Polynomial and Rational Functions

One zero

Three zeros

Four zeros

Two zeros

An inflection point.

A critical value.

None; polynomials do not have horizontal asymptotes.

A zero of the function.

FR: remains constant

FR: oscillating between negative and positive values

F(R): going toward zero

If R: goes toward infinity positive value

There must be two complex conjugate roots due to the fundamental theorem of algebra.

There are no complex roots as there are two real roots given.

There can be three or four complex conjugate roots depending on the multiplicity of real roots.

The total number of extrema is not related to the total number of complex roots.

f(x) oscillates between positive and negative values

f(x) goes to positive infinity

f(x) remains constant

f(x) goes to negative infinity

It causes the graph of the polynomial to shift upward or downward on a coordinate plane.

There is no change in end behavior compared with the original polynomial.

Only the positive end behavior changes direction while the negative end remains unchanged.

The directions in which both ends extend are reversed compared with the original polynomial.

A zero with odd multiplicity touches but does not cross through the x-axis at that point.

Multiplicity only affects y-intercept, not zeros' behavior on graphs.

A zero with even multiplicity has no impact on whether it crosses or touches but doesn't cross through the axis.

A zero with odd multiplicity crosses through the x-axis at that point.

As x approaches infinity, f(x) approaches negative infinity; as x approaches negative infinity, f(x) also approaches negative infinity.

As x approaches infinity, f(x) remains constant; as x approaches negative infinity, f(x) also remains constant.

As x approaches infinity, f(x) approaches infinity; as x approaches negative infinity, f(x) also approaches infinity.

As x approaches infinity, f(x) oscillates; as x approaches negative infinity, f(x) also oscillates.

The factorization includes (x - 4).

The factorization includes (x + 4).

The factorization includes $x^2 -16$ .

The factorization includes $x^2 +16$ .

$p(0) = -4$ and it's a local minimum point on the graph

$p(0) = 4$ and it's a local maximum point on the graph

$p(0) = 0$ and it's an inflection point on the graph

$p(0) = -4$ but not an extremum point on the graph