Download Rational Function Analysis 1. Reduce R x to lowest terms. 2

Rational Function Analysis
1. Reduce R ( x ) to lowest terms.
2. Determine the x-intercepts by setting the numerator equal to zero.
3. Determine the y-intercepts by finding R ( 0 ) .
4. Determine the equation (x = ___ ) of all vertical asymptotes by setting the denominator equal to zero.
Graphs will never cross the vertical asymptote since these values cause R ( x ) to be undefined.
5. Determine the equation (y = ____ ) of any horizontal asymptotes by comparing the leading terms of
numerator and denominator. The graph of R ( x ) can cross horizontal asymptotes. If there is no
horizontal asymptote, there may be a oblique/slant or non-linear asymptote.
6. If the degree of the numerator is one more than the degree of the denominator, there is an
oblique/slant asymptote. If the degree of numerator is two or more than the degree of the
denominator, there is a non-linear asymptote. The slant/oblique or non-linear asymptote is the
quotient after dividing the denominator into the numerator. The graph of R ( x ) can cross an
oblique/slant asymptote.
7. If a horizontal or oblique/slant asymptote exists, determine if and where R ( x ) crosses the horizontal
or oblique/slant asymptote by solving R ( x ) = HA / OA / SA .
8. Use all the above to sketch a fairly accurate graph of f ( x ) .
Polynomial Function Analysis
−2 x 3 ( 5 x + 9 )
In Factored Form - f ( x ) =
2
( 3x − 7 )
4
1. Determine the zeros of each factor.
2. Determine the multiplicity of each factor. The graph will touch (or bounce off off) real zeros of even
multiplicity and cross through real zeros of odd multiplicity. Fact: The larger the multiplicity, the more the
graph “flattens out” as it touches or crosses through the real zero.
3. Determine the leading term by multiplying the leading term of each factor raised to the degree of the factor. If,
−2 x 3 ( 5 x + 9 )
for example, f ( x ) =
4.
−4050 x 9 .
( 3x − 7 ) , the leading term will be −2 x 3 ( 5x ) ( 3x ) =
Determine the end behavior ( x → −∞, f ( x ) → ____ and x → +∞, f ( x ) → ____ ) using the leading
2
2
4
4
term. This can easily be done by substituting a negative number (for −∞ ) and a positive number (for ∞ ), to
determine if f ( x ) → ±∞ . Fact: The end behavior determined if the graph goes up or down ( f ( x ) → ±∞ )
as x goes right or left ( x → ±∞ ).
5. Use the above information to sketch a graph of f ( x ) .
In Expanded Form - f ( x ) = 3x 3 − 5 x 2 + 2 x − 8
1. Determine the possible rational zeros p/q where p is the set of positive and negative factors of the constant
term and q is the set of positive and negative factors of the leading coefficient. Fact: If there is a rational zero
of the polynomial function with integer coefficients, f ( x ) , it must be one of these values.
2. Graph the function and use the graph to determine which of the above possible rational zeros are likely rational
zeros of f.
3. Use synthetic division to determine if the likely zeros found in the previous step are rational zeros f. Fact: If x
= c is the likely zero under consideration and the remainder is zero after synthetic division, you can draw the
following conclusions: (a) x – c is a factor and (b) x = c is a zero.
4. Repeat the previous step until you have determined all the rational zeros. Remember to use the resulting
reduced polynomial (the quotient) from the previous step each time you do this step so that you are factoring f
as you go.
5. Once you have factored the polynomial to the product of irreducible linear and/or quadratic factors (or
possibly a polynomial of quadratic type requiring substitution first), apply the quadratic formula to the
quadratic polynomial(s) to find the remaining zeros. The quadratic formula will yield the irrational and
complex/imaginary zeros. Fact: If the polynomial function has integer coefficients, then the irrational zeros
and imaginary/complex zeros will always occur in conjugate pairs a ± bi or a ± b .
6. State the zeros and factor the polynomial into the product of linear and quadratic factors (over real numbers
only
•
Fact: A polynomial with integer coefficients and an odd degree will have at least one rational zero.
•
Fact: The factors  x −  and ( 3x − 1) yield the same zero  x =
•
with when expanding polynomials.
Fact: If P ( x ) is divided by ( x − c ) using synthetic or long division, the remainder will equal P ( c ) . If x = c


1
3
is a zero, the P ( c ) = 0 .


1
 , but ( 3x − 1) is much easier to work
3