Polynomial Function Types

Understanding polynomial function types is key in Honors Algebra II, Precalculus. These functions, ranging from linear to quartic, each have unique characteristics and behaviors that help us analyze and graph them effectively. Let's break them down.

1. Linear functions

   • Represented by the equation ( f(x) = mx + b ), where ( m ) is the slope and ( b ) is the y-intercept.
   • Graphs are straight lines, indicating a constant rate of change.
   • The degree of a linear function is 1, meaning the highest exponent of the variable is 1.

2. Quadratic functions

   • Formulated as ( f(x) = ax^2 + bx + c ), where ( a \neq 0 ).
   • Graphs are parabolas that open upwards (if ( a > 0 )) or downwards (if ( a < 0 )).
   • The degree is 2, allowing for a maximum of two real roots (x-intercepts).

3. Cubic functions

   • Expressed as ( f(x) = ax^3 + bx^2 + cx + d ), where ( a \neq 0 ).
   • Graphs can have one or two turning points and can cross the x-axis up to three times.
   • The degree is 3, indicating a maximum of three real roots.

4. Quartic functions

   • Written as ( f(x) = ax^4 + bx^3 + cx^2 + dx + e ), where ( a \neq 0 ).
   • Graphs can have up to four turning points and can cross the x-axis up to four times.
   • The degree is 4, allowing for a maximum of four real roots.

5. Constant functions

   • Defined as ( f(x) = c ), where ( c ) is a constant value.
   • Graphs are horizontal lines, indicating no change in output regardless of input.
   • The degree is 0, meaning there are no variables present.

6. Zero functions

   • Represented as ( f(x) = 0 ) for all ( x ).
   • Graphs are also horizontal lines at the y-value of 0.
   • The degree is undefined, as there are no variable terms.

7. Monomial functions

   • Formulated as ( f(x) = ax^n ), where ( a ) is a constant and ( n ) is a non-negative integer.
   • Graphs can take various shapes depending on the value of ( n ) (e.g., linear, quadratic).
   • The degree is ( n ), indicating the highest exponent in the function.

8. Binomial functions

   • Expressed as ( f(x) = ax^m + bx^n ), where ( a ) and ( b ) are constants and ( m ) and ( n ) are non-negative integers.
   • Graphs can exhibit a variety of shapes based on the degrees of the terms.
   • The degree is the highest of ( m ) and ( n ).

9. Even-degree polynomials

   • Polynomials where the highest exponent is an even number (e.g., 2, 4).
   • Graphs typically have the same end behavior (both ends go up or both go down).
   • Can have a maximum of ( n ) real roots, where ( n ) is the degree.

10. Odd-degree polynomials

    • Polynomials where the highest exponent is an odd number (e.g., 1, 3, 5).
    • Graphs typically have opposite end behavior (one end goes up while the other goes down).
    • Can have a maximum of ( n ) real roots, where ( n ) is the degree.