Crash Course — Unit 1: Polynomial and Rational Functions

• Change in Tandem
  €” This is the basic idea of functions: as the input x changes, how does the output y respond? Does it increase, decrease, or stay the same?
• Average Rate of Change (AROC)
  €” This is the slope of the line connecting two points on a curve. It tells you, on average, how a function changed over an interval.
• Concavity and AROC
  €” If the AROC is increasing over adjacent intervals, the function is concave up (like a cup holding water). If the AROC is decreasing, it's concave down (like a cup spilling water).

• Identifying Quadratics
  €” A key feature of quadratic functions is that the rate of change of the average rate of change is constant. This is a unique fingerprint for parabolas.
• Polynomial End Behavior
  €” The "leading term test" is your best friend. The highest power (degree) and its coefficient tell you exactly where the graph's arms will point as x goes to positive or negative infinity.
• Zeros and Multiplicity
  €” Zeros are where the graph hits the x-axis. Multiplicity tells you how it hits: an odd multiplicity means it crosses through, while an even multiplicity means it "bounces" off the axis.

• Even & Odd Functions
  €” A quick test for symmetry. Even functions (f(-x) = f(x)) are symmetric over the y-axis. Odd functions (f(-x) = -f(x)) are symmetric around the origin.
• Rational Function Features
  €” To analyze f(x) = p(x)/q(x), you must check three things in order:

  1. 1
     Holes

     Find factors that cancel.
  2. 2
     Vertical Asymptotes

     Find zeros of the simplified denominator.
  3. 3
     Zeros

     Find zeros of the simplified numerator.
• Rational Function End Behavior
  €” This is all about comparing the degrees of the numerator and denominator to find horizontal or slant asymptotes, which describe the function's behavior at the far ends of the graph.
• Function Modeling
  €” You'll use all these concepts to look at a real-world situation (like the height of a thrown baseball or the concentration of a chemical) and decide which type of function—linear, quadratic, polynomial, or rational—is the best fit.
• Polynomial Division
  €” Using long division helps you rewrite rational expressions. It's especially important for finding the equation of a slant asymptote when the numerator's degree is one greater than the denominator's.

Key Formulas / Terms

• Average Rate of Change (AROC)
  For a function f(x) on the interval [a, b], the AROC is (f(b) - f(a)) / (b - a).
• Polynomial End Behavior (Leading Term Test)
  For f(x) = a_n * x^n + ...:

  • Even Degree: If a_n > 0, both arms go up. If a_n < 0, both arms go down.
  • Odd Degree: If a_n > 0, right arm up, left arm down. If a_n < 0, right arm down, left arm up.
• Rational End Behavior (Horizontal Asymptotes)
  Compare the degree of the numerator (N) and denominator (D).

  • N < D: Horizontal asymptote at y = 0. (Mnemonic: BOBO - Bigger On Bottom, y=0)
  • N = D: Horizontal asymptote at y = (ratio of leading coefficients). (Mnemonic: EATS DC - Exponents Are The Same, Divide Coefficients)
  • N > D: No horizontal asymptote. If N is exactly one more than D, you have a slant asymptote found by long division. (Mnemonic: BOTN - Bigger On Top, None)
• Binomial Theorem
  To expand (a+b)^n, the coefficients are found from the n-th row of Pascal's Triangle. For example, (a+b)^3 = 1a^3 + 3a^2b + 3ab^2 + 1b^3.

Exam Traps

• Trap
  Confusing a hole with a vertical asymptote. · Counter: Always simplify the rational function first by canceling common factors. If a factor (x-c) cancels, there's a hole at x=c. If (x-c) remains in the denominator, it's a vertical asymptote.
• Trap
  Misinterpreting multiplicity at a zero. · Counter: Look closely at the exponent on the factor. (x-3) means the graph crosses at x=3 (odd multiplicity of 1). (x-3)^2 means the graph bounces off the x-axis at x=3 (even multiplicity of 2).
• Trap
  Mixing up horizontal and vertical shifts. · Counter: Remember that changes inside the function f(x) affect the graph horizontally and often feel backward. f(x+2) shifts LEFT 2 units. Changes outside the function affect it vertically: f(x)+2 shifts UP 2 units.
• Trap
  Calculating AROC incorrectly from a table. · Counter: The College Board loves giving you tables of data. To find the AROC on [a, b], make sure you are pulling the output values f(b) and f(a) from the table, not just using b and a themselves in the numerator. The formula is (y2 - y1) / (x2 - x1).
• Trap
  Assuming a function is even or odd without testing. · Counter: A graph might look symmetric, but you must prove it algebraically. Calculate f(-x). If the result is identical to f(x), it's even. If it's -(f(x)), it's odd. If it's neither, the function is neither even nor odd.