Polynomial and Rational Functions

cx + d never equals zero.

ax + b has a higher degree than cx + d.

The graph has a vertical asymptote at $x=k$.

cx + d equals zero at $x=k$ but $ax+b$ also equals zero at same $k$.

60 degrees

45 degrees

120 degrees

90 degrees

Both paths are equal because both objects have consistent speeds and distances covered are equal.

Interval of object with a longer path due to a greater numerator affecting altitude.

Interval of object with a shorter path since it has a larger denominator influencing the steepness of the curve.

Both objects travel at the same pace, thus the identical intervals are covered.

y = 3

x = -4

y = -4

x = 3

Undefined

Negative two

Zero

Eight

Exclusively if m is an integer.

Any real number except zero.

None; the equation always has an horizontal asymptote at y=q.

Only when m is greater than one.

Removable discontinuity

Jump discontinuity

Continuous

Infinite discontinuity

Add both numerator and denominator together set equal to zero then solve for x.

Set denominator equal to zero and solve for x.

Set numerator equal to zero and solve for x.

Multiply both numerator by denominator set equal to zero then solve for X.

The location of all previous horizontal asymptotes will shift up or down based on that term’s value

There may be no effect if it doesn't alter highest-degree terms' ratio between numerator and denominator

All previously existing horizontal asymptotes will disappear entirely from its graph

A new horizontal asymptote will always form at y = that term's coefficient.

x = 0

y = 6

x = 6

y = 0