Pre-Calculus Toolkit: Function Families, Quadratics, and Rational Functions

Universal Function Behaviors

Definition of a Function

A function is a relation in which every input (x-value) corresponds to exactly one output (y-value). This is a foundational concept in algebra and pre-calculus.

• Function Rule: For each input, there is only one output.

• Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.

Example: The graph of y = x2 passes the vertical line test and is a function. The graph of x = y2 does not pass and is not a function.

Solving Quadratic & Rational Families

Quadratic Functions

Quadratic functions are polynomial functions of degree 2, generally written as y = ax2 + bx + c.

• Vertex Formula: The vertex of a quadratic function is the highest or lowest point on its graph (a parabola).

• The x-coordinate of the vertex is given by: $x = \frac{-b}{2a}$

• To find the y-coordinate, substitute this x-value back into the original equation.

• Leading Coefficient (a): Determines the direction the parabola opens:

  • If a > 0, the parabola opens upward.

  • If a < 0, the parabola opens downward (reflection).

Example: For y = 2x2 - 4x + 1, the vertex is at $x = \frac{-(-4)}{2 \times 2} = 1$. Substitute x = 1 to find y.

Rational Functions

Rational functions are functions of the form $f(x) = \frac{P(x)}{Q(x)}$, where P(x) and Q(x) are polynomials and Q(x) ≠ 0.

• Asymptotes: Lines that the graph approaches but never touches.

  • Vertical Asymptotes: Occur where the denominator Q(x) = 0.

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

• Horizontal Asymptote Shortcuts:

  • If degree of numerator < degree of denominator: y = 0 is the horizontal asymptote.

  • If degrees are equal: y = (leading coefficient of numerator)/(leading coefficient of denominator).

  • If degree of numerator > degree of denominator: No horizontal asymptote (may be an oblique/slant asymptote).

Example: For $f(x) = \frac{2x^2 + 3}{x^2 - 1}$, both numerator and denominator have degree 2, so the horizontal asymptote is $y = \frac{2}{1} = 2$.

Transformation Logic

Transformations shift or reflect the graph of a function.

• Inside Changes (affect x): Horizontal shifts and stretches/compressions.

• Outside Changes (affect y): Vertical shifts and stretches/compressions.

• For example, in $f(x) = (x + 2)^2$, the graph shifts left by 2 units.

Summary Table: Horizontal Asymptote Rules

Degree of Numerator	Degree of Denominator	Horizontal Asymptote
< Denominator	Higher	y = 0
= Denominator	Equal	y = (Leading Coefficient Ratio)
> Denominator	Lower	None (Oblique/Slant Asymptote possible)

Additional info: The table and explanations are expanded for clarity and completeness based on standard college algebra curriculum.