Precalculus

Zeros:

Zeros are the x-values that make the function equal to 0 (numerator = 0, denominator ≠ 0).

Example: f(x) = (x-2)(x+3)/(x-4) → zeros at x = 2, -3

Domain of f(x):

All x-values that make the function valid (values that do not make the denominator = 0).

Example: f(x) = (x-3)/(x^2 - 9) → denominator = 0 when x = 3 or x = -3 → domain = all real numbers except x = 3, -3

y-intercept:

The y-value when x = 0 (plug in 0 for x).

Example: f(x) = (x-3)/(x+4) → f(0) = (0-3)/(0+4) = -3/4 → y-intercept = -3/4

x-coordinate of hole:

A hole happens when a factor cancels out from both numerator and denominator. The x-coordinate is the value that makes that factor = 0.

Example: f(x) = (x-2)(x-5)/(x-2)(x+5) → hole at x = 2

y-coordinate of hole:

Plug the x-coordinate of the hole into the simplified function (after canceling the common factor).

Example: f(x) = (x-2)(x-5)/(x-2)(x+5) → simplified f(x) = (x-5)/(x+5) → plug in x = 2 → f(2) = (2-5)/(2+5) = -3/7 → hole at (2, -3/7)

Vertical Asymptotes:

x-values that make the denominator = 0 and do not cancel out (not holes).

Example: f(x) = (x-2)/((x-3)(x+4)) → vertical asymptotes at x = 3, -4

End Behavior (Horizontal Asymptotes):

Shows what happens as x → ∞ or x → -∞. Compare degrees of numerator and denominator:

If top degree < bottom degree → y = 0

If top degree = bottom degree → y = ratio of leading coefficients

If top degree > bottom degree → no horizontal asymptote (may be slant)

Example: f(x) = (2x^2 + 1)/(x^2 + 5) → same degree → y = 2/1 = 2 → horizontal asymptote y = 2