AP Precalculus Exam Prep - Reference Sheet Notes

Graphical Behaviors

• Increasing Function: Rate of change is positive, graph is concave up (rate of change increasing), or concave down (rate of change decreasing).
• Decreasing Function: Rate of change is negative, graph is concave up (rate of change increasing), or concave down (rate of change decreasing).
• Points of Inflection: Points where concavity changes (from concave up to concave down or vice versa).

Average Rate of Change

• The average rate of change (AROC) of a function $f$ over the interval $[a, b]$ is given by: $AROC = <br />\newline \frac{f(b) - f(a)}{b - a}$.

Using Tables to Understand Functions

• Equal-Length Input Intervals:
  • If the differences in the outputs are increasing, the function is concave up.
  • If the differences in the outputs are decreasing, the function is concave down.
• For a polynomial of $n^{th}$ degree, the $n^{th}$ differences in outputs will be constant.
  • For a quadratic function, the 2nd differences are constant.

Polynomial Functions

• Complex Zeros: Always come in pairs. If $x = a + bi$ is a zero, then $x = a - bi$ is also a zero.
• Multiplicity: If a factor is repeated $n$ times, it has a multiplicity of $n$. Even multiplicity results in the graph bouncing off the x-axis at that zero.
• End Behavior: Determined by the leading term (highest degree).
  • Even Degree: Left and right sides have the same end behavior.
    • Positive leading coefficient: both sides go up.
    • Negative leading coefficient: both sides go down.
  • Odd Degree: Left and right sides have opposite end behavior.
    • Positive leading coefficient: right side goes up, left side goes down.
    • Negative leading coefficient: right side goes down, left side goes up.
• Limit Notation:
  • As $x$ decreases without bound: $lim_{x \to -\infty} f(x)$.
  • As $x$ increases without bound: $lim_{x \to \infty} f(x)$.
• Even Functions:
  • Graphs have symmetry over the y-axis.
  • $f(-x) = f(x)$
• Odd Functions:
  • Graphs have symmetry over the origin.
  • $f(-x) = -f(x)$

Solving Polynomial Inequalities

1. Put all terms on one side and factor.
2. Create a sign chart with all zeros marked.
3. Check one value in each interval for positive or negative sign.
4. Successive intervals alternate signs unless the zero has an even multiplicity.

Rational Functions

• End Behavior: Determined by the largest degree terms in the numerator (N) and denominator (D).
  • Case I (Top Heavy: N > D): No horizontal asymptote; may have a slant asymptote.
  • Case II (Same Degree: N = D): Horizontal asymptote at $y = \frac{leading coefficient of N}{leading coefficient of D}$.
  • Case III (Bottom Heavy: N < D): Horizontal asymptote at $y = 0$.
• Zeros: Numerator equals 0.
• Holes: Denominator equals 0 AND cancels out with numerator (numerator has equal or larger multiplicity).
• Vertical Asymptotes: Denominator equals 0 AND does NOT cancel out with numerator (denominator has larger multiplicity).

Long Division and Slant Asymptotes

• If the degree of the numerator is exactly 1 more than the degree of the denominator, the rational function has a slant asymptote.
• Use long division to find the equation of the slant asymptote (ignore the remainder).

Binomial Theorem and Pascal’s Triangle

• Pascal’s Triangle provides the coefficients for the Binomial Theorem.
• Expansion of $(x + y)^3$: use 1, 3, 3, 1 from Pascal's triangle for coefficients $x$ gets exponent from 3 to 0 and $y$ gets exponent from 0 to 3.
• $(x+y)^3 = 1x^3y^0 + 3x^2y^1 + 3x^1y^2 + 1x^0y^3 = x^3 + 3x^2y + 3xy^2 + y^3$