Polynomial Functions and Their Properties

Mr. Gerhard introduces graphing polynomial functions, focusing on their smooth and continuous nature. He explains polynomial forms, categorization by degree, and the importance of real coefficients and whole number exponents. The video covers end behavior, influenced by the degree and leading coefficient, and demonstrates direct and synthetic substitution methods for evaluating polynomials.

What are the two main characteristics of polynomial graphs?

• Sharp and discontinuous

• Discrete and finite

• Smooth and continuous

• Linear and quadratic

What does it mean for a graph to be continuous?

• It has gaps

• It is a single unbroken line

• It consists of multiple segments

• It has sharp angles

Which of the following is NOT a requirement for a polynomial?

• Non-negative integer exponents

• Fractional exponents

• Whole number exponents

• Real number coefficients

What is the highest exponent in a polynomial called?

• Root

• Base

• Degree

• Coefficient

What type of polynomial is represented by a degree of 2?

• Quartic

• Linear

• Cubic

• Quadratic

How does a cubic polynomial graph typically appear?

• As a straight line

• As a parabola

• With multiple curves

• As a circle

What determines the end behavior of a polynomial graph?

• The number of terms

• The degree and leading coefficient

• The x-intercepts

• The constant term

If a polynomial has an odd degree and a positive leading coefficient, what is its end behavior?

• One end goes up, the other goes down

• Both ends go up

• It forms a closed loop

• Both ends go down

What happens to the graph of an even degree polynomial with a negative leading coefficient?

• It opens upwards

• It opens downwards

• It forms a vertical line

• It forms a horizontal line

In the polynomial f(x) = 5x^3 + 3x^2 - x + 7, what is the degree?

• 1

• 2

• 4

• 3