3.2 Polynomial Functions and Their Graphs
Ideas in this lesson
- Recognize polynomials and permissible graph shapes
- Transform and sketch power functions
- Explain end behavior using leading terms and windows
- Factor and sketch polynomials using sign tables
- Use multiplicity to describe local zero behavior
- Apply and limit the intermediate value theorem
- Locate all numerical extrema and interpret domain and range
- Reason about possible extrema counts and minimum degree
- Compare parameterized polynomial families
- Find and verify intersections of polynomial graphs
- Prove polynomial parity and decompose mixed polynomials
- Compare powers and construct large-power tables
- Analyze polynomial profit and population models
- Optimize open-box and edge-constrained box volumes
- Model the strength of a beam cut from a circular log
- Optimize a shelter with specified faces only
Recognize polynomials and permissible graph shapes · Worked example
A proposed graph is continuous but has a sharp corner at \((0,3)\). Another proposed graph has a hole at \((4,-2)\). Could either be a polynomial graph? Explain whether being smooth and continuous alone proves a graph is polynomial. Show your work; a teacher must review the construction or explanation. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
Answer: Neither proposed graph is polynomial. Smoothness and continuity are necessary but not sufficient.
Polynomial functions are defined at every real input and are continuous.
Their graphs are smooth, so a corner is forbidden and a hole is forbidden.
A smooth continuous curve such as an exponential is not a polynomial; the graph properties alone do not prove polynomial form.
Written-work review:
Their graphs are smooth, so a corner is forbidden and a hole is forbidden.
A smooth continuous curve such as an exponential is not a polynomial; the graph properties alone do not prove polynomial form.
Written-work review:
- Show the requested algebra or numerical evidence, not merely a selected option.
- Include all requested coordinates, domains, units, comparisons and conclusions.
- For a graph, label axes, intercepts and extrema; use correct end behavior and a suitable scale.