rational graphs

holes

a point on the graph where the value of the function is not defined

• rational functions can have more than one hole

• a rational function does not have a hole when you cannot cancel anything out

excluded values

values in the denominator that make x = 0

vertical asymptote

vertical lines which correspond to the zeros of the denominator of the simplified rational function

• the graph of the rational function will never cross or even touch the vertical asymptotes, since this would cause division by zero

• rational functions can have 1, none, or multiple vertical asymptotes

  • remember, it’s written as ‘x = #’

horizontal asymptote

horizontal line that guides the graph “far” to the right and/or “far” to the left towards infinity.

compare the highest degree in the numerator and denominator of the rational function.

1. make sure equation is in standard form

2. even exponents (ex: x²/x²) divide normally and produce a y = #

3. small over large exponents (ex: x/x²) means y = 0

4. large over small exponents means there is no y-intercept, so ‘none’

x-intercept

to find x-intercept..

1. set numerator of simplified function to 0

2. solve for x

3. WRITE ANSWER AS A POINT - (X, 0)

4. A HOLE CANNOT BE AN X-INTERCEPT

y-intercept

to find y-intercept..

1. make sure rational is reduced/simplified

2. plug in ‘0’ for all x-values in function

3. solve for y

4. WRITE ANSWER AS A POINT - (0, Y)

5. A HOLE CANNOT BE A Y-INTERCEPT

oblique/slant asymptote

a linear asymptote where y = mx + b & guides the end behaviors of a rational function from both ends

• only occurs when degree of numerator is ONE DEGREE greater than the degree of the denominator

• to fine this, use long division

  • quotient will be the oblique asymptote

  • write as ‘y = #x + c’

• YOU CANNOT HAVE A HORIZONTAL ASYMPTOTE AND A OBLIQUE ASYMPTOTE AT THE SAME TIME- ONE OR THE OTHER