Pre-Calculus Chapter 2 Flashcards

Standard Form of a Quadratic Function

• Expressed as f(x) = a(x-h)² + k where a ≠ 0.

Completing the Square

• Method to transform a quadratic into vertex form, involving:
  • Calculating (b/2)² to find a new constant c.
  • Adding this new c, simplifying into a squared expression, then subtracting the new c.

Minimum Value for Quadratics

• Occurs at x = -b/(2a) when a > 0.

Maximum Value for Quadratics

• Occurs at x = -b/(2a) when a < 0.

Continuity of Polynomial Functions

• Graph of polynomial functions is continuous across its entire domain.

Leading Coefficient Test (End Behavior)

• Odd Degree, Positive Leading Coefficient: Left down, Right up.
• Odd Degree, Negative Leading Coefficient: Left up, Right down.
• Even Degree, Positive Leading Coefficient: Left up, Right up.
• Even Degree, Negative Leading Coefficient: Left down, Right down.

Maximum Number of Real Zeros in a Function

• Determined by the function's degree.

Maximum Number of Relative Extrema

• Given by the degree minus one of the polynomial.

Repeated Zeros

• Expressed as (x-a)ᶦ, where k > 1.
  • If k is odd, the graph crosses the x-axis at x = a.
  • If k is even, it touches but does not cross the x-axis at x = a.

Polynomial Long Division

• Involves placeholder zeros, matching leading coefficients, multiplying, subtracting, and repeating with subsequent terms.

Synthetic Division

• Applicable only for divisors of the form x-k.
• Requires switching the sign of k, using coefficients, and sequentially multiplying and adding.

Remainder Theorem for Polynomial Division

• The remainder when dividing by x-k is r = f(k).

Rational Zero Test

• Identifies potential rational zeros as p/q, where p is factors of the constant and q is factors of the leading coefficient.

Possible Rational Zeros (PRZ)

• All combinations of factors of the constant term over factors of the leading coefficient.

Complex Numbers

• Form a + bi, combining real numbers (both rational and irrational) and imaginary components.

Addition and Subtraction of Complex Numbers

• Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
• Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i

Linear Factors of a Polynomial

• A polynomial of degree n>0 has exactly n linear factors.

Complex Conjugates

• If a polynomial has a factor of a + bi, it necessarily includes a zero of a - bi.

Parent Function of Rational Functions

• Represented by f(x) = 1/x.

Finding Vertical Asymptotes

• Determined by discovering the zeros of the denominator.

Finding Horizontal Asymptotes

• Compare degrees:
  • If degree of numerator (n) < degree of denominator (m), asymptote at y = 0.
  • If n = m, asymptote found at y = ratio of leading coefficients.
  • If n > m, no horizontal asymptote.

Finding X-Intercepts

• Derived by finding zeros of the numerator.

Finding Y-Intercept

• Found by substituting x = 0 into the function.

Slant Asymptotes

• Occur when the degree of the numerator is one more than that of the denominator; found via polynomial division.