Final answer:
To determine if a function is a polynomial, check for non-negative integer exponents and real number coefficients. The degree indicates the highest exponent, while the leading coefficient is the coefficient of the term with the highest exponent. Rewrite equations in standard form to identify these properties.
Step-by-step explanation:
To determine whether a function is a polynomial function, one must ensure that it consists of terms that are powers of the variable x with non-negative integer exponents, combined with real number coefficients. For example, quadratic functions are specific types of polynomials of degree 2, which have the general form ax² + bx + c, where a, b, and c are constants and a ≠ 0.
The leading coefficient is the coefficient of the term with the highest exponent. If the function provided matches this structure without any forbidden elements such as negative exponents, fractional exponents, or variables in the denominator, it is indeed a polynomial function.
For instance:
- Linear equations, typically of degree 1, such as y = 6x + 8, are also polynomials with the leading coefficient being the coefficient of x.
- The given equation y + 7 = 3x can be rewritten as y = 3x - 7, showing it is a polynomial of degree 1 with a leading coefficient of 3.
- The equation y - x = 8x² is not linear because it includes an x² term, meaning it's a polynomial of a higher degree.
- The Equation Grapher tool mentioned is useful for visualizing the curve of a polynomial and seeing how the graph shapes change with different coefficients.
When identifying whether equations are polynomial functions and determining their characteristics, like their degree and leading coefficient, one must carefully analyze their form and rewrite them if necessary in standard form (descending powers of x).