Algebra 85 - Building Polynomial Functions

TL;DR

Polynomial functions are built by adding monomial functions, and the leading term determines the end behavior of the graph.

Transcript

Hello. I'm Professor Von Schmohawk and welcome to Why U. In the last several lectures we introduced polynomial functions and said that their graphs can have many different interesting shapes. We showed that polynomial functions are built from a sum of one or more terms called "monomials". Monomials with a single real variable have the form "a times... Read More

Key Insights

• 📈 Polynomial functions are built by adding monomial functions, and the leading term determines the end behavior of the graph.
• 🥺 The coefficient of the leading term affects the shape of the graph.
• 📈 Adding multiple monomial functions can produce graphs with various shapes and changes in direction.
• 📈 The end behavior of a polynomial function's graph can be predicted by the leading term.
• 🍉 The addition of more terms can result in more changes in direction in the graph.
• 💡 The graph of a polynomial function can be sketched to get a general idea of its behavior.
• 💻 Computer programs and calculators can accurately graph polynomial functions.

Questions & Answers

Q: What are monomial functions?

Monomial functions are terms in a polynomial function that have the form "a times x to the nth power", with "a" being a constant and "x" being the variable with a non-negative integer exponent.

Q: How does the leading term affect the end behavior of the graph?

The leading term determines whether the graph will grow in the positive or negative direction as x becomes very large in either positive or negative values.

Q: What happens when two monomial functions are added?

Adding two monomial functions produces a polynomial function, and the value of the resulting function is the sum of the individual functions at each value of x. The graph of the polynomial function may have multiple points of intersection with the x-axis.

Q: How does changing the coefficient of a monomial term affect the graph?

Changing the coefficient of a monomial term can affect the shape and direction of the graph. A smaller coefficient may reduce the effect of that term compared to the other terms.

Summary & Key Takeaways

• Polynomial functions are built by adding monomial functions, with the leading term determining the end behavior of the graph.

• The coefficient of the leading term affects the shape of the graph, such as stretching, compressing, or reflecting it.

• Adding multiple monomial functions can produce graphs with various shapes, including changes in direction.