Overview
This lecture reviews key concepts in pre-calculus related to polynomial functions, including local extrema, concavity, average rate of change, function symmetry, end behavior, zeroes/multiplicities, and solving inequalities.
Graphs of Polynomial Functions
- The domain is the set of input values over which the function is defined.
- Local maxima and minima are highest and lowest points in a local interval.
- Point of inflection: where concavity changes in the graph.
- A function is increasing where tangent line slopes are positive; decreasing where slopes are negative.
- Concave down sections look like an upside-down cup; concave up like a right-side-up cup.
Intervals of Increase, Decrease, and Concavity
- Use tangent lines to identify intervals of increase/decrease.
- To find concave up/down intervals, observe the "shape" of the graph and the rate at which the tangent slope changes.
Average Rate of Change
- Formula: (F(b) - F(a)) / (b - a) for interval [a, b].
- Increasing average rate of change = concave up; decreasing = concave down.
- For stepwise data, compare changes in output over equal input intervals.
Zeroes and Multiplicity
- Zeroes (roots) occur where the function crosses/touches the x-axis.
- Multiplicity: Even multiplicity means the graph touches but doesn't cross; odd multiplicity means it crosses.
Polynomial End Behavior
- Determined by leading term's degree and sign.
- Even degree: same end direction; odd degree: opposite ends.
- Positive leading coefficient: ends up; negative: ends down.
Function Symmetry
- Even function: f(-x) = f(x) (symmetric about y-axis).
- Odd function: f(-x) = -f(x) (symmetric about origin).
- If neither holds, the function is neither even nor odd.
Solving Polynomial Inequalities
- Find zeroes and their multiplicities.
- Place zeroes on a number line, test intervals, and determine sign changes.
- Include or exclude endpoints based on inequality type (≤/≥ vs < or >).
Key Terms & Definitions
- Local Maximum/Minimum — Highest/lowest point in a local interval.
- Point of Inflection — Where the graph changes concavity.
- Concave Up/Down — U-shaped (up) or n-shaped (down) sections of a curve.
- Average Rate of Change — Change in output per unit change in input.
- Multiplicity — Number of times a root is repeated in the function.
- End Behavior — Direction in which the graph moves as x → ±∞.
- Even/Odd Function — Even: symmetric about y-axis; Odd: symmetric about origin.
Action Items / Next Steps
- Practice problems similar to those left unanswered in the lecture (intervals, rate of change, factoring).
- Review and memorize key formulas for rate of change and identifying function symmetry.
- Complete any assigned homework and revisit graphing concepts, especially identifying local extrema and concavity.