opening: If $a > 0$, the parabola opens upward; if $a < 0$, it opens downward.

• Intervals of Increase/Decrease: For $a > 0$, decreasing on $(-\infty, h)$, increasing on $(h, \infty)$; for $a < 0$, increasing on $(-\infty, h)$, decreasing on $(h, \infty)$.

Example

• Given $f(x) = -3x^2 - 9x - 6$:

  • $a = -3$, $b = -9$, $c = -6$

  • Vertex: $h = -\frac{-9}{2 \times -3} = -\frac{9}{-6} = 1.5$; $k = f(1.5)$

  • Axis of symmetry: $x = 1.5$

  • Y-intercept: $-6$

  • Opens downward ($a < 0$)

Polynomial Functions

Definition and General Form

Polynomial functions are expressions of the form:

• $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$

• Degree: The highest power of $x$ with a nonzero coefficient.

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

Key Concepts

• End Behavior: Determined by the degree and leading coefficient.

• Zeros: Values of $x$ where $f(x) = 0$; may be real or complex.

• Multiplicity: The number of times a zero is repeated.

• Descartes' Rule of Signs: Used to predict the number of positive, negative, and complex roots by counting sign changes in $f(x)$ and $f(-x)$.

Example Table: End Behavior

Degree	Leading Coefficient	End Behavior
Even	Positive	Both ends up
Even	Negative	Both ends down
Odd	Positive	Left down, right up
Odd	Negative	Left up, right down

Example

• Given $f(x) = x^4 + 3x^3 - 5x^2 + 2x - 4$:

  • Degree: 4 (even)

  • Leading coefficient: 1 (positive)

  • End behavior: Both ends up

  • Use Descartes' Rule of Signs to analyze roots

Rational Functions

Definition and General Form

Rational functions are quotients of polynomials:

• $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$

Key Features

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

• Horizontal/Oblique Asymptotes: Determined by the degrees of $P(x)$ and $Q(x)$

• Intercepts: Y-intercept at $f(0)$; x-intercepts where $P(x) = 0$

• Intervals of Positivity/Negativity: Solve $f(x) > 0$ and $f(x) < 0$

Example

• Given $f(x) = \frac{x^2 + 4x + 4}{x + 2}$:

  • Vertical asymptote: $x = -2$

  • Horizontal asymptote: Degree numerator = 2, denominator = 1; oblique asymptote exists

  • Y-intercept: $f(0) = \frac{4}{2} = 2$

Graphing and Transformations

Parent Functions and Transformations

• Parent Functions: $f(x) = x^2$, $f(x) = x^3$, $f(x) = \sqrt{x}$, $f(x) = |x|$

• Transformations: Shifts, stretches, compressions, and reflections

Example

• Given $g(x) = -3(x - 1)^2 + 7$:

  • Shift right by 1, up by 7, reflect over x-axis, vertical stretch by 3

Summary Table: Key Properties of Functions

Function Type	General Form	Key Properties
Quadratic	$ax^2 + bx + c$	Vertex, axis of symmetry, direction, intervals
Polynomial	$a_nx^n + \ldots + a_0$	Degree, leading coefficient, end behavior, zeros
Rational	$\frac{P(x)}{Q(x)}$	Asymptotes, intercepts, intervals

Additional info: These notes expand on the provided questions by including definitions, formulas, and examples for each function type, as well as tables for classification and end behavior.