Polynomial and Rational Functions

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

One zero

Three zeros

Four zeros

Two zeros

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.

The factorization includes (x - 4).

The factorization includes (x + 4).

The factorization includes $x^2 -16$ .

The factorization includes $x^2 +16$ .

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 does not change it; they remain consistent with those of any odd-degree polynomials.

Doubling coefficients reverses direction so if originally up on both ends now down

Doubling leads flattening out ends approaching horizontal lines rather than rising falling

It makes one side rise sharply while another falls more gently compared before

It will always pass through the origin.

It touches but does not cross the x-axis.

It moves in opposite directions on either side of the y-axis.

It goes in the same direction.

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.