AP Precalculus Exam Prep - Unit 1 Notes

Graphical Behaviors

• Increasing/Decreasing:

  • Graph of $f$ is INCREASING: Rate of change of $f$ is POSITIVE.
  • Graph of $g$ is INCREASING: Rate of change of $g$ is POSITIVE.
  • Graph of $h$ is DECREASING: Rate of change of $h$ is NEGATIVE.
  • Graph of $k$ is DECREASING: Rate of change of $k$ is NEGATIVE.

• Concavity:

  • Graph of $f$ is CONCAVE UP: Rate of change of $f$ is INCREASING. The graph of $f$ is INCREASING at an INCREASING RATE.
  • Graph of $g$ is CONCAVE DOWN: Rate of change of $g$ is DECREASING. The graph of $g$ is INCREASING at a DECREASING RATE.
  • Graph of $h$ is CONCAVE UP: Rate of change of $h$ is INCREASING. The graph of $h$ is DECREASING at an INCREASING RATE.
  • Graph of $k$ is CONCAVE DOWN: Rate of change of $k$ is DECREASING. The graph of $k$ is DECREASING at a DECREASING RATE.

• Exam Tip: Always check if the statement is talking about the function/graph of $f$ or the rate of change of $f$.

Points of Inflection

• Points where the graph of $f$ changes from concave up to concave down (or vice versa).
  • A graph can have multiple points of inflection.
  • At points of inflection, the graph switches concavity.

Average Rate of Change

• The average rate of change of a function $f$ over the interval $[a, b]$ is given by: $AROC = \frac{f(b) - f(a)}{b - a}$.
• The rate of change of $f$ refers to the slope of the graph at a single point.
• Exam Tip: For rates of change, imagine riding a roller coaster along the graph; the steepness of the car represents the rate of change.

Using Tables to Understand Functions

• Over equal-length input-value intervals (the $x$-values are evenly spaced):
  • If the differences in the outputs ($y$-values) are increasing, the function is concave up.
  • If the differences in the outputs ($y$-values) 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: Zeros

• Complex Zeros: Always come in pairs. If $x = a + bi$ is a zero, then so is $x = a - bi$.
• Multiplicity: If a factor is repeated $n$ times, it has a multiplicity of $n$.
  • If a zero has an even multiplicity, the graph will bounce off the $x$-axis at that zero.

Polynomial Functions: End Behavior

• The leading term (highest degree) determines the end behavior of a polynomial function. Determine the right end behavior first, then the left end behavior.

• Largest Degree Even: Left side has the same end behavior as the right side.

  • Leading Term Positive: Right side goes up ($\lim_{x \to \infty} f(x) = \infty$).
  • Leading Term Negative: Right side goes down ($\lim_{x \to \infty} f(x) = -\infty$).

• Largest Degree Odd: Left side has the opposite end behavior as the right side.

  • Leading Term Positive: Right side goes up ($\lim_{x \to \infty} f(x) = \infty$).
  • Leading Term Negative: Right side goes down ($\lim_{x \to \infty} f(x) = -\infty$).

• Example: $f(x) = -0.5(x+1)(x-2)^2$

  • Has zeros at $x = -1$ and $x = 2$.
  • There is a bounce at $x = 2$ because $(x-2)$ has multiplicity of 2.
  • The leading term is $-0.5x^3$.
  • Since the leading term is negative and the degree is odd, the right side goes down and the left side goes up.
    • $\lim_{x \to -\infty} f(x) = \infty$
    • $\lim_{x \to \infty} f(x) = -\infty$

Even and Odd Functions

• Even Functions: Graphs have symmetry over the $y$-axis.
  • If you plug in a number and its opposite, you get the same answers: $f(-x) = f(x)$.
• Odd Functions: Graphs have symmetry over the origin.
  • If you plug in a number and its opposite, you get opposite answers: $f(-x) = -f(x)$.

Solving Polynomial Inequalities

1. Put all terms on one side and factor first.
2. Create a sign chart with all zeros marked.
3. Check one value (the far right interval is usually easiest to check) for + or -.
4. Each successive interval on the sign chart will alternate signs unless the zero has an even multiplicity.

• Example: $(x + 3)(x - 1)(x - 4) < 0$
  • Zeros at $x = -3, 1, 4$
  • Solution: $(-\infty, -3) \cup (1, 4)$

Rational Functions: End Behavior

• End behavior is determined by the largest terms (biggest degree) in the numerator and denominator.

• Case I (Top Heavy): Degree of Numerator > Degree of Denominator

  • No Horizontal Asymptote; Slant Asymptote

• Case II (Same Degree): Degree of Numerator = Degree of Denominator

  • Horizontal Asymptote: $y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$

• Case III (Bottom Heavy): Degree of Numerator < Degree of Denominator

  • Horizontal Asymptote: $y = 0$

• Examples:

  • $f(x) = \frac{4x^3 + 3x + 5}{x^2 - 4x + 1}$ (Top Heavy) - $y = 4x + 13$
    • $\lim_{x \to -\infty} f(x) = -\infty$
    • $\lim_{x \to \infty} f(x) = \infty$
  • $g(x) = \frac{3x^3 - 3x^2 + 4}{5x^3 - 2x + 5}$ (Same Degree) - $y = \frac{3}{5}$
    • $\lim_{x \to -\infty} g(x) = \frac{3}{5}$
    • $\lim_{x \to \infty} g(x) = \frac{3}{5}$
  • $h(x) = \frac{2x + 4}{x^2 - 3x + 2}$ (Bottom Heavy) - $y = 0$
    • $\lim_{x \to -\infty} h(x) = 0$
    • $\lim_{x \to \infty} h(x) = 0$

Rational Functions: Zeros, Holes, and Vertical Asymptotes

• 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).

• Factors in the denominator will NEVER be a zero of the function.

• Factors in the denominator ALWAYS become the location of a hole or a vertical asymptote.

• Example: $r(x) = \frac{(x+5)(x+3)}{(x+3)(x-1)(x-4)}$

  • Zero: $x = -5$
  • Hole: $x=-3$
  • Vertical Asymptotes: $x = 1$ and $x=4$

Long Division and Slant Asymptotes

• If the degree of the numerator (N) is exactly 1 more than the degree of the denominator (D), the rational function has a slant asymptote.
• Use long division to find the equation of the slant asymptote ($y = mx + b$).

Binomial Theorem and Pascal's Triangle

• Pascal's Triangle helps to quickly use the Binomial Theorem.
• Example: Expand $(x + y)^3$
  • Coefficients from Pascal's Triangle: 1, 3, 3, 1
  • $(x + y)^3 = 1x^3y^0 + 3x^2y^1 + 3x^1y^2 + 1x^0y^3$
  • $(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$