Negation in Transformations and Squaring

Transformations with negation

  • -f(x): reflect across the x-axis; y-values are negated.
  • f(-x): reflect across the y-axis; x-values are negated.
  • In data tables: multiply y-values by -1 (or x-values by -1 for input reflection).
  • Mapping notation: negative in front of x corresponds to f(-x).
  • If you factor out a -1, remember the square applies to the entire factor; (-1)^2 = 1, so the sign may disappear when squaring.

Squaring and sign properties

  • When squaring an expression with a leading -1, the minus sign vanishes:
    ((−1) A)2=A2.((-1)\,A)^2 = A^2.
  • Therefore, (−f(x))2=f(x)2.(-f(x))^2 = f(x)^2.
  • If you factor -1 from inside a squared expression, the result is unchanged because the -1 is squared to 1:
    ((−1)⋅f(x))2=f(x)2.(( -1)\cdot f(x))^2 = f(x)^2.
  • Caution: f(-x) and f(x) are not generally equal; squaring may or may not make them equal depending on the function (even functions vs general case).